four hours later, I'm like, oh, I should have asked them this. Well, don't worry. Here's our contact information. We're here to help. And so if you have questions that come up or clarification on anything, you know, please feel free to reach out to any of us. We are here to help. Hey, well, I think that's all the health keeping items and introductions that I had. Jody, do you want to take it away? I will take it away, Hannah. Is correct. I'm going to speak about conservation plan of work and the dart report. I'm not going to get into the nuts and bolts of how to put a conservation plan of work together. If that's something that you're really interested in, I would suggest that you attend a leadership academy. We do talk about that. Or you can visit with me independently. You could also talk to Hannah. So Hannah, if you could forward to the next slide. Keep going. So I promise you, my presentation is going to be about 10 minutes long. And then we will open it up for questions if that's okay with Hannah. So first I'd like to talk about the conservation plan of work or as I refer to it as the annual plan of work, APW. Next slide, please. My recommendation is don't go to loan. I've heard a lot of conversation about supervisors requesting that the district manager or the district clerk develop the plan of work and then presented to them and they rubber stamp it. The district board should be involved in the planning of this conservation plan of work. However, we live in reality and you only meet with your supervisors typically once a month. So I have a couple of suggestions. Go ahead and develop a rough outline and then send that off to your supervisors ahead of time before you have that plan of work meeting. But ask them for their assistance. So you might want to ask for things such as what are some of the tasks that go underneath the goals and objectives? Benchmarks because we have to evaluate our plan in some fashion. Are there any resources that are needed? Who is the responsible person? Are there associated costs, timelines, things like that? One thing that should not occur on the annual plan of work is that your name is under every responsible party column. This needs to be spread out a little bit, involve your supervisors. If you have a district tech, a watershed coordinator, bring them into the fold. This can't fall just on your shoulders. You might want to use the previous year's annual plan of work as a guide. Now, saying that doesn't mean you take the 2023 annual plan of work and you just put 2024 on it. There needs to be a review of it and evaluation. Did you meet your goals? Obviously, the plan of work isn't going to drastically change, but there should be some perhaps some different objectives or tasks that go in, maybe some different resource concerns. Where do those resource concerns come from? If your district held a local work group with NRCS, there should have been some resource concerns that came up during that work group process. Or maybe you have conversations with your conservation partners. Perhaps you held a meeting where some community stakeholders showed up, gave their two cents on what they thought was a concern. So really involve your stakeholders from your district in this plan of work. Next slide, please. You may also contact your program coordinator for assistance. Unfortunately, we just have one program coordinator at the time. That's Hannah. She's in area five. That doesn't mean if you're in a neighboring area that you can't call Hannah and ask for some help. I can also assist in this process. So don't feel like you need just to wade through this without any idea of what you're doing. You might also want to use the smart goals planning process. That's something that we talk about quite frequently at the Leadership Academy. It's a very step by step based process. It's fairly simple to fill out. So if you're looking for some guidance, go ahead and get that smart goals process as part of what you're doing for your annual plan of work. We have an Excel template on the SCD Google Drive. Everybody should be familiar with how to access that. I do know that in my signature block as well as Hannah's, we have the link to the SCD Google Drive. So please, if you're not familiar with it, just one of these days when you have a little bit of downtime, click on that and just peruse it. But on the Google Drive, we have an Excel template that it's basically a form with the smart goals planning process that you can download and you can just fill in. It's pretty simple. I've gotten a lot of good conservation plan of work using that template. If you want different examples of soil conservation plan, contact me and I can get those out to you. Because there are different ways to do it. We just encourage the smart way of planning because it seems to make sense. Next slide, please. Why do we ask for the annual plan of work? Well, it doesn't come from our office specifically. It's actually stated in the century code. So if you look at statute 4.1-20-24 and go down to letter H under powers and duties of districts and supervisors, it does state that SCDs are required to develop comprehensive plans and that these plans must specify in detail possible acts, procedures, etc. That means not only do you have your goals and I would say three to five goals is reasonable, but your objectives, the task, how are you going to achieve these goals? Do you have benchmarks? How are you going to evaluate it? And when I mean benchmarks, do you want a percentage of individuals to show up at your workshop? Do you want to have three workshops? Do you want to have six workshops? So you're putting some numbers in there or do we want to have it by this deadline? Also, it is requiring that it's current. So be sure that when you are sharing your plan of work, it says 2024 and it has been evaluated or has been reviewed. And then of course, you want it as detailed as possible. Next slide please. If you read further into the Century Code, it does say and to publish such plans and bring them to the attention of the occupiers of the lands within the district. Basically, you want to have this available to the public and there's a couple ways that you can do it. The simplest way is to put it on your website. Then you've already met the requirements of the Century Code. If you can get it onto your Facebook page, that's great. You can have a copy of it sitting in your office desk. You can have it on your computer. If somebody comes up to you and says, hey, I'd like to see your annual plan of work, you are required to show it to them. Just make a copy. It's very, very simple. Next slide please. Okay, we're going to segue over into the district activity reporting tool. So if you have questions regarding the annual plan of work, just hold on to them a little bit longer. We're going to talk about Dart. And first, I'd like to start off with the purpose of Dart. You need a little bit of history for those of you that are new. Dart is the outcome of a legislative management study that was done back in 2017 to 2018. And the reason for the study was the state legislature was asking to see so conservation district results, output, data. They were asking questions, well, what are the districts doing? Should we be funding the districts? What is the SSCC doing? We want some hardcore data. At that time, all that we have were qualitative reports. So way back when a district was applying for state assistance funds, you had to submit a narrative as well as some other documentation. That really wasn't sufficient for the state legislature. So 15 soil conservation districts got together. They worked with Sarah Tungi and a programmer. And basically the result was a quantitative report or Dart. And it is a way of basically demonstrating soil conservation district output. It is also a requirement for state assistance funds. That's how we rank the districts. So it's really critical that when you're filling out the district, the Dart report in the very beginning of the biennium when we start asking for how many acres are in your county? What's the population? What is your mill levy? How many employees? What is your operating budget? Be really, really specific and accurate because once that information is in there, it can't be changed for two years. Another thing I wanted to mention is the nice thing about the Dart report and the state assistance fund, it removes subjectivity. So the state soil conservation committee isn't swaying the report one way or another. It's very, very objective. And we think that's very critical. At this point in time, the legislature is content with this process, which is good. So we kind of went through a period where there was a lot of doubt about the soil conservation districts and the Dart report kind of satisfied that worry at this moment. Next slide, please. Now, what's the value of Dart to your district? Well, it is a really great way to demonstrate progress and show others your output or the progress that you are making. So I would utilize the Dart Activity Report, print it off and send it to your state legislator, your senator, your representative, do it on a quarterly basis or at least an annual basis. Send it out to your community or have it available if they want to see it. Share it with your county commissioner and your county auditor because then they know what you're doing. And when you are involved in that process of wanting to raise your mill levy, they know that you are actively pursuing conservation. Now, Dart is not meant to track every minute of the day. So if you're taking phone calls, you don't put that into Dart. You have to be somewhat reasonable. It's whatever you had input into Dart, that's what you track. This is a way to a guide, so to speak, a way to achieve your conservation goals and objectives. You can use it in conjunction with your annual plan of work. If, for example, you feel like I'm doing so much work to achieve our conservation plans and I can prove it through the Dart report, I urge you to print it off and share it with your supervisors and say, hey, we can use another employee and let me show you why. Or I would like an increase in my salary. I can prove that I'm putting a lot of work forward. So use it as best you can. I don't know if people are familiar with the fact that in Dart you can set parameters and you can print off those parameters. So if you're interested in how we did the last quarter, you can go into Dart. There's a section on reporting and you can put in the specific dates and get that printed off. You can compare one biennium to a next year to year. So it's really, there is value in it, you just need to utilize it. Next slide, please. Along with Dart, where I should say, at the beginning of every biennium, we do send out a fund dispersal form so that Carissa is sending this out to all 54 districts. This is, this notifies you, the district, of the amount that you're going to receive for the biennium. And during that, and in that form, it does need to be signed by the board chair, as well as the district clerk or the district manager. You can agree to all the money that you're being received, that you are receiving, or you can decline all or part of the funds. The most important thing is you need to return the document to Carissa. If you don't return it, we assume that you don't want the money. And I would say 99% of the time, almost everybody wants their money. So it's really critical that you, that you fill this out. Next slide, please. By signing the fund dispersal form, your district is agreeing to submit various required documents. That's really important that you and your board chair understand this. So within the century code, there are certain, there are certain required documents, for example, the annual plan of work. In addition, the century, the century code allows the state civil conservation committee to require other documents, which they do. And those include things like meeting minutes. That's also an open records, open meeting minutes law. We require the supervisor training report. Oh, equipment inventory list, things like that. So know that this is similar to your signing a contract. The state will give you money. In return, we would like to have these documents. And honestly, the documents are there for your benefit, not necessarily for our benefit. So if your computer would crash, if you had a fire, and you lost all your documentation, we would have some very valuable information that we could provide to you. So know that this is something that we are doing within the century code, these requests. Next slide, please. I want to end with some tips for the district activity reporting tool. The thing that I cannot stress enough is input data on a regular basis. So if you do it weekly, bi-weekly, monthly, continue to do that, get in that habit. It's very similar to Facebook. Make sure you update Facebook on a regular basis so it doesn't get out of date. Consistency is the key with Dart. I'll say it again. Consistency is the key. If you always include XXF, whatever, under accounting, continue that practice. Because if you change the way you input things, then your Dart activity report really isn't going to make a lot of sense from quarter to quarter or year to year. So just be mindful of that. We do not have a definition sheet for Dart, and we've been asked, could we possibly put something together? I spoke with Sarah, talking about this, and Sarah's response was, we do not have a definition sheet because we want Dart to remain flexible. So you can determine what this means to your district rather than us determining it. I would say the only thing that we want consistent is when you're filling out the Dart report at the beginning of the biennium, and they ask for operating cash, it's the cash that you have on hand to run your district. That's the one time where we say, please be, everybody be on the same page. It's not the money that you have in reserve. It's not the money you're saving for a truck or a new tree cooler. It's the money you have to get through, to run an office basically. When you're also filling out Dart, do not save the document unless you mean to. If you accidentally save it and you didn't want to, you're going to have to contact Sarah Tonghi by email so she can unlock it for you. Use draft as long as you possibly can so you can continue to make changes. Also when you code employee activities, do it to match the Dart reporting system. So it goes a lot smoother. You're not having to think, okay, what did this employee do and how does it match what we have in Dart? Try to streamline that as much as possible. Next slide, please. Okay, I am finished. And again, if you have questions, I would say start with Carissa. Carissa is very wise and familiar with Dart, more than I am. Also Sarah Tonghi, she helped develop Dart. So again, if you run into a problem where you're locked out or let's say Carissa doesn't know the answer, go ahead and get a hold of Sarah. And I've provided you with her cell phone number, or excuse me, her phone number and her email. Just remember, Dart is all about accountability and transparency. And it's very critical that all the districts are involved in this, regardless of whether you're not, you accept the Dart money. It's just a really good tool to use for your district and to provide others who make important decisions on conservation. I'm finished. And if there's any questions, shout them out, raise your hand, contact me later, put it in the chat. Everybody's doing a good job. And I'll probably be contacting districts, or Carissa will if we don't have an updated plan of work.  If you have any questions, just give us a holler. Thank you. Thanks, Hannah. I'm just going to chime in one more thing about Dart. I can also change your status. If you accidentally put it in final status, I can change it back to draft for you. Also, in Dart, there's the supervisor training report. We still have the old one in there. The new one did not get updated this year, but we will get that updated for next year. And so please remember to use the new form that's located in our Google Drive, instead of the one that's on Dart. Hey, Carissa, what if I'm a new employee and I can't find my model for me? Can you look at books of that? I would maybe contact Sarah for that one. Yeah, Sarah will be the person. Be persistent with Sarah. She's very busy, persistent and patient, I guess. She'll get back to you. And if you run into any trouble and she's not contacting you, just give me a holler and I'll reach out to Sarah for you. All right, I guess I get to continue. So I'm going to be talking about oaths of office and the reorganization form. Next slide. And again, so oath of office is another thing that's in the North Dakota Century Code under chapter 44-01. Oaths of office aren't just for soil conservation district supervisors. They're for really any public office. So the supervisors for soil conservation districts are in two groups. We have the elected supervisors and the appointed supervisors. Elected supervisors are elected for six years, so we only need a new oath of office at the beginning of their six-year term. We don't need another oath until they become re-elected. And the oath is effective January 1st following the time that they're elected. For appointed supervisors, they are appointed annually, so we do need a new oath of office signed by the supervisor every year. And that oath is effective the date that it is signed or they're appointed. Neither type of supervisor can perform any supervisor duties until that form has been signed and is on file with the secretary of state. Next slide. So here's a sample of the oath of office. It is in a fillable form. I do have that linked on the slide. Please be sure to complete it in full. I have an example. So I, Carissa Kittlesen of Fargo, North Dakota, do solemnly swear that I will support the Constitution of the United States and the Constitution of the state of North Dakota that I will faithfully discharge the duties of the office of supervisor of the best soil conservation district, blah, blah, blah. You get the point. Then I would sign it in front of a notary public.  Otherwise, I have this later on. A good tip is to invite a notary public to your board meeting and have them sign off on the oaths at that time. Next slide. So your oath is signed, notarized. What do I do with it next? The original is going to be mailed to the secretary of state to Jess Martinez. I have her address up on the slide. Again, we will share this information. It should be in the email that I sent out earlier as well. And you can email me a copy. So I get the copy. Electronic version is fine. You don't need to mail it to me, but Jess does need the original. Next slide. Just a few tips. Be sure to use the fillable form. When people are signing, it's hard to understand their signature. I know mine is illegible as well. So if we have it typed out nicely, we can be sure that the name is spelled correctly. Complete the form in advance. If you know you're having a board meeting, have the fillable form completely ready to go. So the supervisor just has to sign and have it notarized. And again, invite the notary public to your meeting. Next slide. On to reorganization form. Next. Again, this is one of the forms that need to be completed annually. You'll notice this year it was updated just slightly. Be sure that you are completing the reorganization form in full. There's a little note in gray kind of above the district staff information just to have an asterisk by the board chair just so we have that record on file as well. It might be handy if you do have any staff or supervisor changes to maybe not refill out this entire form, but have it saved and then update it with any changes and then send that to me as well. Next slide. On our new form, we have alternate information. This is on here because of, you know, the possible government shutdown. If the government is shut down, we won't be able to contact district staff. So we want to be able to contact them in case of any sort of emergency. Again, the reorg form can be emailed to me. The reorg form is in our Google Drive. You will need to download a copy of it in order to edit it. I know we've had a lot of questions on that and requests to edit. We want to keep the original in the Google Drive so if you can download it, save it to your own computer, fill it out and then send it back to me. That would be fabulous. Next slide. Just a few tips. Verify everything has been completed correctly. That just is a time saver. I mean, if there are fields that are incorrect and I have to go back to you or if we try to send something out and the email address is incorrect, then it gets returned to me and I'm not getting that information out properly. The little asterisk by the board chair, it's kind of a sneaky little bugger. So just wanted to highlight that and then having that alternate contact information just in case there's some sort of emergency. We won't use that information any other time. It's just emergency use. And I think that's it for me. Any questions? I guess I'm up. So my name is Andrea Bowman and I'm going to visit with you about meeting minutes today. I'm doing an overview of meeting minutes, but before we talk about meeting minutes, we're just going to do a review of the open record and meeting laws. So you have a basic understanding of that and you'll understand how it ties into what your meeting requirements are. So I think you can advance, Hannah. So the open meeting and record guide. Everyone should have a copy of this with you. Some more close all the time. Every supervisor should have a copy of the open records guide and the open meetings guide for reference. And so essentially these two guides summarize the sunshine laws in North Dakota. These laws make it that all government records and meetings are open to the public unless there's a specific law that authorizes a record to be withheld or a meeting to be closed. These laws apply to all agencies, businesses, or nonprofits. Thank you, Carissa, for sharing that link. That received public funds. So soil conservation districts because you are receiving taxpayer funds, public dollars are subject to the open record and meeting laws. And while maybe you don't sit in the hot seat like some other organizations that are subject to these laws, it is important that you follow them. Remember that anyone has the right to attend your meeting, but it's not open. It's not a public meeting. They're there to attend, but you don't have to allow them to participate. So the link is in the chat. There is a guide, like I said, that goes with the open record and meeting guides. There's also, those guides are just double-sided. There's also about a 40-page manual that goes with each of them as well. The Attorney General's website is a great option. You can advance, Anna. So when we look at the open record laws, if you look, if you happen to have a guide in front of you, the first thing that they both do is define what a record, this one defines what a record is. And a record includes all recorded information regardless of physical form. So this includes paper, email, computer files, photographs, audio tape, or recording video text message that has a connection with how public funds are spent or with the public entity's performance of its governmental functions. So any time you are doing business as an SCD, and that could be just text messages back and forth, if you're talking business, that is a record and it could be requested. So keep in mind, you know, it's not about the platform you're sharing the information on, it's about the information you're sharing. So if it's on your personal phone, that doesn't matter. It's public information that you're sharing about the board. Hannah, you can advance. So I'm not going to talk about open record so much anymore, move into the open meeting laws. And the open meeting law says that all government records and meetings must be open to the public, again, unless otherwise authorized by a specific law. And both of these are based on the public has the right to know government functions, how government functions are performed, and how public funds are spent. So when we look at the meetings, the definition of a meeting is any gathering of a quorum of the members of a governing body of a public entity regarding public business. Now remember, this includes committees, subcommittees, informal gatherings or work sessions, and discussions where a quorum of members are participating by phone or any other electronic communication. So that's the definition of a meeting. The open meeting guide goes on to explain how you have to notice your public, your meetings, and when they're going to happen. And all of this is covered more in depth in the ethical leadership micro lesson. So if you want to tune into that for a recap on open record and meeting laws, you can as well. Hannah, you can advance to the next slide. So on this open meeting guide on the backside, it tells you what your meeting minutes must include. Can you define a quorum, please? Okay, so a quorum of the meeting is whenever there are enough members present, typically it's over 50% of your membership or supervisors to conduct business. So for example, if you are at a non-district event and you have a quorum, say there's five supervisors there and you start talking about business, then you are actually having a meeting. You have a quorum present of your district and that is a meeting and it therefore should have been published. So everybody knew that you were meeting. So if you have questions about any of this, I'm happy to visit with you one-on-one. Afterwards, please feel free to contact me by email or a phone call. Happy to review that. So getting back to the open meeting guide, I want to save Hannah time at the end here for her portion. It actually states in there, it says that you are required to take minutes and these are the records of what happened at your meeting. These are the items that are listed. Of course, you need who was there. It's kind of the who, what, where, and when. Who was there? When did you meet? What did you talk about in a description of each motion that was made at the meeting? So in order to fulfill that part, you would say, you know, Bob moved to purchase the no-till drill might be an example. You're going to know who it was seconded by. You don't have to list all of the discussion. You just need to say that the motion was approved or not approved. If it is something controversial, maybe a big dollar amount, and you do a roll call vote, you need to make sure that you're listing each member's vote, each member supervisor by name and what their vote was. And this again applies to all government bodies, including committees and subcommittees. Sometimes we forget those. But Hannah, you can advance to the next one. You know, the meeting minutes are their notes. Therefore, they're a record so people can track what happened at your meeting. So they need to reflect on any business that happened. They're a great resource for new supervisors coming in, as well as staff to go back and look at the history on what has happened. So you only need to record the facts. You don't have to talk about opinion. You might say there was lengthy discussion on something, but you certainly don't have to list all the discussion. And like Dodie talked about earlier, they're really there for accountability and transparency. And you need to submit a copy to the State Soil Conservation Committee. Now, we had some discussion the other day, whether it should be a draft or an approved set of minutes. And I think consistency is important. So whichever you are comfortable with, any of you have questions on that, you can certainly visit with one of us. Just make sure if you're submitting draft minutes that they are labeled as draft so that it's clear that they are not official minutes. However, it does state in the open meeting guide that if someone were to request meeting minutes in draft form, those need to be provided. And again, that's just the State Soil Conservation Committee. They're dispersing public funds. And so their need to be responsible to make sure that the funds are being used appropriately. And that's why in combination DART and the meeting minutes help them ensure that. You can also put your minutes on your website or share them with partners. You can advance to the last slide, Hannah. There's a handout that will be shared after this. I believe it's in the Google Drive. It goes over meeting minutes of what they should include. And this is just an example. It's easy to start with a template that you can use for every meeting and just fill in. Sometimes I get in trouble when I copy and paste things because I forget to change dates and things like that. But it's really easy to start with a template and go from there. Take your agenda and build off of there any action that's going to happen. Read minutes from other districts what they're like. Maybe give them to someone that wasn't at the meeting and make sure that they can understand what happened. Some entities are required to publish them in the newspaper, so conservation districts aren't. I was reading some the other day and they weren't very complete. I couldn't even tell what matter they were discussing. Make sure that they're clear enough that everybody knows what is happening. Are we required to have and record a motion and second to adjourn? That's a good question. Every board handles this a little bit differently. If every item of business on your agenda has been addressed, typically the chair has the ability to adjourn the meeting. Now for common courtesy to everyone on the board, lots of organizations will ask for a motion to adjourn. If you have not can address everything on the agenda, then you are going to want a motion to adjourn. So I hope that's clear. If you want to err on the side of being cautious, you can have a motion to adjourn. The other thing I wanted to note, you may want to record the minutes so that you can go back and listen and fill in details that maybe you missed. That's fine. If it is an executive session, and we don't really have time to talk about that now, but remember there's only certain reasons why you can go into an executive session, but if you do go into executive session, that does need to be recorded either audio or visually and you need to keep that on file. So I'd encourage you not to do that on a personal device, like your personal cell phone. You want to do that. It might be a good investment for your district to purchase a little device to record meetings so you can store that safely in the district office. Hannah, I think that's all I had. If anyone has other questions, I'd be happy to. Dodie has her hand raised right now. Do you want me to take that now? Yes. Actually, it's more of a comment than a question. It was something that you mentioned about sending in draft minutes or approved minutes. I have requested draft minutes because oftentimes a district may take a month off and then I don't get the minutes for two months, sometimes three months. And by the time I read the minutes and I do read the minutes, my question is a moot point. So I'm requesting draft just so I have a general idea of what's going on and it's more current. And we are not providing these minutes to anybody. We realize that if minutes need to go out, they need to be approved. So if there was a request for minutes for a county from us, we would go back to the district and request approved minutes. It's up to you and your board, what you want to do. As Andrea said, just be consistent. I'm okay with that, but I just wanted to explain my reason for asking for the draft minutes. And I believe there's a question for you, Andrea. Before we jump to the question, Dodie, do you want to explain why you take the time to read people's SED minutes? Are you going to catch them making a mistake or what's going on? The only way I can keep up to speed on what's going on with 54 soil conservation districts is to read the minutes. I like to know what's going on with conservation. A lot of times I can help districts with funding opportunities or maybe a speaker. I recently offered a speaker name for a district that is going to be doing a workshop down the road. So it's just a way for me to keep in touch with the districts because it's hard for me to to visit all of them. And that's really all it is. I do appreciate it. And if your board doesn't want to send draft minutes, that's fine. I don't have a problem with that. Yeah, in case someone can't see the chat, the question was just to clarify an executive session that those do need to be recorded. And they do. If you look on the back side of your open meeting guide, it has procedures for closed meeting or executive session. One, you want to make sure you have a legal reason to go into executive session. And just because you don't want other people there isn't one. It has to be one of the six listed. So you convene in an open session, you announce why you're going into executive session. You pass that motion to hold the executive session. And number four listed is record the executive session electronically. Remember, the only thing you can talk about in executive session is the reason you went into executive session and you cannot take action in executive session. No motions can be made in executive session. Those need to be made in the open meeting. And if you have any other questions on that, please feel free to ask. Another question brand new here, how do you get the minutes approved at the next meeting? Good question. So you will present a draft of the minutes to the board before the meeting so they can review them. And on your agenda, you will have like the sample one here, we have minutes approved. You could do secretary's report. It can be called different things. When you come to that item of business or that line item, then the board chair will ask for the minutes to be approved. If there's any questions, they may be asked there or additions that need to be made, corrections, it can be made there. Recorded meeting, are recorded meetings official documentation as well? Yes. Any recording is an official document and can be requested. Now what I've been told is that if it's an executive session, you might want to consult your local state's attorney as to whether you would have to share executive session. They have to have a valid reason to request executive session minutes is my understanding. Again, if you ever have questions on any of this, your local state's attorney is your resource and the attorney general's website has a lot of great information. Earlier, Andrew, you mentioned how some SEDs, they record the audio of a meeting just to go back and fill the plane. Do they need to keep a copy of that audio once they've updated their minutes or when they need to read back on it? I think that it's always probably good practice just to keep it as long as you know you have it. I hope you feel popular, Andrea, because you obviously there is a need and people have questions about meeting and if that really is the music. Good thing we have you on today. Thank you. Yes, so we're just here to help you succeed. So please feel free to ask questions. I'm going to round out the four of us and everything and I'll wrap things up by overviewing the equipment inventory sheet and supervisor training reports. To start with, I have received questions as to why the NDSNCC requests an equipment inventory list and how I've explained and answered that question is part of the committee job is to increase SED capacity and an equipment inventory list is one of the foundational record for any business. We need it for insurance purposes and in general it's just good to have on hand to help you be aware of what inventory you have and it can help you assist in making financial planning and purchasing decision. Like Jody mentioned earlier that it may be requested copy to store in case the file they're lost or need to serve as a backup but we don't compare one. SED is the list to another SED collection or that information. We like to be the best of the service to all of you, right? So what could be included on your equipment inventory list? Basically whatever information will be helpful to your SED when making a decision including that equipment purchases repair or maintenance. What is showing on this slide are suggestions for what you could include on your equipment inventory list and that they include you know the date that you purchased that item what the item is in the item description. General notes those could be maintenance notes or to just whatever that whatever kind of note that you would appreciate being able to add an expert for it, right? It can include an item description, general notes, the serial or the number, purchase price, where you bought that item, the physical location that it stored, the equipment condition, maintenance schedule or photo and so on and so forth. We or the NPSF we see doesn't have any set requirements for what to include or not to include on this list. We leave that up to you as an individual district. So in summary include whatever information that would help your district function at its highest level and be at a reference in case documentation would be lost down the road. I'm all about tips and tricks and so a couple tips that I wanted to share as far as equipment inventory list goes is that if you are sorting from square one this can feel very overwhelmed, right? And so I would encourage you to break it down. Maybe the first year your list just includes a list of here's all my equipment, right? My tree planter, barber, barrier layer, you probably look my words here, but you just list the equipment right enough or we have on your list. And then you decide that the next year maybe you add photos of each of the equipment or write a note about what type of condition that it's in. Now don't feel like you need to create that list all at once either. Break it up month by month or season by season. Make it more manageable for you. I would also encourage you that once your list is created, updated or for needle, right? It's a lot easier to add new equipment and the needed information as it is purchased than running around trying to collect and gather that information down the road. And finally, you might be chuckling at me and wondering, well, how can I make this a board bonding experience? Well, don't feel like you need to create this list on your own. Involve your supervisors. Maybe after a monthly meeting, have them assist you in recording some of that information. They also might have ideas about what information they would like to see recorded for future reference. You shouldn't have to think of everything, right? Well, speaking of those supervisors, I'm going to talk about supervisor training report next. I know how much supervisors look forward to those annual training requirements. Kitty, I know they're not always everybody's favorite thing, but everyone on the soil and watershed leadership team is working to provide additional training opportunities to make these requirements easier. We understand how the busy everybody is, and so we're trying to think outside the box to figure out how can we provide training in a timely and considerate way, right? Okay, let's start with why do supervisors need to compete their annual training? The tough love response, and I think something that we have hit home at multiple points throughout this webinar, is that it's required in the North Dakota Century code. End of story. It's kind of hard to argue with that. I prefer taking the approach of let's get folks on board with this idea. And so with that in mind, I would just say that the purpose of an SCE is to provide and promote conservation related practices to area constituents, right? And training is meant to keep supervisors up to date with relevant conservation practices, along with strengthening their managerial skills that they need to supervise the employees that they hire. Now, we want to keep learning and once we stop learning, that's when you start running into trouble, right? Okay, so now that we kind of understand the why and hopefully everybody is on board, what is the yearly training requirement exactly? This differs depending on if you're a new or returning supervisor. For new supervisors, they're expected to fulfill the following training requirements. So that's a year one training. It's kind of like an introductory training. And then they're also expected to fulfill their annual training requirements. Those are for new supervisors, a little bit extra training that that is asked of them. For a returning or current supervisor, whether they're elected or appointed, they are asked to fulfill an annual training requirement. Okay, now, what exactly are you talking about when you say an annual training requirement? Well, we've got options, right? If supervisors are looking for a one-in-one approach, they can attend one of these five options, right? That includes a leadership academy, level one or two, and that's available whether you're a new supervisor or you've been around the block a time or two. You can attend the NDASCD convention, an area annual meeting, or if you're looking for a team approach at your monthly board meeting, everyone can watch and discuss the micro-lessons that are sent out. You must watch at least four of the eight videos and discuss the follow-up questions of the board for this option to be out, okay? You don't get to go over the grades fully with due book reports. One person doesn't get one person doesn't get to watch the micro-lesson and give a book report at the annual meeting or at your monthly meeting. You need to watch it together as a group. There's also a mixed vague approach where supervisors can mix and match three of the following to meet their annual training requirements. These include attending a local event hosted by the SCD or Extension. They can watch a conservation webinar. They can individually watch a micro-lesson or listen to an agriculture-applied-more-than-dirt podcast episode, okay? So the one disclaimer is that you can do no more than two out of the category. Now, I know I just spoke a lot of words at you and everything and I like examples. So I wanted to run through a couple examples to help you wrap in mind around what I just talked about. We have three supervisors. Alrighty, so Meet newly elected supervisor Hannah. She's excited to be serving on our local SCD board. To complete her annual training, she attended the leadership academy level one to fulfill her initial training required, right? Now, to fulfill her annual training requirement, she attended a local shop talk that was discussing covered crops and she watched an excellent webinar about soil conservation districts in terms of opportunities and then she attended another educational event about soil health facts, okay? The next supervisor that we have is returning elected supervisor Jody. She's been serving on our board for 20 years and it's hard for her to commit to all the events. As a result, she attended a shop talk on cover crops. She lived into a riveting podcast episode of agriculture-applied-more-than-dirt where they discussed challenges that SCD supervisor faced and finally she attended a lead local event hosted by her local NDSU extension office. That's one option or the other option is that you can go, Chris has left. She is a returning appointed supervisor that's been serving on her local SCD board for four years. She likes to check things off all at once and so how she met her training requirement did by attending the NDASCE convention. Alrighty, so those are three ways that those different training requirements will be available and hopefully gives you a better idea of what that looks like. I've got one minute so I'm going to quickly go through these final tips and tricks is make sure to review training requirements with superintendents. I've created a handout that summarizes what I just went over. Of course, it doesn't have the fun commentary that I was able to add in but I can trust that you can add your own square. Help them understand that this is really a requirement to help them grow as supervisors and make the power of them to increase their local SCD copy. Keep a training tracker sheet in supervisor fighters and figures. I understand that employees are really busy and it can be hard to run down supervisors. Go ahead and print off a copy of that trapping sheet and include it in that binder or that folder and I can serve as a gentle reminder to supervisors that training is something that they're supposed to be prioritizing. Finally, make it a board bonding experience. I know, I know, I am the conveyer one of the bunch. But seriously, events are so much more fun to go to if there's a core load goal. If you're going to a neighboring SCD event, invite your supervisors to come up or encourage your supervisors, maybe ones that have been around for a while to welcome a new supervisor and then have them invite them to an event. I was just at an SCD meeting and we were talking about podcasts and training and all of a sudden the supervisor was like, oh, well, I listen to podcasts and webinars all the time. You know, I'm waiting for the water to on thought. I watch a webinar and it just led to this great discussion about what, how there are some really great educational resources out there and they were going to start sharing webinars and podcast episodes. So if training doesn't have to be something boring, it can be something that can be fun and kind of create, bring folks together. So without being said, are there any last burning questions that we need to address? Hannah, while you were presenting, I answered a couple questions and I just want to run this by you. Since you are leading the webinar, it was asked if we're going to send out the presentation slides and I said that they'll be uploaded to the SCD Google Drive along with the webinar. Is that correct? Yes, and I think HRSA and I had discussed, right, HRSA emailing out the PowerPoint to everybody. Yeah. Okay. And then there was a comment, what if your supervisor or supervisors don't want to do training? I did respond and say that the State Soil Conservation Committee is going to be addressing that sometime this year. You can also call me and I'm more than willing to talk to the chair of your board about the training requirements. Oh, good question. Lido, we are at the top of our hour and everything like that. Again, I will look back to the original page that has all of our contact information and everything and we are all more than happy to help answer any sort of questions that you have or resource within the health. And, you know, they'll be to reach out. Thanks, Anna. I just wanted to add something to what Jodi kind of started the webinar talking about in that, you know, some of these things are required for the state legislature. Right now it's a positive relationship there. But let's not let that change. You know, we need to keep advocating for districts and sharing the good work you're doing. And so by supervisors not wanting to attend training, that doesn't shed a positive light on districts. And keep in mind we'll be looking at some different leadership at the state level. So we need to keep those lines of communication open and share with your local representatives what's happening in your district. Thank you, Hannah, for organizing this. And I appreciate the districts that attended. Thank you, everyone. But good luck finishing up all of the, I know there's finishing up the documentation and getting that into Chris and everything. She does a lot of hard work kind of reaching out and kicking all that stuff organized. So we really appreciate the work that she does and we appreciate the work that each and every one of you do. Yes, I second that. See everybody. I was Andrea, I was just talking with Lynette yesterday about the legislature.


Okay, so let's start. So we are discussing various examples of topologies. Last was product topology of two spaces, another one subspace topology. So the notation, we have all those x topological space. So it's set with a topology, topological space, and there's a subset which is called y. Y subset of x, okay? We have a subset of a topological space. We want to define a topology on y. So t, y, the topology on y, these are all intersections. Y intersection with u, such that u is open in x, okay? U open in x, u in t. So this is topology of x. So this is a topology, and this is a subspace topology. So this is a definition. Well, one has to prove that the topology, this is very easy because what is the three conditions? The empty set is, of course, the intersection of the empty set. The whole set, y is intersection with the whole space, x. And then we have arbitrary unions. Well, for arbitrary unions, what do you have? Arbitrary unions, no? Union alpha and j of open sets. So the open sets are y intersection, intersection u alpha, no? Y intersection u alpha. Where these, so these are the open sets. These are open in the whole space, no? These are typical open sets. And of course, this is set theory again. So it's y intersection union u alpha, alpha and j. And so this is also open in x. This is open in x, okay? This is for the union. And the same for the intersection, y from 1 to n, finite intersections, no? Y intersection of ui, I change index, no? This is a finite, finitely many, u1, un, okay? Yeah, I don't know, u alpha, so I cannot. So this, of course, is equal. So these are again open in ui. So this is again y intersection ui, i from 1 to n. And again, this is open in x, finite intersection of open sets. So this is more or less proof, okay? Which is easy set theory, all those. So this is the topology on y. And this is a subspace topology. That's the definition. Okay, so as for the product topology, if you have a basis, so there's a little lemma observation, more lemma. If b is the basis for x, for the topology of x, then I should use the notion then by. So what we take is, we take not all intersections, but only with the basis. So y intersection b, b in b is the basis for y, for the subspace topology. It's the basis. So how do we prove that something's the basis? I will not write the proof, but I'll make a picture, no? But this is proved by picture. We apply some lemma as the basis. For each open set, for each open set, each point in this open set. We should be able to find the basis element that contains the point and it's contained in the open set. Then the lemma says, this is the basis, okay? And it generates a topology, which we have. That's the lemma, okay? Which we always use in this situation. So what we have? We have x and we have a subspace y, a subset y. And we have an open set, okay, and a point. So what are the open sets in y? The intersections, so this is an open set in y. This is the u, okay? So this is u intersection y, u is open in x, no? And we have a point, we take a point, okay? Then we should be able to find, so this is x, small x, okay? So we should be able to find the basis element which contains the point and it's contained in this open set, no? And then we apply the lemma. However, this point, of course, is in u also, okay? And we have a basis for this topology of x. So we find the basis element which contains x is contained in u. The basis element in the basis b, okay? This is b in b, this basis element, no? And then it's clear that we intersect b with y, okay? And then, of course, x is contained, so we get here this part here, no? In b intersection y, oops, b intersection y. And this is contained in u intersection y. So this is a proof, okay? Trivial proof, okay? You make a picture and look at the picture and then you write down what you see. Okay, so now I give one example here of subspace. And this is a little bit strange, but it's important. It will be important, example in the book. So here's an exercise, or example, better example. So i is the interval, 0, 1 in R. So this is a real interval, okay, i. That's why we don't call an index at i, because in general i is this interval. That's the notion of the book. And then we take i cross i. Of course, i cross i, we have here the subspace topology, okay? This is the standard topology, and we have the standard topology on the interval. Then we have the standard topology here, which is the product topology. That's not very interesting, okay? So what you take now, so this is contained in r cross r, that's clear, and we have the standard topology if you want. But that's not what I want now. What I want is, so this is ordered, okay? So we have the, what did we say? Dictionary order on i cross i, dictionary order. It takes a dictionary order on i cross i, okay? And with this topology, this is called the ordered square in the book, okay? Then we get the ordered square. That's the definition of the ordered square. And it's written in the book, i squared ordered, that's an o, okay? The ordered square. So this is a notation. This is one of the many examples of the book, which we will see again and again. I will discuss it more and more. So we take here the dictionary order on i cross i, okay? First coordinate, second coordinate. And then we take this topology, the ordered topology, and this is the ordered square. The ordered topology, okay? Dictionary, we take, of course, the ordered topology here. And then we get the ordered square, okay? That's an important example and that's a notation. Of course, you can do something else here, okay? You can take here the ordered topology, okay? We can take here the ordered topology of the dictionary order, okay? That's the only we see of the dictionary order. You can take here the ordered topology. And then we can take the subspace topology, okay? Then take the subspace topology. Then take the subspace topology, as we defined, on i cross i. So we have the subspace topology of this ordered. And we have directly the ordered topology, okay? And the observation is they are not equal. One is an interesting example. The other is not so interesting, okay? So what happens is, there's an exercise. One of the exercises in the book, but it's important so I do. So observation, you would maybe guess they are equal, okay? They are not equal. So the exercise is the subspace topology is strictly final. Then the dictionary, okay? The subspace topology, in this sense on i cross i is strictly final. They are not equal, these two topologies, but they are comparable. It's strictly final than the order topology of the order square, okay? Then the order topology. So this is the situation. So we make, how to see this, okay? So we have the order square, we make p cross. We have the order square, i cross i. That's i cross i, okay, the square. And then we have to prove, we have to prove two things, final, no? The subspace topology is finer than the order topology, okay? So we're taking open sets in the subspace topology. Well, we can work with the basis, okay? The basis for the subspace topology. First, we have the basis here, which are intervals, okay? These strange intervals. There are two types of intervals, but we need only one type of interval, okay? So intervals and we intersect. So what are intervals? Intervals are, for example, these types of intervals, no? This, this, and we have this type of interval, okay? In R2, in R cross R. And then we intersect, and what do we get? We get, we get this one, okay? So this is open. That's a closed interval, the intersection, okay? That's a closed one. This is open in, in which, in the subspace, in the subspace topology. This is open. That's a closed interval, which we get, okay? I, I started with the wrong, okay? I should, if you prove finer, I should start with the other topology. This is smaller topology, okay? But this is open in the subspace topology, but I can continue here. But, so they're not equal to topology, but not open, but not open in the, what is the other one? Order topology, okay? This is open in the subspace, that's clear, no? This closed interval, you get this intersection with this larger open interval in R2, okay? So it's open, by definition of subspace topology. That's an example of subspace topology, okay? But I say it's not open in the other topology, in the order topology, why not? Well, of course, in these points, we have no problems, okay? If we take such a point and we take an open set in the order topology, the problem is this point and this point. These two points are problem, no? Because if this would be open also in the order topology, then given this point, we should be able to find the basis element of the order topology which contains this point and is contained in this interval, right? That's always the same condition, no? So again an interval, we're in the order topology, but on this small, we don't see R2 now. We have only this space, the order topology, okay? These two points are not there now, okay? So I need an interval which contains this, right? Well, I have no problem with the lower bound, obviously, no? As a lower bound, I take anything here, right? That's clear, but I have a problem with an upper bound. This point I cannot take, that's not an element of I cross I, right? I need an upper bound. So what I have to do, I have to go to the right. So with the upper bound, I have to go to the right. And what does it mean? I have to take some point here, I don't know which one. But I have to go to the right, okay? But then I get the interval between this point and this point, okay, as an upper bound. I get everything, all these intervals here and up to this point, okay? The interval between this and this contains all these lines in between. So no basis element contains this point and it's contained in this part here, okay? So they're not equal to the topology, so this, that's what I write. This is open, so again, this was the example. This is open in the subspace topology, that's trivial. But not open in the order topology. So they're not equal to the topology. I should have started with the other one, okay? So now I show, this was not final. This was strictly final, if you prove final, okay? So now I prove final, okay, again. But this is clear, so now I have to start with an element of what, of, no, I lost. I have to, with the topology, which is course, okay? That's the order topology. So I take an open set of the order topology. How do they look like? Well, I take two points here and I take such an open interval, for example, no? And it's clear that this is also open. That's the basis element, okay? For the order square, this is open also in R2, okay? This is also an interval in R2. Or I have the same, of course, if I take a point here and maybe a point here now, then it's this one, no problem, open in R2, okay? So this is no problem. So then I can take points with different coordinates, intervals, okay? So this goes up, then you have all this stuff here and this is the interval, right? And it's clear, this is open R2, given any point here now. For example, this point, we should be able to find a basis element of the subspace topology, which contains this point and this content here. Well, if this takes this point, there's no problem again, okay? This is clear. Here we can take again this, but also for the other points. You may think, the only points, maybe look at this point, okay? This is here and now, but this interval here is intersection of something like this, okay, in R2. So the intersection of this with this is this, okay? So this is also, okay, we find the basis element. So the difference is the basis elements, maybe you find this easy. The basis elements for the subspace topology. So what are they? They are open intervals like this. There are also these intervals, okay? Which is intersection with the larger interval here. And of course, we have also, for example, these, okay? And these are sufficient because these are, we take intersections in R2, of intervals in R2 with this. And we get this kind of stuff. And for the order topology, it's different, no? The basis, this is not this. So we get again, of course, these intervals, no problem. But we don't have this kind of stuff, okay? What we have is this, then, if you don't want to stay on one line, then we have to go and forget immediately such a kind of construction, okay? Because the upper point of the interval, well, we have to go to the right, okay? If you want this point here, we have to go to the right. We cannot go up because we are not in R2, we are in I cross I, okay? So these are two topologies in I cross I. By the way, this topology is not interesting, why not? Which is this topology for the subspace topology? What is this topology?  Of course, you have also these, no? These are open also, no? So what is this topology? You can give, describe in a different way. Yes, discrete times standard. So this is subspace topology, so this is equal, okay? To I discrete topology times I standard, okay? That's exactly this topology. I discrete with a discrete topology. So this is a discrete topology, and this is, if you don't say anything, it's standard topology, standard. And you get this topology. Well, you get basis, the basis of the product is basis times by basis element, okay? Here you have points, okay? And then times basis elements of I, what are basis elements of I? We need also these intervals. Now, because it's bounded, we need open intervals. Half open intervals, if you have the largest or smallest element, no? And that we have here, okay? We have the largest or smallest element. So this is standard, this is just this product, okay? And it's not so interesting. What is interesting is this one. This one is interesting, okay? This is many open sets, okay? Too many to be interesting, okay? Just discrete times standard and many open sets. So you can forget more or less this, it's just an exercise. But this one you will see, okay? And this is I squared, ordered square, okay? That's the notion, that's the ordered square, ordered square. And the basis looks like this. And this has much more open sets than this, and this we will see, okay? So this is an example for subspace topology, order topology. This is an exercise, okay? So you should under, this comes back, you should remember this, okay? This was an exercise. The exercises are very often more interesting than the proofs. I will give tomorrow some exercises for home tomorrow, okay? So this is a subspace topology and some example. And now we have closed sets and limit points. This is very formal again. So closed sets and limit points. This is like real analysis, limit points. Maybe not limit points, it's not so clear. So what is a closed set? So X is a topological space. So the closed sets are complement of open sets, okay? So A in X is closed if, that's the definition. X minus A, the complement, that's the complement is open. Well, it's equivalent in some sense. You may work with open sets or with closed sets, no? The axioms for closed sets is empty set and X are closed. This is the complement of X, this is the complement of the empty set. Then arbitrary unions of open sets are open. Finite intersections of, finite unions, sorry, finite unions. It's just the opposite, no? Finite unions, that's the morgan of closed sets are closed. And arbitrary intersections of open sets, of closed sets are closed. So this is the morgan, okay, again. I mean, this first is trivial, this is trivial. Finite unions of closed sets, so union, finite union of closed sets. So what are closed sets? Closed sets are complements of open sets. So it's X minus, no, Ui, I from 1 to n, no? So this is a closed set here, this, I mean, this is open, and this is closed. So finite union of closed, and then we have the morgan, what does it say? This intersection of X minus intersection of Ui, is that okay? That's what it is, okay? And this is open, but we need to find that intersection. Otherwise, we have a problem, okay? And the same, of course, for the other morgan.  We have an arbitrary intersection, arbitrary intersection. So this is again X minus union U alpha, alpha in J. And here we have no problem because arbitrary unions are open. So this is open. And that's the morgan, two times, okay? So these are the axioms for closed sets. So we have, we prove two lemmas, which are almost trivial, but used often. So the first lemma is, so if, why, we have subspaces, no? We define subspace, subspace topology. So Y in X is subspace, let Y in X be a subspace. This means that X is a topological space, Y is a subset with a subspace topology, okay? So this is a short way to write that. And then a subset A in Y is closed. So what you suspect is, open sets are intersections with open sets from the big space, no? And closed sets are same. So if and only if, then I is closed, if and only if A is intersection, C, C closed in X, yes. So the open sets in the subspace are intersections with open sets in the large space. And the closed sets are intersections with open sets, okay? So maybe, the proof is very easy, but so I will write the proof. Well, you take compliments, no? I might take compliments, okay? So let's write the proof, okay? So we have two directions anyway, so we have to be careful. So suppose A is this one. Suppose A is Y intersection C, where C is closed in. So what do we have to prove? We have to prove that A is closed in, yeah, A is closed in Y. So Y minus A is open in Y. That's what we have to prove, okay? So this is closed in X, what does it mean? That means X minus C is open in X, no? That's the definition, compliment. And this implies that X minus C intersection Y is open in Y, yes. That's the definition, no? This is open intersection Y, these are exactly the open sets in the subspace, intersections with open sets in the large space, okay? However, what is this? This is Y minus C intersection Y, if you look at this now. You have to throw away C, okay? So Y minus C, C intersection Y, okay? Because we are in Y, so that's clear that this is the same. And this is Y minus A, okay? So this Y minus A, and this means of course that Y minus A is open in Y. So the compliment is closed in Y, no? So this implies that A is closed in Y, since its compliment is open, okay? So this is one direction, okay? That's one direction, and conversely, maybe we did this, so conversely. One has to concentrate here, no? What to write, conversely is very depends on logic. So which direction did we prove? We prove this direction. Now suppose A is closed in Y, okay? So conversely, suppose A is closed, and we have to prove that it is intersection of a closed set in the whole space with Y. No? That's what we have to prove for the other direction. So A is closed in Y. This means Y minus A is open in Y. And that means that the open sets we know, so this means Y minus A is equal to U intersection Y where U is open in X. The open sets are intersections. So what have you proved? No, no, I don't see anything. So this means, what is U? U is open in X, no? So X minus U is closed in X. I have to write no, closed where, no? Closed in Y, closed in X, otherwise we have problems, no? And Y intersection X minus U is equal to Y intersection. No, sorry, Y minus Y intersection U, no? This is what we have to throw away, Y intersection, avoid U, okay? So Y minus Y intersection U, so I want, this is A, no? Of course, this should be A, that's clear. So this is Y minus, maybe I'm exaggerating, intersection A, so this is A. I'm saying, that's what I want, this is A, what is U? U intersection Y is, sorry, I wrote something wrong, no? Minus A, U intersection Y, no? Y minus A, so it's okay. It's two times the complement. Okay, so here one has to concentrate, okay? On taking complements, open where, no? So one, it's not so pleasant to write, okay? You have to concentrate a lot on these kinds of proofs, okay? The same is true for the next lemma, which is also. So this finish is a proof, no? So the next lemma, but they are used often, so, in particular, in particular the next one, so maybe I prove that also. So what's the next lemma? The next lemma says, I call everything lemma here. The book is proposition, sometimes, whatever, I don't know. So Y subspace of X, well, I mean right, subspace of X. Again, as before, sorry. I have to define the closure before, so definition. Before I can write this lemma. So the closure, what is the closure? So I is a subset of X, X is a topological space. So this is a subset and X is always a topological space now. So we have the interior, which is interior, interior of A, also written in this way. This is a union of all open sets, which are contained in A. So U, open in X, U contained in A. So arbitrary union of open sets is open. So this is open, okay? Open in X, this is open in X, clearly. Arbitrary union of open sets, no, it's open in X. In fact, it's the largest open set, obviously. It's the largest open set contained in A. The largest open set contained in A, that's what it is. That's the interior, I didn't write, the interior of A. And you write interior of A or A, interior, okay. More interesting is the closure. So then we write sometimes closure of A, more often this notation, the closure of A. So this will be the closure of A, so what is this? So now we take the intersection, we take dual to this intersection of all closed sets. So C closed in X, such that A is contained in C now. So arbitrary intersection of closed sets is closed, so this is closed, okay? So closure of A is closed, it is closed in X, certainly. It is closed in X, and in fact, it is, what is it? Yeah, exactly, the smallest, because it takes intersection of all closed sets which contain A. The smallest closed set containing A, that's what it is, okay? The largest open which is contained, the smallest open which contains. And if you change roles here, it makes no, it's in general not well defined, okay? Because arbitrary intersection of closed sets will not be open, and arbitrary union of closed sets will not be closed in general, okay? So you can define these two, but you cannot change, what is not well defined in general is, so this is the largest open set contained in A, the largest closed set contained in A, it's not well defined in general, okay? The largest closed set contained in A, you cannot define, okay? Because you, and what is also in general not defined is the smallest open set containing A, that's also not well defined, okay? Well, you can try to write, but then you get, have to take arbitrary union of closed sets and arbitrary intersection of open sets. That's in general not open and not closed, okay? It's defined, but it's not open and not closed. So it's not, okay. So now comes the lemma, which I wanted to write immediately for getting to define. So this is the definition, and now this lemma, which I start again, so what does it say? So why is the subspace of X again? Why subspace of X? So that means X is a topological space, Y is a subset, and we have the subspace topology, okay? Which we defined.  A subset of Y, then the closure of A in Y is, so what, you take the closure in the whole space and intersect with Y. So it's A closure intersection where this is the closure of Y in X. So this symbol is also in reference to the large space, okay, not where. So Y is closure of A in the large space, okay, always in X. Okay, so let me prove this also. It's also concentration, again, one has to concentrate, okay? It's not so pleasant, always, because one is tired, one has to apply definitions, okay? What is the definition of this? What's the definition of this, and then apply, apply, no? And then it's almost canonical. In any case, it's useful, so we have two closures. We have the closure of A in Y, and we have the closure of A in X, no? So it's better to give a name for the proof, to the closure of A. We have a name for the closure of A, so let B be the closure of A in Y, and we have to prove that B is equal to this, okay? So a claim is, of course, that B is nothing else than A bar, so this is the same. And so we have two inclusions again, no? As usual, here we have these two inclusions, maybe not. Okay, let's see. So A bar is closed in X, by definition. That's the closure and the whole space. So the intersection, this means, of course, that A bar intersection Y. So I don't write then, okay, we go, this means this and so on, implies, implies, no? So A bar intersection Y is closed in Y. That's a proceeding lemma, no? Proceeding lemma, that's what you're proceeding. Of course, A is contained in A bar. This is the largest, the smallest closed set containing A and it's also in Y. By the definition of closure, exactly, of closure now in Y, okay? Because now we are in Y. So the closure of A is B in Y. B is contained in A bar intersection Y. Because this is closed, no? And B is intersection over all closed set with contain A, over all, okay? So B is, okay, so this is one inclusion, which one? Well, this one, okay? This one, so we need the other one. Conversely, conversely, it's not so clear. For the other inclusion, no? That means conversely. B is closed in Y. That's the closure. B is closed in Y. So by the proceeding lemma again, okay? The lemma before precedes with one here with two here. So I'm not completely sure, no? Purchheader with one, I suppose. In Italiano, in Italian, it's this one. Yeah, it should be one also here, then I suppose. By the preceding lemma, maybe not, I'm not sure. One is sufficient, it seems, no? By preceding lemma. So we have a closed set in Y, so what is that? B is equal to, the closed sets in Y are C intersection Y where C is closed in X. That's what we proved before, no? I'm writing two large, it seems. So what we know is that A is contained in C. This clear A, of course, is contained in B, you know? That's a closure. So A is contained in C. C is closed in X. This is closed in X, which I'll write again. This is closed in X. And this implies that A bar is contained in C again, no? Because this is closed in X. A bar is the smallest closed set. And then finally, we have A bar intersection Y is contained in C, intersection Y. And that was B, and that's it. That's the other inclusion, so they're equal. So one has to concentrate on notation, no? If one is tired, it's not so, but everything is canonical, okay? No ideas, but applying definitions, definitions. But we have to use the notation, and we have to think all those, at which point are we, okay? So this is this lemma. So now, so I start again here. I mean, this definition is somewhat awkward, no? Of the closure. The largest, the smallest closed set containing the intersection over all closed sets, no? You don't want to find all closed sets containing and then take the intersection, no? So what we apply is the following proposition. And so A is a subset of X. X is the usual topological space. And X is the point in X. X is in X. So I, then, the following. X is in the closure of A. You want to decide if this point is in the closure of A, no? If and only if every open set containing X intersects A. So now I introduce some notation, which is very useful, okay? If and only if every, and now I write neighborhood. I call this neighborhood. But one has to see what is the neighborhood, okay? They have not one, two possibilities at least. Every neighborhood, it's a long word, but we will abbreviate later this. Every neighborhood, U of X, and what is the neighborhood?  That's an open set containing X. Open set, that's a neighborhood for us. An open set containing X. That's a definition. There's another possibility of neighborhood, more general. It might be a set containing X, and X is in the interior. So X, there's an open set, which contains X as in, contains in the, maybe the larger is not open itself, okay? But for us, neighborhood is open. Every neighborhood of X intersects, what means intersects? Intersects means has non-empty intersection with X, no? Has non-empty intersection with A. So that's the shortest way to say it. Every neighborhood of X intersects A. Then X is in the closure. And the second part is, we can take the second part is, if B is a basis, B a basis of X, yes, X, of the topology of X, B a basis of X, then X is in the closure of A. If and only if every B in B, which contains X, we don't have to check it for all neighborhoods. Every B in B, which contains X, intersects A, okay? So we have to check only for basis elements. Intersects A, as usual. One can always restrict to basis elements. That's useful because basis is easy to describe in general. We find the nice basis for topology, but the topology is not nice to describe in general. So it's good to work with the basis. So proof. And two, I will not prove. I mean, one direction follows from this anyway. If any neighborhood intersects, then every B in B, which is open intersects. So one direction is, if you prove this, one direction is trivial, and the other is very easy. Again, the property of the basis is given open set, given a point, there's a basis element, which contains a point, is contained in the open set now. So you apply just this, and then it is the same as this. So I, if, let's suppose X is not in the closure of A. If X is not in the closure of A, then of course, U, which is a complement, X minus closure of A, this is open, no? And it contains X, because X is not here. So it's a neighborhood of X. It's a neighborhood of X. And now I abbreviate neighborhood, okay? Because this is too long, one time, okay? But this is neighborhood, it's clear, no one can understand. It's a neighborhood of X, which does not intersect A. Because it's X minus a larger set than A, okay? Which does not intersect A. Since A is contained in A bar, no? Since A is contained in, this is the smallest close set which contains A, okay? The smallest close set which contains A. So this proves, what does it prove? This proves? Yes, it's something. So if X is not in the closure, then there is a neighborhood which does not intersect. So if every neighborhood, if X is not in the closure, then there is a neighborhood which does not intersect. If every neighborhood intersects, then X is in the closure. If every neighborhood intersects, then X is in the closure. So this proves this direction, okay? This proves this direction. Well, you have to just, it's the language, no? If X is not in the closure, then there's a neighborhood which does not intersect A. If every neighborhood intersects A, And then x must be in the closure. So this is the solution. OK. And the other one, conversely, again. Well, I will not prove this, OK? That's a very easy exercise. And so conversely, for the other inclusion. Conversely means for the other, not inclusion, for the other implication. Inclusion, implication, whatever is the case. Conversely, let u be a neighborhood of x. Neighborhood of x, which does not intersect. Suppose there's a neighborhood, u, which does not intersect a. This means that x minus u is closed. Let u be a neighborhood of x. And it does not contain x, no? And x is closed, and it contains a. Neighborhood of x, which does not intersect a. u does not intersect a. So a is contained in x minus u. Yes. And this means that's the closure of a. So this is closed. And then a bar. Because this is the smallest closed a bar, no? It's contained in x minus u. But u is enabled of x. So x is not here. It's not here. So x is not in. It's not here, so it's not here. It's not in a bar. So if x is in a bar, then every neighborhood intersects. So this is the other direction, OK? This proves this direction. So we have both. And now we can distinguish between two types of points in the closure. So this is a definition. Important for us, definition. So a is subset of x. Again, the same situation as before. So x in x, we have a point x, a small x in capital x. So x is the limit point of a. So we won't define limit point. A is the limit point of a. If every neighborhood of x, neighborhood, again, of x, maybe u of x, it's not important for the definition. But neighborhood u of x intersects a. That would be boundary in the boundary, no? That's the proposition. So these are exactly the points in the, sorry, in the boundary, in the closure of a, OK? These are the points in the closure. And now I add, for limit point, intersects a in a point, in a point different from x, in another point. Well, it's not clear why this is called limit point here, OK? That doesn't look. So we have this neighborhood. And it has to intersect a. But not in the same, maybe in x also, OK? But that's not sufficient. We need another, one other point, no? In a. And that doesn't look like a limit, OK? So we have to comment on this a little bit. But it's called limit point. That's the definition of limit point. And so there's a lemma, then, that's the closure of a. It's a and union a prime. So this is not a standard definition, but sometimes, OK? We will not, we use it only here. So a prime are all limit points of a. So this is the closure is a itself, a is contained in a closure. And view at all limit points. Proof is easy. So two inclusions again, no? This and this, as usual. So a is contained in a bar, by definition. The, what? Yeah, there's a smallest closed set containing a. And a prime is contained in a bar by the proposition, OK? So this is by the proposition. Intersects, every, every neighbor of x intersects this a, OK? In a point different, but in the closures anyway. So this implies that a bar, what do you want? A union a prime certainly is contained in it. And we have to prove the other inclusion. Conversely, again, conversely, for the other inclusion. Suppose, well, the other inclusion, which one is it? That's a bar is contained in this one. A bar is contained here. So let x be in a bar. X, and let's add x is not in a. Suppose x is in a bar, but not in a. Then we have to prove it's in a prime, of course, OK? Same proposition, no? So this is then every neighborhood of a of x intersects. That's a proposition, no? By the proposition. In a point different from a, x. Because x is not in a, OK? In a point different x, since x is not in a. So then x is a limit point. So x is in a prime. So x is in a prime. Sorry, x is in the limit point. So it's in a prime. And that proves the other direction, OK? It's not in a. Things, yes? Since x is not in a, x must, so then x is in a prime. So this implies, so this implies then that a bar is contained in a union a prime. If it's not here, it's here. If it's not here, a point here. If it's here, it's good. If it's not here, it's here. Sorry? Yes? No, no, no, no, no, no, no. Sorry, sorry. So you are asking if points of a can be limit points, OK? So for example, q, the closure. So q is in r, OK? The closure of q is r y. So every point is in the closure of r, you know? So given a point, given a neighborhood, this neighborhood contains an interval. This is the basis. And each interval contains rationals, OK? Since each interval contains rationals. In each open interval, we have a, interval is in general, if you don't say open interval. Well, I should say. Since each open interval contains rationals, what are the limit points of q? There's also r, no? So each point in q is the limit point of q. Because each interval contains this point, but it contains other rationals, OK? Not just one rationals, no? So also the limit points of q, every r, OK? So a intersection, a prime is not in general disjoint or something, OK? The points of a can be limit points, itself, OK? This is not the disjoint union. I don't know. What's the question more or less, or not more or less? So you can ask again. Take the integers, no? Then the closure is this, and the limit points. These are all isolated points, OK? OK, so this was the example. I will give a nice example as an exercise, maybe. It's the ordered square, no? And you have certain subsets, and you want to find the closure. And that's very surprising, because it's a strange topology. But I will not do it now. Tomorrow, I'll give you some exercises. Of course, why is it called limit point? That's not so clear here, no? One other point, limit seems limit. There's some limits, OK? Limit means some kind of convergence in general, no? Limit point. But here, you have one other point only, OK? Which is very weak. So there's a condition, which is we, this is Hausdorff. That's the first condition for topological space. So topological space X is Hausdorff. So that's the property. That's a name, and that's a, OK? X is Hausdorff. It's a Hausdorff space. It's a Hausdorff space. What is Hausdorff? That's an adjective, and it's a Hausdorff space. And maybe that's a, but I'm trying to, it's a Hausdorff space, OK? So what? If every two points, X distinct points, X, Y in X, have disjoint neighbors. If every two distinct points, X distinct from Y in X, have disjoint neighbors. Neighborhoods, now this is now neighborhoods. N, B, H, D, S, OK? So the picture is this. We have two disjoint points, X, Y. And then we have disjoint neighborhoods. So these are the disjoint neighborhoods. That's Hausdorff. So why introduce Hausdorff at this point? Because then we have lemma proposition. I come back to limit points in three minutes, but here's a proposition. So what is the impression of the standard topology R? R2 in analysis. Now points are closed. Points are closed. I mean points can be open if you have the discrete topology or some strange topology. If there's a standard topology on the reals or on R2 or on Rn, then points are closed. Finded sets are closed, no? Finded unions of closed sets, no? And this is the property of Hausdorff spaces. So if X is a Hausdorff space, so every finite set, every finite set, every point, in particular, every point in a Hausdorff space, so in particular, every point, in particular, sorry, every point in a Hausdorff space is closed. It suffices to prove that every point is closed, no? It suffices because finite unions of closed sets are closed, no? Finite unions of closed sets are closed, not arbitrary. It suffices to prove that every point, every point is closed. Well, every one point set is closed. Every one point set, singleton, yes, x, this is small x, is closed, x and x, of course, OK? So this is a property here, Hausdorff space. It suffices to prove that every one point set, x is closed, x and x. So what does it mean? That means the closure of this is again this, no? The closure of this is this again, no? If you take the closure here, we remain in this set, OK? So take any other point. So let y be in x, y different from x. So these are all the points, no? x, and we have no different point y, OK? So we're getting tired. So you want to prove that y is not in the closure, OK? That is, you have to find the neighborhood of y which does not intersect x. So we have to find the neighborhood here which does not intersect x. So x is Hausdorff, OK? So this implies that y has a neighborhood, u. So this is u, well, maybe not u. Don't give a name, u, v, two neighborhoods. y has a neighborhood which does not x. Well, we have two neighborhoods which are disjoint, even, OK? So it says a neighborhood which does not contain, because we have disjoint neighborhoods, no? If here we have a second neighborhood, if you want, OK? We don't care too much about the second one. y has a neighborhood which does not contain x. So this implies that y is not in the closure, OK? So y is not in the closure. But that means the closure of x, there's no other point. So the closure of x is x. So x is closed. So x and x, this is closed, no? The closure is closed. x is closed. So that's the proof. But you see, we don't need the full strength, OK? So that's nice to remember. So I will, this is not correct, OK? We have three minutes, right? I have three minutes here of my clock. Two, two and a half. This is OK, this is, I checked. So this is right. I don't know why they have this. So what, so we have, this is a house door, no? Every two different points have to show a neighborhood. Now there's this house door. And this is also T2. This is called T2 sometimes, no? And we don't need T2. What we need is, even given two points, each one has a neighborhood which doesn't contain the other. So the picture is this now, OK? We have disjoint, we have neighborhoods. Each one, each point has a neighborhood which doesn't contain the other. But maybe they are not disjoint. That's T1, yeah? So these are separation axioms. We will see much later more. But these are the separation axioms. And the nice thing is then, T2 is important, OK? The house door, house door we will see, and T1 we will not see. Because it's just, well, it's also important sometimes. But so we have a proposition if you want. That's an exercise. So X is T1, OK? It satisfies this separation axiom. Even though the points are closed. That's the equivalent to the fact that points are closed. That's exactly T1. T2, yeah, it would be enough to have T1 for points are closed. So proof, proof. So one, this we proved already, no? As before, right? Well, we use only T1 in the proof before. We use that Y as a neighborhood which does not contain X, right? In the proof before. That T1 is sufficient. We don't need T2. We just have T1, right? And the other direction is an exercise. Well, this is trivial. This is trivial. So what we have to prove is T1. Points are closed. So let, given two points again, X, Y, distinct points, no? X different from Y. So we have to find the neighborhood of Y which does not contain X. But points are closed. So Y minus X, because this is closed. Points are closed. So U minus a point is a neighborhood of Y. So this is open. This is a neighborhood of Y. What do we want to prove? Which does not contain X, of course. Which does not contain X. So this means it's T1, OK? So this implies X is T1. It's symmetric in X and Y here, obviously, OK? So Y has a neighborhood which does not contain X. And this is symmetric completely. So X has a neighborhood which does not contain Y, OK? It's symmetric. This condition is symmetric in X and Y. So T1 is just points are closed, OK? Of course, now the general game, so I finish. Two minutes over. Sorry? So this is, points are closed, no? And so this is open. The complement of closed is open. Ah, what is U? Yes, thank you. There's no U, no? Careful with the X. Yes, yes, of course. You are right. No U. So we take X, the whole space, no? X is closed. X minus X is open. So this is the neighborhood of X, of Y, of the other point, which, yes, thanks. So tomorrow I will ask you an example. So this game in point set topology now is, well, maybe it's the same T1 and T2, no? If it would be the same, it would not be called T1 and T2. So they are different, no? So then we have an example. We want to see an example which is T1 but not T2. It's not the same condition, no? So you can guess what this example, we know an example already, which is T1 and not T2. This is a finite complement topology here. Finite complement topology of an infinite set, OK? So maybe tomorrow we start with this. So it's easy to recall T1, no? Because its points are closed. And T2, it's a little bit stronger. But in many cases, T1 is sufficient, OK? In almost all of the cases, it's T1. I don't know. I don't remember in this moment a single situation where you need T2 and T1 is not sufficient. Let's see, let's see. So we have a sequence which converges to this point. And you say it converges maybe also to this point, OK? But so T1 means if you have these joint neighborhoods, it's clear this sequence has to decide where to go, OK? If you have this side of neighborhood, this type of neighborhood, we have some problem, yeah, right? Yes. Yeah, it might be in the intersection always, OK? Yeah, that's a good remark, OK? So we have for uniqueness of limit, yeah, that's why we should have talked about that anyway, no? In house-toff spaces, if you have a convergent sequence, then the limit is unique. That's an important property of house-toff spaces, OK? And as you say here, it seems that T1 is not sufficient, maybe, OK? Exactly. So that's the situation, OK? Where you want house-toff, T2 and not T1. Well, we discussed tomorrow, two minutes, if we have an example, OK? Yes. So for today, five minutes over.


The Attorney General of the Federation and the Director General of the State Security Service have both denied season the international passports of the leader of the Islamic movement in Nigeria, Ibrahim Elzazaki and his wife, Zina. Their respective offices have also denied preventing them from traveling. The spoke India-separate counter affidavits tended in response to the fundamental rights enforcement suits filed by Sheikh Elzazaki and his wife before the federal high court in Abuja. Elzazaki and his wife had accused the Attorney General and the SSS of seizing their passports and preventing them from traveling aboard for medical treatment. They are also seeking 2 billion error in damage. Hearing in the suit has been fixed for January 19th.