Bone is living tissue, okay, has its own blood supply and is a store of mineral calcium. It's also very protective, so it's the bone that houses your brain, covers your ribs and your heart, your chest and also provides support for your organs as well as the muscles. Now, so having said that because it is living tissue, then that means it is subjected to the various things that would affect any living tissue. So anything offensive, you know, poor diet, poor nutrition as you would see, probably in kids not getting enough sunlight, being malnourished will affect the quality of the bone. Here the bone is not strong enough in which you have conditions like rickets where the children's leg may be either bokeh and they actually look very ill. As you get older you have the effect of the hormones on the body, especially in women that are postmenopausal, estrogen is very protective for the bones. So after menopause, withdrawal of estrogen tends to make the bone very weak in what we call osteoporosis. Another way for you to look at it is consider it as porous bone. So osteo being bone, porosis being porous, all right. So in between that tells you a lot.  So as you use the bone it gets the message that it needs to do more work, so it gets stronger. The stronger the bone is, the better the muscles are in good condition and the feed of each other. Okay, because it is living tissue, calcium and other things are also very important. So vitamin D, via exposure to sunlight, calcium from your diet as well as vitamin D from your diet are very, very important. And once upon a time we used to believe that as Africans we could never be affected by vitamin D deficiency. Over time studies have shown that that is not correct. So especially nowadays that a lot of people have to stay indoors to work and don't go out to work as much as they would so not in our sunlight. So vitamin D deficiency is a common problem and COVID also brought that to the fore. But I mean this very, very good is protective for organs and also very helpful for bone. So our question is more like the chicken or the egg which one came first and I'll explain why. Now a person can be obese if one of the factors I mentioned exercise. So essentially one model about explaining obesity is that you take more than you expend. All right, so you have more energy and then the body will store it as fat since you don't use it. Okay, so now when a person doesn't exercise or happens the bones get weak, muscles get weak. So that is one angle. The other angle is that when the person is obese there's increased weight for the same amount of bone to work with. What do you think is going to happen? You and I can carry a table together. If four or five people join us it's easier but if you ask one person to do it. So the weight increases I still have the same amount of bone to carry the weight. Definitely. So you have certain things have been it's been implicated in certain conditions like osteoarthritis in which case you would find out that the joints gets worn a lot quicker. So you would advise for weight control. Other things that come with obesity what we call the metabolic syndrome in which case hypertension, diabetes, a distortion of how fats are stored in the body now would also affect the bones and how they work as well as the muscles and the blood supply and other things like that because they tend to clog up the blood supply. So those guys get affected. Lot rests on us that is the health care providers, doctors, nurses, as well as the governments, parastatals who are charged with that. There needs to be sensitization, getting people to be aware that okay this are the possible problems. These are the things you should look out for. When you are young this are the potential issues that you might have. When you get a particular age you need to start checking your BP regularly. You need to check your sugar. You need to be sure that things are fine. You should exercise. So it's not just waiting until you get sick. You should inculcate the habit of trying to live healthy and then also ensure that I mean you get regular checks as and at when do you understand what the family history is like. I mean if your folks are diabetic or hypertensive then it behoves on you at some point that you need to start checking out those things so that you don't develop issues and even if that comes up you recognize it on time and can seek help. So there's a lot, I mean like what you guys are doing currently is also very helpful and getting information out there. What are the common health problems that we see in Nigeria? What are the common problems we see in the youth health wise? What can we do about them? What advice can we give and then promote that and I mean make it a regular point of contact for them so that they understand that okay, none of us created ourselves. The human body is a working machine and it's just like your car can develop a fault at any point in time. So part of what you're talking about is like you maintain you service your car you want to go on a long journey you go and service it as has from the regular services so what stops the human body from getting the same thing?


If you want to use your stream deck as a calculator for your Mac or PC, then I've got just the video and the icon pack for you, so keep on watching. Hello, and welcome to Take One Tech. My name is Alec, and today we're talking about the stream deck again, and this is one of a series of videos that I'm doing where I'll be sharing some of the uses that I have for the stream deck, and I'll also have icon packs to accompany all of these as well. So today we're looking at possibly not the most power user of applications, but it's the humble calculator, and what I've done is I've created an icon pack so that I can have all of the controls for the calculator in my stream deck. Now, technically, yes, you can do this with the number pad, but I have added a few extra little shortcuts on there as well, and, well, you can't really beat the feeling of a nice clunky desktop calculator, can you? So let me show you what I'm talking about. If I come over to my stream deck, there you can see I've got my e-cam live controls. I'll drop a little video up into the top corner for the video I did all about how I use e-cam live and how I've got all of my shortcuts set up on stream deck, but I've just added a handy little shortcut to the calculator app here, and if I click on this one, then there you go. This is what we're talking about. It is having all of the sort of number pad and controls for the calculator right on your stream deck. So the way I have done this is that obviously I've programmed all of these shortcuts. They're pretty simple, and it's a good introduction to what you can do with stream deck, really, because it sort of gives you an overview of how you set things up, how you add keys, how you add the hotkeys and things like that, and so that is what we're going to be going through. But if I can just give you a quick demonstration of exactly what this is doing, so here I've brought up on the screen the Mac application, the calculator application. You could equally do this with a Windows calculator as well, but as you'll see, as I type into the stream deck, one second, how can I activate the calculator? If as I type into the stream deck, then it is inputting what I'm typing into the stream deck on the calculator. There you go. And then the other things that I've added in here, so we've got all of the functions that you've got on the standard calculator, but then there's a couple of others such as copy and paste. So some people don't realize that if you've just typed a calculation into the calculator, then just pressing copy, you don't have to highlight anything, we'll copy that text. So if you do need to copy it into an application, it's quite handy to have it open there. Right, so what I'll do now is I'll show you exactly how to build this out on your stream deck. So if I go over to my screen sharing, I've got the stream deck application open, which is just a little menu bar item up at the top here. So you click on that to open the stream deck, and then you can see this is basically replicating the layout that's on the stream deck, and here is the layout of all of those keys. Now, what you could do is you could set this up as a profile, at which is whereby in stream deck, you can have a number of profiles, and then you can have it so that when you open a specific application, it will load that profile. So if I created a profile here called calculator and built this out in that, and then I could just click on here and select the calculator, and what that would do is every time you open the calculator, stream deck would switch over to that icon set. Personally, I don't actually use profiles. I prefer to have things in nested folders, and the reason for that is that I've got a lot of apps that I've created shortcuts for and stream deck icons for, but I'm often using these applications at the same time as each other, and sometimes I don't want it to flick from one set to another just when I hop between applications. An example of this would be, I spend a lot of time on Teams meetings, Zoom meetings, things like that, and we're often discussing things that may be things in Excel, for example, and so I have got my Excel power user icon set for stream deck, so I'll put a link in the top corner when I get round to actually making the video on that one, but what I wouldn't want to happen is if I'm on a Zoom call for it to switch over to the Excel icon set when I'm in Zoom, and so I prefer to just have these things organized by nested folders. It's sort of maximum of, say, three folders deep, but it just means I can hop around between different applications and get in and out of them quite quickly without actually worrying about the profile. So there may be other viewpoints and ways to do this, but this is just the way that I feel comfortable doing it. So for now, what I'll do is I'll just, for the purposes of demonstration, I'll just create another folder in here to give us a blank screen, and so here we go. We've got a sort of blank screen to work with, and what we'll do now is we'll just actually build out that calculator screen. With stream deck down the right-hand side, you've got a number of different sort of groupings of the actions you can add. So we've got some system actions, some specific to stream deck, and then there's other applications as well. There are also a number of other applications that you can add to this. So just down at the bottom, actually just a little bit off the screen, there's a little button that says more actions, and you can click on that. And what that will do is bring up a whole list of other apps and things like that that you can add to your stream deck, and that will just import those different shortcuts and things like that into your stream deck. But for now, we'll just work with the ones we've got. So for this setup, all we actually need is one of these system actions, which is the hotkey one. And so we're basically just drag this into here. We can keep dragging more of these because we just want to build out the set of icons, the set of buttons. So in fact, actually, if I press copy, then I can just actually paste this because they're all the same. Well, there we go. By the magic of technology, I've saved you sitting through all of that pasting of buttons there. And now you can see what we've got is we've got a set of buttons now, which sort of matches what this one is going to look like. So I'll just talk you through now exactly how you go through and add in all of those as well. So this is basically where the number pad is gonna be. So all of the actions that I added in were hotkeys. So basically for you to add in a keystroke. And it's quite simple. We're just gonna add in the keys zero to nine here. So assign hotkey if I click on the zero, oh, sorry, the seven for there on the number pad. So instead number pad number seven, but you can see what we've got here is we've got it's writing that title over the top of it. Well, what we want to do is have that pretty little picture of the number seven. Well, what I've done is I've created the icon pack and I've actually made that available. So you can get that on my website for a couple of dollars. And it's basically just two sets of icons. We've got the numbers in white and in black. I'll just use the black ones for now so they show up a bit better. In fact, actually, I'll use the white ones since the other ones are black. So here you go. What you do is you just go down to the image. In fact, let me just make this a set of, as a list, it might be easier. Now that you've seen what they look like, this is the set of icons. So here we've got the number pad number seven and all we're gonna do is just click in this little key here and then we're gonna go and drag the seven image or the key image and we'll drop that in there. But now we've still got this text over the top of it. So what we need to do is just drop down here where the title is and then you can just highlight this show title. It's obviously what this is because it's got a big number on it but when I'm creating other icons and icon sets in the Stream Deck application, I always do leave a title but I just make sure to hide it rather than just delete the title out of it because then that means when you're looking, if it's not quite so obvious as a big bold number seven then you can always see what it is. So there we go. We've got the number seven. So what I'm gonna do now is just duplicate that for the rest of the keys. So there we go, that's the number pad built out. If I just pop the calculator up again, just in case you've forgotten what the calculator looks like, there's still a few more that I'm going to add and so the next one would be here. I'm gonna add the period and I'll just grab that one from here and drop that over on there as well. Next thing I need to do is add the divide, multiply and so on. So I'll just go and add those in. So for the divide, then that would be the slash key. So the backslash and then for the multiply, that would be the asterisk key. So we'll come in here and click asterisk. Whoops, wrong one. Asterisk. For the minus, we're going to have obviously just the minus key and for plus, we have a plus key. I'm just using the number pad, but obviously you could use the keys along the top, row of numbers as well. And then here we've got equals, so I'll just add that one in. And then now what we want to do is add the icons for all of these. So again, just let's go through and take out the titles and then I'll add in the keys for these ones. Okay, there we go. So that's nearly it for the main keys that you've already got on the calculator. There's another one that's worth having on there and that is the backspace because some people don't realize you can actually just press the backspace and it will delete what you're typing. So if you can see the calculator down here, if I just, in fact, let me just set this up first. If I add in a keystroke and that's just the backspace key. If I just type something into the calculator and then we type in something in here and then if I press that backspace key, then it just deletes one character at a time. So that's always a useful one to have in there. The next one that is actually on the calculator but isn't necessarily obvious for how to clear is the obviously the C button and that is C or AC and that is just simply pressing escape. So for this one, we click to assign keystroke and click the escape key. So that little symbol down there is the symbol for escape. So if I just drag this one down I'll put it onto there and then remove the title again. So you can see we've nearly built it out now. Then next we'll add in our percentage. So again, keystroke, shift and on my keyboard it's five. And then the last two that I've added in is just simply copy and paste. So click to assign shortcut. So command C for that one and command V for this one. So I'll go and drag the copy icon across here. Whoops, I've dragged it to the wrong place. I'll drag it to here, that might help. So that's copy. So the last one I've got up here is actually a shortcut to the app itself. Now I've put keystroke there but that is not quite what that should be. So let me just delete that for one moment and I'll add in it's this one, this open. So this opens an app. So what we'll do here is we'll just go to find the calculator. So that just brings up the typical Mac dialog where you can select the app and I'll just pop that in there. So there you go, it's brought up the calculator app. Now it has actually bought in the icon for the app. So when you do add in a shortcut for an app it does automatically bring in the icon. So let's just hide the show title there. Let's take that off. But if you want, I've also included the same app basically but it's just a slightly bigger version. So you'll see it takes up the full key by adding that in there as well. Incidentally, the reason why you would have that open key just there for the calculator is two-fold really. First of all, obviously, if you don't have the calculator open then when you click on that it will just open the application for you. But also if you are in another application and you're wanting to use this calculator then you will need to make sure that you actually switch over to the calculator app first because otherwise these inputs of Stream Deck is basically just giving you keystrokes. So unless you've got something with a bit deeper integration like the ECAM ones for example where even if you're in another program then if you click on the ECAM sort of change scene button or select a scene as long as ECAM's live is open it will perform that action. Whereas if you're using these keystroke ones in this list here then it's basically just clicking that keystroke. So if I happen to be in a different application like Excel or Pages or whatever it happened to be if I were to press this number one button on the Stream Deck here it would just input the number one in whatever application I was in. So you can also use the open function in Stream Deck as a kind of like app switcher. So you can switch between apps. So this would have the effect even if the calculator is open but you were in another application if you click on this open calculator button it just switches back to the calculator and then you can use the buttons on the Stream Deck. So there we go. We've got that calculator now looking pretty much the same as the one that you saw there except this is the black version and this is the white version. The black ones do sort of show up a bit better on the Stream Deck I think but the white are just more sort of the same as the actual buttons on the calculator app itself. So I can just show you that what that looks like on the Stream Deck now actually. So here we've got the original one that I showed you in black and if I switch over to the white one it's a bit hard to see actually on the... There we go, that's a bit better if it's just not pointing to the light. It is quite clear with the white version but I think definitely the black version is certainly more crisp and clear. But anyway, both of those icon packs are available. If I go back to my e-cam live I can switch back over to my desktop. So both of those icon packs are available on the website. So I'll put a link to that in the description but it's quite easy to remember it's just to take one tech.io. And so if you go over there and click on the store then you'll see that the icon pack is available there for download. If you have found this video useful then please don't forget to give it a like and a subscribe down below and switch on notifications so that you'll be notified whenever I post any new videos. As I've said, I'm gonna be doing a whole series on Stream Deck and how I use it for productivity. If you've got any questions about how I'm using the Stream Deck then obviously please feel free to go and leave a comment about that as well. I'll be happy to answer any questions or if you've got any ideas for any other use cases that you'd like to see reviews on then I'm always happy to look at other ideas as well. That's about all for me today. So I hope you found that useful. Don't forget to check out some of the other videos that I've linked to just over here and also don't forget to subscribe to the channel. Have a great day everyone.


This is Saturday, but I got 14,000 views on my YouTube channel yesterday, and I am kind of freaking out. I've never gotten more than 8,000 in one of my videos, and it was the first time I posted a YouTube short, so I'm like flabbergasted, like, oh my god, what is happening? This is really great, and like, kind of motivating, even though my regular long form videos aren't doing so good.  I packed these for Snack Asteroid, and I didn't eat them, so I'm going to eat them now. Just half an apple, sliced up with some cinnamon on top. How messy my room is, but this is my little outfit for SoulCycle, and I'm running a little bit late, so let's go.  How, how does that happen? Guys, it's so cold, my nose is literally turning red, but I've never been out this early on a Saturday. It's so weird, because the streets are like empty. What's happening? Am I in New York? Just finished SoulCycle. It was so much fun. Amanda's going to give me a ride. She's my sugar mama. I'll catch up with you guys. Make a smoothie, something to eat, and yeah, I'm trying to think what am I going to do for the rest of the day. I really have no idea what I'm going to do. I need to wash my hair. My hair, I'm getting like a lot of length retention, and it's really growing. I'm just going to breed it up now. I need to step by Sephora, because I'm out of this, and this is like my absolute favorite. Like, that I like to put, and honestly, I just like put it on the ends, and then I'm going to put it on the top of my head. I'm going to do this, and this is like my absolute favorite.  Weird enough to do what I need to do. The second one is always so much better than the first, so I'm going to do this one over again. I'm matching hair ties, so we're black, blue, white. I am dressed and ready for the day. Even though honestly, after SoulCycle, and then hanging out with my friend, washing, blow-drying, and styling my hair, all I want to do is sleep.   I am going to take a little bit of a nap before my hair appointment. It is 3.36. I think I just got stuck in the elevator. I'm in the elevator. Alright, thank you. I'm stuck in the elevator all by myself. Isn't that great? Please send help. I'm free. This elevator ain't stopping me.  This elevator ain't stopping me. Nah, yeah, yeah. Deadass. I'm literally so tired after being stuck in that elevator for an hour. Oh god, this lighting is terrible. After being stuck in the elevator for an hour and then running on my errands, I got home at 8 o'clock and I was supposed to get my nails done at 8 o'clock. I ended up canceling my appointment because I was just like, I'm beat, y'all. I'm beat. The devil has won today. I'm going to change and go to bed. Not the exact date, but I'll probably put it on the screen. I am just so tired and exhausted yesterday. I just have been in a bit of a funk, you know? I think being locked in the elevator for an hour and a half just put me in a bad mood. I wasn't able to get my nails done because I just was so tired as you guys saw last night. I'm waiting for my nail tech to give me like an alternative appointment time, but I still haven't heard from her. So I might go down to just like my cheaper corner place because one, be cheaper and two, take walk-ins. So I can just walk in there and get my nails done. So that's something I might do today.  That always makes me feel good. So I'll probably do a little laundry and then some reading this morning because I'm pretty behind on my school stuff. And then I might be ending the vlog today just because my performance really dirty and I kind of want to film a queen with me. But if you're still here, keep watching. We will figure our lives out. I just heard back from my nail tech and she's booked until next Saturday. So we're going to my corner store salon that always has appointments. The one thing is, I'm just not sure if they can take off the like a pre-extension. So I might have to do that myself. Damn. We are so Gucci. Okay. Let's go to the salon. So I'm bringing a snack with me. My small water bottle. I also feel like I've been dehydrated lately. So I think that's also where I'm kind of feeling crappy. My yoga notebook just in case if I have to go straight to my class after getting my nails done and then in my Columbia folder I have two readings I need to get through for Tuesday. I figured I'd probably be reading while I wait. I literally look like a hoo-ha from Google. I don't know.  But whatever. It's 52 degrees but the sun is shining. So it feels a lot warmer. So I'm just going to wear this jean jacket. Add everything in my little tote bag. And we're ready to go. Let's get our nails done. Bye baby. Bye. Mommy just want to step on the carpet to fix it right here. She already has her shoes on. But I love you. I'll see you soon. Okay. And let me close this so he doesn't like fall out the window either. Okay. So I just finished up the nail salon. It's literally two o'clock. Y'all, I got there at 10.20 this morning. How crazy is that? But I am obsessed with my nails. If you follow me on Instagram, I hope you voted for the winner. I do a poll just about every month to see like what nails I'm going to get. So if you like nails like me, make sure you follow me on Instagram. I'm at my yoga studio and I'm just going to end the vlog here. Let's get on a high note. I'm really happy about my nails. And they look so cute. And I'm honestly just going to swap. Okay. I was looking at it in the vlog, but I'm not because I just going to teach you my class. And she bought me a salad from Sweet Green. One of my yoga teachers came and took my class and bought me a salad. How sweet is that? So now we're ending for real. Thank you so much for watching. Like, comment, subscribe, and I'll see you guys next time.


War is never going to be fought in a perfect scenario. The warlords do not miss. By hitting that at which you aim, you have a true capability. And that's something that we live by here in Tutu. Is that we are meticulous about the details. We know why we hit. We make every single round count. Every one. We'll hit with whatever we shoot. Tutu put a heavy emphasis on training incidental missile. And what we call that is a javelin team in every rifle squad. With 54 incidental missile trained in the battalion, we now have a javelin team in every rifle squad who is capable of eliminating tanks at a javelin's max effective range. The training we conducted out here improved our ability to be ready to go anywhere at a moment's notice. As well as making old tactics of light infantry new again. We're ready to deploy anywhere. And when we employ our weapon systems, we hit.


Weeds. Everyone has them, probably always will. So what's all the fuss about weed control? Well, in a word, money. Estimates show that the weeds in Utah alone cost citizens and businesses somewhere around 40 million dollars a year. Sound hard to believe? Well, let's take a look at how weeds cost us so much money. To begin with, some weeds are poisonous.  Other weeds cause a wide variety of allergic reactions. And the cost adds up, not only for medication and doctor visits, but also for lost time at work. Poisonous weeds also cause expense in the livestock industry. In fact, all animals that graze are threatened by these weeds. In 1988, 70 horses died in Utah as a result of eating poisonous plants. So it's easy to see how poisonous weeds cost us money, but that's still just the beginning. In fact, non-poisonous weeds are an even bigger problem. That's mainly because there are simply more of them. And the problem with all weeds is that they keep other plants from growing. The fact is, there is only so much nutrient in the soil, so much sunlight, so much water, and so much space. The more weeds there are, the hotter it is to raise any other plant. So for a farmer, weed control is an absolute necessity. Left uncontrolled, a weed like Johnson grass can completely take over a cornfield. Okay, so poisonous weeds drive up the cost of livestock products, cost our citizens thousands of dollars in medical bills, cost our businesses thousands of dollars in lost production, and non-poisonous weeds cost farmers and citizens thousands because of weed control and higher prices. And there's one more cost associated with weeds, a cost that you can't put a price tag on. Safety. Some weeds can grow quite tall and can block out vision on our highways. When people can't see at intersections, they're obviously in great danger. That's one good reason why weed control is a big concern for the maintenance division. We have to take responsibility for the enormous amount of land in our right-of-way. Land that in many places is infested with both poisonous and non-poisonous weeds. Beyond that, if we don't control weeds in our right-of-way, it's only going to be a short time until the weeds spread to all the neighboring property. Of course, we have selfish reasons to control weeds, too. Anyone who's worked in maintenance for even a short time knows how important drainage is. Thriving weeds can block our drainage ditches and channels, and the buildup of water will eventually undermine our roadways. Well, what about all the mowing we do? Doesn't that take care of the weed problems? No, unfortunately, it doesn't. Just as with grass, mowing only cuts weeds. It doesn't kill them. And if mowing is done at the wrong time, it can actually make the problem worse by spreading seeds, which of course leads to even more weeds. The only solution then is to treat plants with chemicals so that the entire weed is killed. But this is not as simple a solution as it might sound. In order for a chemical control program to be effective, you have to identify the exact types of weeds to be treated, the time of the year to treat the weed, the type of chemical to use, and the rate to apply it. But before we look at each of these, there's a surprising fact you should know about weed control. Weeds are easiest to kill when they're most healthy. That's because weeds are like chemical factories. When they're growing vigorously, they draw nutrients and moisture from the soil and pump it, so to speak, to every location in the plant. The roots, leaves, stalks, everywhere. When we apply chemicals to a healthy plant, the plant pumps the chemicals and moisture from the soil to all its parts, consequently killing the entire weed. On the other hand, weeds that are not healthy are nearly impossible to kill with chemicals. That's because the factory has essentially shut down. So remember that. If there's no moisture in the soil, the plant isn't working, and spraying chemicals during a drought situation is just a waste of time and money. With that in mind, let's go back and look at the factors for a good weed control program, beginning with weed identification. The first thing you should do is get a copy of weeds and poisonous plants of Wyoming and Utah. This is an excellent resource to help you identify the weeds in your area. As you can see, there are pictures of each type of weed at its various stages of growth. So you can identify the weeds in your area almost any time of the year. Once you've determined the types of weeds and their locations, it's a good idea to plot them on a map. By noting the mild post marker or any other item on the feature inventory such as guardrail or drainage pipe, you can easily keep track of the weeds you have to control. You should also be sure to get a copy of the Utah weed control handbook. In it, you find the type of chemical to use for each type of weed, the rate to apply the chemical, and the best stage of growth to spray the plant. To further help you, you may want to write that information on your map, so you don't have to look it up every time. And that brings us to the next factor, the time of year to treat the weed. With chemical control, timing is everything. As a general rule, the earlier in the plant's life you can spray it, the better chance you'll have of killing it. Now, as we've already seen, the handbook will tell you the best stage of growth to treat the plant. But it doesn't tell you whether that will be in April, May, June, or what have you. That's because the times the plants reach their various growing stages are different all over the state. So it's up to you to determine the time of year to spray. And that's important, because the best time period to treat some weeds is very short. For example, the handbook tells you to treat dire's woad in the rosette stage. And in Salt Lake County, that's about a two week period. So as you can see, weed control isn't something you get to when you can. If you don't treat weeds at the right stage of growth, you're wasting the chemical and your time. Along with seasonal timing, you should also be aware that the time of day can make a big difference. As I said earlier, if the plant isn't drawing moisture, it's not going to draw the chemical either. Okay, we've looked at weed identification and timing. Now let's look at the type of chemical to use. Here again, the handbook is your best reference. It not only tells you the best type of chemical to use for a specific weed, but also how much to use. But selecting the right type of chemical is only the first step. Before going any further, always read the label on the container. Safety has to be your first concern, both for yourself and the property surrounding the areas you'll be spraying. The label will clearly indicate whether you should wear safety glasses, gloves, an aspirator, an apron, or all of these things. Whatever the label suggests in terms of safety gear, wear it. There's just no sense taking chances with potentially dangerous chemicals. You should also check the label to see if there is any problem with compatibility. Some chemicals simply cannot be used with other chemicals. Now as I said, you have to be concerned about the surrounding property. The chemicals you're using will not only kill weeds in the right of way, but also plants on private property. And careful spraying won't solve the problem all by itself. That's because a lot of chemicals drift. So the first rule is, don't spray when the wind is stronger than four miles per hour. It's just too great a danger of the wind carrying the chemical where it could harm other plants. The second rule is to use low drift whenever the label recommends it. As you would expect from its name, the low drift agent will keep the chemical from being carried in the wind. And that brings us to the rate of application. To put it simply, the correct application rate is just that. Correct. Never apply more or less than the rate shown in the handbook.  And applying too much is just a waste. Dead is dead. You should also know that applying too much or the wrong type of chemical can lead to bigger problems. Here's an example where the wrong use of chemicals has left the slope bare. It certainly doesn't look good, but the real problem is that there's nothing left to stop erosion. So again, always use the right type and the right amount. Of course, you have to know how to use the spraying equipment properly too. There's another course covering the operation of our weed sprayers, but there's one point worth repeating here. Always calibrate the sprayer whenever you change chemicals. Basically, all you have to do is collect the discharge for one minute, measure the amount you collected, and then do some calculations to be sure that the equipment will spray the amount you want. And that brings us to the end of this program on weed control. As you've seen, there are a lot of good reasons for all the fuss, but with your help, we can drastically reduce the amount of money weeds cost us each year.


Welcome to module 31 of NPTEL point set topology part 2 course. So, having studied the some general properties of filters, today we shall study convergence properties of filters. For that, we start with a topological space not just a set. Remember that filter can be talked on any set whereas, now we have a topology x comma tau be a topological space and f be a filter on x. A point of x is called a limit of f or you can say it is a limit point no problem limit of f that is with respect to the topology tau. So, limit will not be spoken unless you have it above that is important. If the neighborhood system the entire neighborhood system n x the neighborhood set of all neighborhoods at the point x that itself is contained in the filter. In this case we also say that f converges to x. Similarly, we define x to be a cluster point of f if every member of n x intersects every member of f. So, here you can see that this is a weaker condition than that one because if n x is contained inside f then this properties also prove because f itself has that property finite intersection of finite limit numbers is nonemptive. So, that is why so that is a weaker condition this every member intersects every member of f. Indeed all these are copied from what happens for the case of when you take a sequence take the corresponding limit of a sequence or a cluster point of sequence and look at what is happening in the associated filter. So, that is the motivation for these definitions. Some observations which are immediate from the definition are the following.  The larger filter so called larger in this sense also converges to x. So, all the points can limit limits of f are limits of f prime also. Conversely, if f prime converges to x then you cannot say that f converges to x, but f will be x will be a cluster point of f. So, that is a weaker corner system. For this reason the bigger filter f prime is called the sub filter. So, this is some terminology which you have to a strange terminology which you have to live with because this is borrowed from what happens to sequence and subsequence. If you have a sequence which converges then every subsequent will also converge to the same point. If a subsequence converges then the point of convergence of subsequent sequence will be only a cluster point of the original sequence. So, that is what is happening here. Secondly, every limit point is a cluster point that I have already told you because cluster point is a weaker condition. But the converse is not true just like in the case of sequences. So, we shall not bother to discuss it any further. Limit point is a cluster point, cluster point need not be limit point. The third observation is let x be a limit of a filter. If a is any member of f then x will be in the closure of that member x belongs to a bar. So, this is very easy to see, but I have added explanation here. If not what happens? If x is not in a bar will mean x is inside a bar complement, a bar is closed so a bar complement is open that means that a bar complement is in nx but nx is inside f. So, a bar complement is in f but a and a bar complement, a complement bar, a bar complement either way they are disjoint. So, therefore, that will be a contradiction to the property finiteness property of f. So, that cannot happen. Therefore, if something is a limit then it is easy to check that for each member it must be in the closure of that member for each member of the filter. There is one more important remark. In fact, it has three parts here and we will use it. So, pay attention to this one. Now, I am looking at a net which is a generalization of a sequence. Start with a s from d to x that be a net in a topological space s, x is topological space s and let us take fs to be the associated filter. Then the first thing you can check is that x belongs to x is a limit point of f. The task take any point that is the limit point of this s in the sense of when the net is a for a net. If we do not leave x is a limit point of the corresponding filter. The second thing is a cluster point of f similarly, if we do not leave that point is a cluster point of fs. So, this is precisely what I have already told you that both the definition of limit point and cluster point have been copied from what happens to the associated filter. The third one is very important here. The association s to fs has the naturality property. For any function f from x to y, we have f check of fs is equal to f of f you know curly f of f composite s.   So, 1 and 2 are easy to verify. The third one is actually a general property has nothing to do with the net. So, I should have done it earlier. In any case, let us check that this is also true. So, I will take some time to check that f check of fs is equal to f of f composite s. So, here is the proof. Start with any subset b of y function f is from x to y. So, things are happening inside y. So, take a subset of y that subset b is in f check of fs if and only if this is by definition of what is f check of fs. If and only if there exists a belonging to fs such that f of a is contained inside a b. So, that is the definition how to generate f check of fs. It is contains all the supersets of all the images of fs under the map f. That is the definition. So, b is inside f check of fs if and only if there exist a such that which is inside fs such that f of a is inside b. Now, this is same thing as there exist d belonging to d such that sd the what you call the right ray you can say sd is contained inside a. So, this part is same thing as a belonging to fs. Second part is same thing.fa is contained inside b.  So, that is same thing as saying there exist a this such that f composite s of d which is nothing but f of i which it is containing that bit. But now this is a condition for f composite s. So, this just means that V is inside the SOC active filter of F composite S. So, that is all. So, it is purely point set topological point set thing here that is no topology in this one, but this will be useful. So, I so this is important and this this the map S to FS is going to be very important. It is a one way bridge from Nets to filter. The filters have much more generality you cannot always come back to the Nets here. So, now let us come back to study of limits and cluster points. A point in X is a cluster point of a filter F. If it only if there is a sub filter F prime containing F, I will read F prime but I will read it as sub filter F prime containing F of which it is a limit. It is a limit for every sub filter this may not happen, but if it is a limit this will be sorry it happens that for every sub filter also it is a cluster point. So, now we are putting a I am actually giving you a converse of that one also. So, if part we have already seen in the earlier remark to see the only if part suppose X is a cluster point of F. This just means that NX the neighborhood of system union F this has finite intersection property. See within NX finite intersection property between neighbor systems within F it has finite intersection any member here will intersect any member here the condition for being a cluster point. So, NX union F will also have finite intersection property. Hence we can take F prime to be the filter generated by this family. Any family with finite intersection property will generate a filter namely you now take all the sets which are supersets of some members of that that will be our F prime. I have to produce one F prime there exists some filter there may be many more no problem. The next theorem is about function take two topological spaces and a map a from X to Y a function actually sorry just a function from X to Y. Let X belong to X be any point then the following conditions are equivalent. So, I told you it is a continuity of functions now giving characterization in terms of filters. So, start with any function take a point in the domain F is continuous at that point in the first condition. Second condition is for every filter F on X which converges to X F check of F converges to FX. So, this is a statement. The third statement is for every net S from D to X S converges to X implies F composite S converges to FX. So, these are three statements this statement I would have proved when doing NETS and so on. But I deliberately postponed it because these things can be done much easier once you go along with the filter source. So, the three conditions are equivalent is the statement of the theorem. So, we will as usual we will 1 implies 2 implies 3 implies 1 that is what our plan. Assume that F is continuous at X now take a filter F which converges to X this is the same thing as say N X is contained inside F. Let now we belong to N of FX I have to show that we belongs to F check of F that is what I have to show. Start with a neighborhood of FX by continuity of F it follows that there is an open substitute a neighborhood of X such that F of U is contained inside V that is a continuity. But now U is in N X but N X is inside F. Therefore, F of U will be inside F of first of all all members of F therefore it is in F check of F. So, what we have shown is that F U itself is in F check everything above everything above bigger than that will be also in F check because F check is a filter. So, V is in F check. So, we have shown that the neighborhood system at F X is inside F check of F. So, that just means that F X F check of F converges to F F X is a limit. So, 1 implies 2 is over. Let us do 2 implies 3. For this we use remark 7.41 both 1 and 3. So, let me just show you 7.41 what it was just now we have done that but let me show you that one namely 1 and 3. The association here is a limit point of if and only if X is a limit point of F S. So, this is the statement for from a net to a filter. A limit point of S if and only if it is a limit point of F S that is the first one. The third one is association F check of S equal to this one. So, both of them will be used what I am going to say what I just told you that is all. So, let me go back to the proof of this thing 2 implies 3. Start with a net S from X D to X. Let S converge to this point X. Then we know that the filter F S converges to X by 2 this implies F check by 3 actually F check of F S converges to F X Y 2 that is fine. F check of F S converges to F S. So, by this remark that we have seen F check of F S is nothing but F of no the filter associated to F of F S composite S. Therefore, again by this remark we get F composite S converges to F. F X is a limit point limit of this filter the same thing as that F composite S the function converges to F. So, that is the 2 implies 3. Now, let us go to 3 implies 1. So, here we prove it by contradiction namely if F is not continuous X then we will construct a net now which does not converge to F X, but it converges to X. So, that is the way we are going to prove 3 implies 1. Start with assumption that F is not continuous at X. This just means that there is one neighborhood V of F X such that no neighborhood of U no neighborhood of X F of U will be contained inside V which is same thing as saying that F U intersection Y minus V is not at it. So, it is not contained inside V means Y minus V intersects F V. So, this happens for every neighborhood U of N X every neighborhood U of X. Now, I choose my directed set D to be N X itself with its usual inclusion. Reversed inclusion you have to remember that. And S from D to X I am going to define the net now is such that each SU is chosen from this non-empty subset U intersection F inverse of Y minus V.  So, check that point. So, by the example of choice there is such a function. So, that function is going to be our net.  To show that S converges to X we have to show that S is eventually in every neighborhood of X. So, if W is any neighborhood choose W itself after that means what W less than or W prime which is same thing as W prime is contained inside W. S of W prime is inside W prime by the choice but W prime is inside W. So, this means eventually this entire net is inside W. Thus we have shown that S is eventually in every neighborhood of X this means that S is converging to X. On the other hand it is very clear that F composite S as a net what is its property S composite S of U is inside Y minus V for all U V is fixed. So, V is a neighborhood of F X some after something you should have been inside eventually inside this one no but it is not possible because all the time for all U its SU F of SU F S of U is inside Y minus V is outside. So, this means that F of F composite S is not converged to F X. So, these were some fundamental properties. Now, we shall prove some useful corollaries to these things about products and convergence properties for products. Take a family X, Xj of topological spaces and take X to be the product. Xj is a topological space is F is a filter on the product. Take a point X inside the product the filter converges to this point if it only if look at all the projection maps look at the corresponding the filter the image filter pi j check of F. You take pi j of F and then take the filter generated by that that is pi j check of F. So, pi j of X is a point of Xj, pi j check of F is a filter on Xj what we want is for every j, pi j check is a limit of pi j check of F. So, this is if and only if one way we have already seen namely because pi j's are continuous functions general statement we have proved if F converges to X then F composite F check of F converges to F F that we have already seen. So, we have to see only the converse part. So, only if part we have seen now we have to see only the if part. Let us now suppose that pi j check of F converges to pi j of X for all j what does this mean the neighborhood system at pi j of X this is inside the component Xj the topology space Xj that is contained in the pi j check of F this filter this should happen for every j. But what we want to show is the neighborhood system Nx in the product space of X that is contained inside F. For this it is enough to show that the sub base for Nx you have to remember what is the topology? Topology is product topology the product topology neighborhoods at a point are generated by a sub base pi j inverse of Uj where Uj range is over all the neighborhoods of Xj Xj is simply the pi j of F. So, Uj's are in N of pi j of X you take pi j inverses take finitely many of them as j ranges over different indices here take the intersection. So, that will be those will be the basic neighborhood. So, this is a sub base if this sub base is inside F finitely many intersections will be there and anything bigger than that will be also there. So, whole Nx will be there. So, that is why this is enough to show that for each j if you can fix j here this is all j's are taken if that is contained inside the if it only for each fixed j it is contained. So, I have to show that pi j inverse of U belongs to F if U belongs to N of pi j of X. So, that is very easy for each Uj inside the neighborhood the 26 whatever we have we have put this one what is this one pi N of this one is contained inside this one. So, what is the definition of this pi j check of F? So, that is what you have to show when is a member here is here that is what I have to interpret. So, that implies that this Uj is inside pi j check of F if and only if there are finitely many members of F such that their intersection pi j of F i intersection F 1, F 2, F k then to take pi j's of that take the intersection that is contained inside Uj because these things are first of all there in the generating thing sub base to base and then every super steps must be there this is the condition for members of pi j check of F. So, if such a thing is there then intersection of F i's pi j of that is contained in this one. So, intersection of F i's contained in the pi j inverse of I am applying pi j inverse on both sides here I can just take intersection which is pi j of this is contained in the intersection of pi j's that is not equality need not be equality this is contained inside this one pi j of that one pi j of that one this whole thing is contained in the pi j inverse of therefore, pi j inverse of Uj is inside. So, next theorem is a topological space is hostile if filled only no filter F converges to two distinct points of F. Remember when we defined a limit of a filter we did not say that the limit or something there may be many limit points in general, but you take a hostile space it can have at most one limit point that is not the point the point is if filled only if. So, we get a we get a characterization of hostile spaces maybe this is the fifth or sixth characterization remember that we had several characterizations of hostileness hostileness being very important. So, this will keep coming again and again. So, a topological space X is hostile if filled only no filter F converges to two distinct points inside X. So, let us prove this one suppose X is not equal to Y and they are points of X and F converges to both X and Y then what happen n X is inside F and Y is inside also F.  So, this is all darkness both U and V are inside F. So, that is cannot happen because then empty set will be inside F or you can just say U does not intersect V that cannot happen that far. So, converse suppose X is not hostile then I will produce a produce a filter which will have two distinct points which are limits. So, not hostile just means that there are points X and Y not equal to each other such that every member of n X intersects every member of n Y. If there is one member here one member which does not intersect will be hostile not hostile means precisely this one. This just means that n X union n Y has finite intersection property as soon as something has finite intersection property we know that it is contained in a filter namely the one is generated by this filter generated by this family. So, that filter will contain we will have both X and Y as limits because both n X and n Y are there that is all. Finally, we have this big theorem here like X be any topological space then the following conditions are equivalent like host darkness has many characterization we want to characterize compactness also X is compact is the first condition every filter on X has a cluster point every filter has a convergent sub filter that is for filters again I have promised you to do similar things for nets also. So, two different characterizations here similar one every net in X has a cluster point every net in X has a convergent subnet. So, A, B, C, D, E are equivalent namely these are so four more characterizations of compactness in X what we have already proved is B implies and implied by C. Look at here every filter has a convergent sub filter. Convergent sub filter means it is a cluster point and conversely this part we have seen similarly net also this part also we have seen. So, B implies and implied by C D implies and implied by D these much we have seen. So, we shall now prove A implies B implies D implies A automatically C and E will be taken care.  So, that is the contra positive of A implies B.  If U X meets every member meets every member then you are done. So, there is one member here and one member here we do not meet that is the negation of that F has no cluster points for every X it must happen. For some neighborhood this happens then that will be cluster point for every one member U X here and one member here U X is a neighborhood of X and V X is a member of F they are they do not intersect they are not meet.  So, that is an open covering when you range X over X. So, take a finite sub cover here that is what I would take but then you look at intersection of all V X size corresponding to each X I here your V X size here finitely many of them intersection of V X size what is it I see that is empty because union of V X size is the whole of X. If there is a no if there is a member here which will be inside each V X size it will be in one of the V V correspond U X size here each V X I U X are disjoint. So, there is there cannot be intersection of V X size empty which is absurd because these are finitely many members of a filter. So, that is not okay. It is not very hard to prove it directly also you can try to prove it instead of you know reduction and absurd instead of assuming F is there is such a filter then you prove that X is not compact. So, I have used a compactness and if as we say has filter which is no crucial point then I have arrived at a contradiction. So, B implies D now. So, I have to go to net now starting with a net S consider the corresponding filter F S by B this filter has a cluster point. But this is same thing as S as a cluster point as union 7.41 which we have just done now. So, B implies D is all day almost done for us. Finally, I have to show that D implies A. So, this we have to work a little harder. So, let us so please pay attention. So, this way I am saving time for instead of proving B implies A and then D implies A, D implies A, B implies D is very easily done. So, D implies A I have to start with a family of closed subsets of X with finite intersection property. We have to show that the entire intersection of members of C all of them taken together that is nonempty that is equivalent statement for compactness. So, this is what we have to show. As in the previous example, now consider D the family of all intersections of finitely many members of C which is directed by the reverse inclusion. Where is this one? Get this one the family of all intersections of finitely many members of C. In particular, it will contain this C also C 1, C 2 are members of C, C 1 intersection C 2. C 1 intersection C 2. In other words, you saturate this C put by putting all finite intersections. So, take all finite intersections of these members. They are all closed subsets by the way and you take the reverse inclusion. So, that is the partial order there now. Let me case that we have done several times like in 7.5 and so on. So, that is the directed set curly D for us. So, what is the relation? D is less than or equal to E if and only if D contains inside it D and E are subsets of set of this topological space X and there are members of this now members of this family. Since C is contained inside curly D and I want to show the intersection of members of C is nonempty, I am going to do something more intersection of all the members of D is nonempty because this is a larger intersection. So, intersection set will be smaller. Intersection is taken on a larger family. So, intersection will be actually smaller. If this is nonempty, if this itself is nonempty, then that will be also nonempty. So, intersection of D such that D is inside curly D, this is nonempty is what I am going to show. Now, look at each D inside D, they are nonempty by definition because I have taken finite intersections of members of C which has finite intersections of it. So, they are nonempty. So, we can take the net to be from X D to X to be such that S D is inside D. So, once again here we are using axiom of choice. So, such a function is there. So, let that be our S now. Suppose X is a limit, a cluster point of S, suppose it is a cluster point of S. We claim that X is inside D for all D inside capital, inside this curly D. This we have actually seen, I will repeat it because you might have forgotten how it came. Suppose it is not true, suppose X is not inside some member, D belongs to D, then X minus D is a neighborhood of X because each member in the curly D is a closed subset, that is the point. It is a neighborhood of X. Hence, by the definition of a cluster point, there exists an E belonging to this D, curly D, such that for all members which are following A, following this D, that means E is contained inside. Remember, this is E contained inside D. All the S E's will be inside X minus D. X minus D is a neighborhood and the X is a cluster point. So, it should have. But then this implies D is inside S and would imply D contains E and S E is inside E all the time. So, if S E is inside E, S E is inside D, X minus D is contained as X minus D. Therefore, S of E is inside E as well as inside S of E, X minus E also, that is observed. How can it be in X minus A as well as S E? X minus D was a neighborhood who contained X minus E, right? Whenever D is contained inside. So, these two are contradictory. I have started S E inside E but S E is now X minus E, that is it. So, for this net which is cleverly chosen, this will fail. So, what we have proved is that X belongs to D for all D inside D. So, intersection of this one is not, all right? So, that completes today's plan, Model 31. Thank you. We will meet again next time.