So today, I'm going to talk about building privacy-protecting infrastructure, what it is, why we need it, and how we can build it. We look at VACU, which is a communication layer for web free, and also look at how we're using CK technology to incentivize and protect the VACU network. We also look at Syrokit, which is a library we've been building to make it easier to use signals proofs in different environments. And at the end of this talk, I'll hope you'll come away with a better understanding of the importance of privacy-protecting infrastructure and how we can build it. So first briefly about the VACU and me. I'm the director of research at VACU, and we build public good protocols for the decentralized web with a focus on privacy and communication. We do applied research based on which we build protocols, libraries, and publications. And we also do custodians of protocols that reflect the set of principles. It has its origins in the status app and basically trying to improve on the underlying protocols and infrastructure. We build VACU, among other things. So privacy is the power to selectively reveal yourself. It's requirement for freedom and self-determination. Just like you need decentralization in order to get sensory persistence, you need privacy to enable freedom of expression. To build applications that are decentralized and privacy protecting, you need the base layer, the infrastructure itself, to have these properties. We see this a lot. It's easy to make trade-offs at the application layer than doing them at the base layer. You can have custodial solutions on top of a decentralized and non-custodial network where participants own their own keys, but you can't really do it opposite. And if you think about it, even something as basic as buildings can be seen as a form of privacy-protecting infrastructure is completely normal and obvious in many ways. But when it comes to the digital realm and mental models and way of speaking about it, it hasn't sort of caught up yet for most people. I'm not going to go into too much detail about the need for privacy or what happens if we don't have it, but suffice to say it's an important property for an open society. When we have conversations, through peer-to-peer offline conversations, we can talk privately, and if we use cash to buy things, we can do commerce privately. On the internet, great as it is, there are a lot of forces that makes it the natural state of things, not the default. Big tech has turned users into commodity, a product, and monetized users' attention for advertising. To optimize for your attention, they need to sell your habits and activities and hence breach your privacy, as opposed to more traditional business models where you're buying a product and incentives are maybe more aligned. And to sort of build these, sensor-persistent and privacy-protecting and enable these types of apps that enable freedom of expression, we need credible, basically, infrastructure that have these properties. So infrastructure is what lies underneath. There's many different ways of looking at it, but I'll keep it simple, as per the original web-free vision. So you add a theorem for computing consensus, swarm for storage, and whisper for messaging. And Vaku has taken over the mantle from Whisper and is a lot more usable today than Whisper ever was for many reasons. On the privacy front, we see how your theorem is struggling. It's a big UX problem, especially when you try to add privacy back on top. It takes a lot of effort, and it's a lot easier to censor. And we see this with recent actions around tuning to cache. Compare this with something like Ccash or Monero, where privacy is there by default. There are also problems when it comes to the pay-to-pay network inside of things, for example, with Ethereum-validated privacy and hostile actors and jurisdictions. If someone can easily find out where a certain validator is physically located, that's a problem in many parts of the world. Being able to have stronger privacy protection guarantees will be very useful for high-value targets. And this doesn't even begin to touch on so-called dApps that make a lot of sacrifices in how they function, from the way how domain works, to how websites are hosted, and the reliance on centralized services for communication. We see this time and time again where centralized single points of failure, they work for a while, but then eventually fail. And in many cases, an individual user, they might not care enough, and for platforms, the lure to take shortcuts is strong. And this is why it's important to be principled, but also pragmatic in terms of the trade-offs that you allow on top. And we'll touch more on this when it comes to design goals and around modularity that Evacue has. So Synod Proofs are a wonderful new tool, and just like smart contracts enable programmable money, Synod Proofs allows us to express fundamentally new things. And in line with the great tradition of trust minimization, you can prove statements while revealing the absolute minimum information necessary. And this fits the definition of privacy, the power to selectively reveal yourself perfectly. And I'm sorry I don't need to tell anyone in this room, but this is truly revolutionary. And technology is advancing extremely fast, and it's often our imagination is limit. So what is Evacue? It's a set of modular protocols that compete to be a communication, has a focus on privacy and security, and being able to run anywhere. And it's a spiritual successor to Whisper. And by modular, we mean that you can pick and choose protocols and how you use them, depending on constraints and trade-offs. So for example, bandwidth users versus privacy. It's designed to work in resourcistic environments, such as mobile phones and web browsers. And it's important that infrastructure meets users where they are and support their real-world use cases. Just like you don't need your own army and a castle to have a private bathroom, you shouldn't need to have a powerful, always-on node to get reasonable privacy and sense of persistence. We might call this self-solventy. One way of looking at Evacue is it's an open service network. There are nodes with varying degrees of capabilities and requirements. For example, when it comes to things like bandwidth usage, storage, uptime, privacy requirements, latency requirements, and connectivity restrictions. We have this concept of adaptive nodes that can run a variety of protocols. And a node operator can choose which protocols they want to run. Naturally, there will be some nodes that do more consumption and other nodes that do more provisioning. And this gives rise to the idea of a service network where services are provided for and consumed. So there are many different protocols that interact. Evacue really protocol is based on libpdb.gov.sub for pdbmessaging.  Light push for nodes with short connection windows to push messages into the network. Store for nodes that want to retrieve historical messages. And then on the payload layer, we also have things like support for noise, hand shakes, and key exchange. And this means that as developers, you can get end-to-end encryption and expected guarantees out of the box. We have support for setting up secure channels from scratch and all of this paves the way for providing things like signals double-ratch that a protocol level much easier. We also have experimental support for multi-device usage. It's worth noting that similar features have existed in, for example, the status app for a while. But with this, it makes it easier for any platform using Evacue to use this. There are a lot of other protocols as well related to things like peer discovery, topic use, and so on, and you can check out our specs for more. So when it comes to the Evacue network, there are a few sort of immediate problems. For example, when it comes to network spam and incentivizing service nodes, we want to address these while keeping the privacy guarantees of the base layer. I'm going to go into both of these. So the spam problem arises on the gossip layer when anyone can overwhelm the network with messages. The service optimization is a problem when nodes don't directly benefit from the provisioning of a certain service. This can happen if they are, for example, not using the protocol directly themselves as part of normal operation, or if they aren't socially inclined to provide a certain service. And this depends a lot on how an individual platform choose to use the network. So since the peer-to-peer relay network is open to anyone, there's a problem with spam. And if we look at some existing solutions for dealing with spam and traditional messaging system, a lot of entities like Google, Facebook, Twitter, Telegram, Discord, they use phone number verification. And while this is largely civil-resistant, it is centralized and not private at all. Historically, Wisp use-proof of work, which isn't good for heterogeneous networks, and things like peer-scoring are open to civil attacks and doesn't directly address spam protection in an anonymous peer-to-peer network. And the key idea here is to use RLN for private economic spam protection using CK Snarks. I'm not going to go into too much detail of RLN here. We have some bright apps on back.dev by SNAS, who's been pushing a lot of this from our side. And I believe there's also another talk by Taylor at PC tomorrow afternoon, who's going to go into RLN in more detail. But I'll briefly go over what it is. It's sort of the interface and circuits and talk about how it's used in vacuum. So RLN stands for rate-limiting nullifier. It's an anonymous rate-limiting mechanism based on CK Snarks. And by rate-limiting, we mean that you can only send N messages in a given period. And by anonymity, we mean that you can't link messages to a publisher. We can think of it kind of as a voting booth, where you're only allowed to vote once every election. It can be used for spam protection and peer-to-peer messaging systems, and also rate-limiting in general, such as for a decentralized capture. There are basically three parts to it. So you register somewhere, then you can signal and find there's a verification and slashing phase. You put some capital at risk, and this can either be economic or social. And if you double-signal, you get slashed. So here's what the private and public input to the circuit looks like. The identity secret here is generated locally, and we create an identity commitment that is inserted into a Merkle tree. And we use Merkle proofs to prove membership. Registered members can only single once for a given epoch or external nullifier, for example, every 10 seconds in Unix time. And RLAN identifier is for a specific RLAN app. We also see what the circuit output looks like, and this is calculated locally. So Y here is the share of a secret equation, and the internal nullifier acts as a kind of unique fingerprint for a given app user epoch combination. How do we calculate this Y and internal nullifier? This is done using Shamir's secret sharing. Shamir's secret sharing is based on the idea of splitting a secret into shares, and this is how we enable slashing of funds. In this case, we have two shares. If a given identity A0 signals twice in an epoch external nullifier, A1 is the same. And then for a given RLAN app, internal nullifier also stays the same. X is the single hash, which is going to be different. And Y is public output, so we can reconstruct the identity secret. With identity secret revealed, this gives access to, for example, financial stake. So this is how RLAN is used with the relay gossip protocol. So node registers and locks up funds, and after that, it can send messages. It publishes a message containing the share knowledge proof and some other details. Each relay and node listens to the membership contract for new members, and also keeps track of relevant metadata and the Merkle tree. And metadata is needed to be able to detect double signaling and perform slashing. Before forwarding a message, it does some verification checks to ensure that there are no duplicate messages, that the share knowledge proof is valid, and that no double signaling has occurred. It's worth noting that this can be combined with traditional peer scoring, for example, things like duplicate messages or invalid CK proofs. In line of Vacus' goal of modularity, RLAN really is applied on a specific subset of PubSub and content topics, so you can think of it as kind of an extra secure channel. So where are we at with RLAN relay deployment? We've recently launched our second testnet, and this is using RLAN relay with smart content on Gully. It's integrated with our example P2P chat app, and it does so for three different clients, Nevaku, Govaku, and JSVaku for browsers. And this is our first P2P cross client testnet for RLAN relay. Here's a screenshot of a short demo. In the interest of time, I'm not gonna do the demo here. I'll actually do it tomorrow afternoon. But basically, this shows a user registering in a browser and then signaling through JSVaku, and it then gets relayed to a VACU node that verifies the proof. And when more than one message is sent to a given epoch, it detects a spam and discards it. Slashing hasn't been implemented fully yet in the client and it's work in progress. If you're curious and wanna participate, you can join the effort on our VAC Discord. We also have tutorials set up for all the clients that you can play around with, and that's the QR code right there. As part of this, and in order to make this work in multiple different environments, we also been developing a new library called Syrokit. I'll talk about this a bit later. So going back to the idea of the service network, let's talk about service credentials. So the idea behind service credentials and private settlement is to enable two actors to pay for and provide services without compromising their privacy. We do not want the payment to create a direct public link between the service provider and the requester. We call it the sort of the vacuum service network illustration with adaptive nodes where nodes sort of choose which protocols to run. And a lot of these aren't very heavy and they sort of enable them work by default. So for example, they're really protocol. Other protocols are much heavier to provide, such as for example, storing historical messages. It's desirable to have additional incentives for this, especially for platforms that aren't community-based where some level of altruism can be assumed. For example, status communities or wallet-connect cloud infrastructure. So you basically have a node Alice that's often offline and it wants to consume historical messages on some specific content topic. And you have another node, Bob, that runs a server at home where they want to store historical messages for the last several weeks. Bob is happy to provide this service for free because he's excited about running privacy-protecting infrastructure but he's using it and he's using it himself but his node is getting overwhelmed by free-loaders and he feels like he should pay something to continue to provide the service. So Alice deposited some funds in a smart contract which registers it in a tree similar to certain other privacy element mechanisms. A fee is taken or burned. In exchange, she gets a set of service credentials and once she wants to do query with some criteria she sends this to Bob. Bob responds with size of response, cost and receive address and then Alice then sends a proof of delegation of a service token with a payment. As a payment. Bob verifies the proof and results the query. The end result is that Alice has consumed some service from Bob and Bob has received payment from this. There's no direct transaction link between Alice and Bob and gas fees can be minimized by extending the period before settling on chain. This can be complimented with altruistic service provisioning for example by splitting the peer pool into two slots or only providing a few cheap queries for free. The service provisioning is general and can be general for any kind of request response provisioning that we want to keep private. This isn't the perfect solution by any means but it's an incremental improvement on top of the status quo and it can be augmented with more advanced techniques such as better non-reputable node reputation, proof of correct service provisioning, et cetera. And there's a lot more details to this and we are basically currently in a raw spec proof concept stage of this and we expect to launch a test in this later this year or early next. So SeruKit is a set of Serenade modules written in Rust and designed to be used in many different environments. The initial goal is to get the best of both worlds with circum-solidity and JavaScript on one hand and the Rust ecosystem on the other. And this enables people to leverage circum-based constructs from non-JavaScript environments. For the RLN module, it is using circum circuits via Arc Circum and Rust for scaffolding. It exposes a CFFI API that can be used for other system programming environments like NIM and GO. It also exposes an experimental Wassam API that can be used through web browsers. And that could be the infrastructure running in many different environments such as NIM, JS, GO, and Rust and mobile phones and web browsers. So this is a requirement for us. Circum and JS Strings is access to things like DAP developers, tooling, generating verification codes, circuits, and Rust Strings is that it's system-based and easy to interface with other language runtimes like NIM, GO, Rust C. It also gives access to other Rust C ecosystems such as Arcworks, which opens the door for using other constructs like Halo 2. This becomes especially relevant for concepts where you might not want to do a sort of a trusted setup or where circuits are more complex or custom and performance requirements are higher and so on. In general with Circuit, we want to make it easy to build and use Synology Proofs in a multitude of environments such as mobile phones and web browsers. Currently, it's too complex to write privacy-protecting infrastructure with Synology Proofs, considering all the languages and tools you have to learn, all the way from JavaScript, Solidity, and Circum and how to write the circuits to Rust, Wassam, FFI, and that's not even touching on things like secure key storage or mobile depth. Luckily, more and more products are working on this, including various DSLs. And it would also be exciting if we can sort of make a useful tool stack for JS less CK devs to sort of reduce cognitive overhead similar to what we have with something like Foundry. I also want to mention a few other things we're doing. One thing is protocol specifications and we think this is very important for peer-to-peer infra and we see a lot of other products that claim to be doing peer-to-peer infrastructure but they aren't very clear about guarantees or are stable something is or the actual semantics of it. This makes it hard to have multiple implementations to collaborate across products and to analyze things objectively. Related to this is publishing papers. We put out free so far, related to VACU and all and really. And this makes it easier to interface with academia and there's a lot of good reasons out there and we want to build a better bridge between academia and industry. Another thing is network privacy. So VACU is modular with respect to privacy entities and there's lots of knobs to turn here depending on specific deployments and requirements. For example, if you're running the full really protocol you currently have much stronger receive anonymity than if you're running a filter protocol from a bandwidth or connectivity restricted node. We aim to make this plugable depending on specific user needs. So for example, mixness such as NIMM they come with some trade-offs but are definitely a very useful tool in this arsenal. A good mental model to keep in mind here is the anonymity trail Emma where you can only pick two out of three when it comes to low latency, low bandwidth usage and strong anonymity. We're currently exploring things like Dandelion-like additions to the really gossips of protocol which would provide for stronger send anonymity especially in a multi-node botnet attacker model. As part of this we're looking into things like different parameter choices and more generally possibilities for low latency usage. This could make it more amenable for latency sensitive environments such as validated privacy especially under very specific threat models. In general, FEMA is that we want to be rigorous about the guarantees we provide under what conditions and for what precise threat models. And we have a lot of specs around this and adversary models and so on. Another thing that I mentioned earlier is this noise payload encryption and specifically things like allowing for pairing different devices with QR codes. And that's sort of, it's very useful for developers because we live in a multi-device world and we want to use things from different devices. As a summary, we're going over what privacy protecting infrastructure is, why we want it and how we can build it. We've seen our CK is a fundamental building block for this. We looked at VACU, the communication labor web-free and how it uses Synology Proofs to stay private and function better. We also looked at CIRCIT and how we can make it easier to do Synology Proofs in different environments. Finally, we also looked at some other reasons we've been doing and all of the things mentioned in this talk and a lot more is available as write-ups, specifications or discussions on our forum or GitHub. Here's some links. If you find any of this exciting to work on, feel free to come up to talk to me. We're also hiring and have started expanding into other privacy infrastructure technology such as private and provable computation with CK Wasm. And the QR code is a link to our Discord. Thanks, any questions? Hello. One of the main challenge I perceive in crypto messaging is the fact that producers are incentivized to pay to have a massage broadcasted, but there is no incentive then for relayers to distribute it to consumers that much. And this makes centralized solution usually the outcome as you want to have something like, I don't know, MetaMask to be able to relay things to users with Intra and everything, which make it in return a censorship sensitive where then the nation state can pressure these centralized actors into not distributing the message. And if you're relaying only on peer-to-peer, you might still need then some very old-fashioned dexer layer which can also be centralized to, hey, this message is here. So how do you specifically in your protocols solve this, making sure that there is incentive for the distributor to receive a message? So it repeats the question. So if you're in a decentralized messaging protocol, you have consumers that want to receive a message and as you put it yourself, sometimes they don't really have a lot of resource, they don't necessarily want to verify their identity and everything, but there is no direct incentive for broadcasters to have the private message that these consumers want to send this message to the consumers. They're not getting paid for it. Yes, so I think we're splitting this problem up a bit. It's definitely a general problem if you have some multi-hop model, if you want to keep that private in a centralized set and in general that leads to very heavy models. The way we are basically simplifying the problem and cutting it up is looking empirically at the problems we see today and a lot of those problems, as you kind of mentioned with Metalskin, things like that, come exactly when you are in a restricted environment and you have to use some kind of service, some request-to-spawn kind of thing there and this is exactly what we're trying to do with the things like the service credentials and these things and there's also a lot of things going on when it comes to discovery and finding and ranking these types of nodes but when we look at the relay network, just like if you look at GossipSub, usually the participants in the network are incentivized to sort of relay messages and furthermore, because of four animated reasons, we are encouraging the use of a single PubSub topic which means that you're relaying messages in a network in general because you don't necessarily know the contents of it. So it's kind of a by-product of normal operation. So it depends a bit on what layer you look at it, if it's sort of the normal operation or if you're looking at the edge kind of thing and then the reputation thing is another aspect of it. So if there is no direct incentive in the network, it's just a side effect of producers wanting more users. Kind of, like there's these levels of it, right? There's different types of incentives and the relay network itself, empirically it's not the big problem right now. It is definitely a resource area and it's not something that we are looking into and we have some ideas around it but it's not actually a high priority if you look at empirically the problems we're having. The problems we're having are more around these kind of edges where there's service provisioning as well as how you discover things and web browsers and reputable nodes and these types of things. So that's higher priority and empirically but things might evolve and yeah. Do we have any more questions? I have two questions. First is all based on semaphore. Second question is, is there any gas payment involved in zero knowledge verification if there is, how do you prevent from exposing your own chain identity from gas payment? Yes, that's a good question. So all in is, you can look at it as kind of semaphore but then more things when it comes to actually preventing double signaling. So semaphore plus some other stuff to do that then do the slashing and so on. As for the CK, like CK verification this is actually something that's a bit different with this peer-to-peer infrastructure because it's not happening normally on the blockchain. It's actually you're verifying it on each peer-to-peer node sort of that's running this thing. So that doesn't require gas usage. When it comes to slashing, so if you've detected the double signaling then you have to go on chain obviously and interact that and that will also verify the proof but you can use the relays and these types of things. But as part of normal operation you're not actually interacting with the blockchain. It's all happening peer-to-peer. Unfortunately time is up. But thank you Oscar. Again, I'm going to repeat the fact that you can go talk to Oscar outside of the room so please pull up on your conversation there. So again, thank you.


guys yeah it's been a hot minute or has it been two hot minutes I think it's probably been a long time it's almost been a month since I've recorded a video holy cow actually that's not true I did record a video but there was a bit of drama around it I mean I shouldn't say a bit a major amount of drama around it that I took the video down so it's been a month overall since I've visited with you before and like a lot to catch you guys up on so if you're curious you know the drill stay tuned so oh my gosh anyways I it's Sunday I'm hoping to get this uploaded so that you guys can see it Sunday afternoon sometime I know the Patriots are on this afternoon and I'll have plenty of time I'm not planning on I'm going down to visit Jane I know none of you don't even most of you don't even know the story yet but let me just tell you since our since the last video let out first I'll get the drama out of the way this is going to be sort of a chatty thing all over the place but I'll get the drama out of the way I had recorded a video about a wig from a company called I am wigs and they had sent me two wigs that were squished together in this funky box and I contacted them I said I wanted you you know why don't you just take it back because I wasn't going to record it if this is the way you send it to you know a so-called influencer what is it going to be like when you send it to people that aren't on YouTube type of thing so they apologize they went all over the place oh no is the new shipping company blah blah blah blah and so they were sending me these messages one after another and I was just ignoring it and then one day I got two new wigs in the mail and they were packaged beautifully they were packaged absolutely beautifully and the wigs other than the gray one was not my color the wigs were surprisingly high quality I thought so I said oh right I'm going to give these people a second chance and I recorded the video and I published the video and I did it on Instagram and I did it on YouTube and and then I realized oh it's Marlene Fab and Glam who had posted something about someone being a scam site and I was like I you know I contacted Marlene and she said yeah these people steal the photos and they sell poor quality wigs and all this stuff and they just scam people and I then got really upset I reached out to the company and they said no no that's not true blah blah blah and I'm like going holy you know I don't want to be in the middle of this right so I took my video down and I took my Instagram stuff down and that kind of really shook me because when when I talk about something you know even in that video I talked about the first two wigs coming could you know squished in the box and all that I explained it but when I talk about something it comes from me comes from come how I feel about it you know it's I'm not like an influencer that's making a ton of money on YouTube I don't make thousands and thousands of dollars a month on YouTube I'm lucky if I make two hundred dollars you know maybe so I'm not I'm not anyway near that level and I never I never will be so when I talk about stuff it's because I really like it or I don't type of thing so I was so so bummed and that kind of I think threw me for a major loop because there I was publishing this video and this as far as I know this company ended up being a scam company and that kind of burst my my confidence level right up being up to judge people or being up to judge products it really burst my confidence level so I said alright I just after I took it down you know I had a good cry had a good little meltdown onto the next thing you know and then one thing happened after another I think I mentioned in my last video that was published that I had joined the gym so I had joined the gym and I joined Planet Fitness mainly because I wanted to be able to go to gyms all over the place and I knew that Jay and I were going to be going away camping and this was for labor I think it was Labor Day weekend yeah Labor Day weekend and I wanted to be able to take advantage of those locations which successfully we did so we did go camping and I recorded all these little bits and pieces of video which I'm probably never going to do anything with but because I intended to do a vlog but then the whole thing with the wig thing just kind of burst me really really set me back so anyway I'm going to close some pictures because we did go camping we had a great time we were up in old Orchard Beach area a little beach called Pine Point Beach and I'm going to insert a video I know I shared this before we went there for sunset and there's they there's a drummer is there they're just it's it's like I think it's I'm not too sure if they're all Native American or what the whole story is they didn't ask but they would they drum and the moon comes up and it was just really mesmerizing that part of the beach is so really nice and flat you can walk a real long time anyways we stayed at Bailey's which is a favorite campground in that area for us it has a beautiful pool and we just it was hot hot hot Labor Day weekend after a crappy summer Labor Day weekend was super super hot it was unreal but so we just really had a great time Labor Day weekend and I'm not going to put the vlog together at this point in time because it's too far back but we did go to the gym while we were there and I found a gym in Portland in South Portland and we went to Planet Fitness both Jay and I and about this time he had already heard he was going to have hip surgery so we knew his hip surgery was going to be September 18th so he was really trying to you know strengthen a little bit stuff like that especially with his arms he's been in so so much pain for so long that it's been it's been insane for him you know he hasn't been able to walk like he wants to walk he hasn't been able to go off like he wants to go off so many different things impact your life you know when you're in pain right that impacts your lifestyle your things that you like to do well his pain level was way way up there so we were both like yes yes you know finally get the hip replacement surgery done because the beginning of the year we're gonna he needs to be very mobile because we're gonna be doing some different things the beginning of the year and I personally can't drive the RV it's too big so he needs to be mobile and and you know fully together so he had his surgery on the 18th it went extremely well the doctor said he came out of it with flying colors all of this stuff they send you home from the hospital that same day he had an interior I guess in the front you know so I guess that was most the easiest way to do it and they sent they sent him home it came home Monday night I brought him home it was pouring rain so it was a little bit difficult getting in the house as you can well imagine but we got him in when that was all cool and then he couldn't get comfortable just couldn't get comfortable on the easy chair then he couldn't get comfortable in bed it was a really rough night for him Monday night but I figured Tuesday would be better but we knew that the first two days it would be rough well Tuesday morning he got up wanted to get up and then I got up because he had to go to the bathroom and I wasn't gonna let him go to the bathroom alone so I had to walk or it was getting ready he said to me that he felt a little lightheaded I said okay just sit there and don't move I'm gonna go out into the barn the shop as we call it when my father used to have the machine shop actually I'm filming out here I'm gonna go out there I'm gonna get a bucket and you can just I'll help you just pee in the bucket I don't it have to get up and go to the bathroom so I ran out here to get the bucket and ran back in I wasn't even gone four minutes three or four minutes I mean if five minutes would be an absolute stretch when I went back in the bedroom he was on the floor and I was like I was like oh my gosh and at the time I went back in the bedroom and oh my gosh what did you do what happened what happened I told you to stay put he was just starting to come come to he had passed out and apparently he he he had thought he was just lying back but he went this way and he he doesn't even remember hitting the floor he doesn't even remember going he just thought he was going on the bed lying back and he argued with me for a good 30 minutes I'm trying to get him up I couldn't as he said he wanted to try to crawl I said Jay you can't crawl he's had hip replacement you can't do any of that so finally I just started to cry and he said alright call somebody because I went to call 911 or whatever they just they're gonna come with the fire department I said alright I'll call the police department non-emergency line so I did that I called the non-emergency line explained what happened they said we're gonna have to initiate rescue they're gonna have to come over you know so they did long long story shot he went to the hospital they checked his neck they checked his x-rays he fractured his new hip my gosh and he was still dizzy and and lightheaded and so his blood pressure when he was lying down his blood pressure was okay but when he sat up it went way down when he tried to sit up or stand up it was like way way way I mean way way down dangerously low so they admitted him to the hospital he couldn't come back home so he was in essence in the hospital since Tuesday morning Monday was a surgery Tuesday he went in since Tuesday morning today is Sunday yes Friday he was finally stabilized they figured out he was very dehydrated there's some other condition I can't remember the name of it that drops your blood pressure especially when you dehydrated and stuff like that so they finally were able to stabilize him and all that he still couldn't get up he still couldn't do anything on his own so he's in rehab I went to rehab about five o'clock Friday and and so yesterday he had a little bit he had his assessment close today Sunday so we won't really know what's gonna go on until Monday but Jay's in rehab his spirits are super good and he's able to be a little bit more mobile obviously than when he was in the hospital so I'm like thank goodness thank goodness thank goodness you know I just want him to be able to come home to be able to walk you know to be able to be strong and be Jay again he's a big guy I couldn't get him up he's six two I think and he's probably about 240 and he's got a lot of muscle on him so he'd be too heavy I just couldn't get him up off the floor and it was just one thing after another oh my gosh this is probably gonna be a long drag-out video that's where he is he's being taken care of I feel a hundred percent better although being alone at home is not something I'm used to so I'm adjusting and I'm kind of liking the fact I don't have to cook for anyone because I just don't cook I haven't cooked since he was here but yeah so I also haven't gone to the gym all week because I've been going to the hospital every day and I had a real estate transaction and I know some of you asked me to tell you more about the prices and stuff like that I'm gonna link my real estate YouTube channel somewhere maybe in the description or up there so you guys can if you're interested you can see anyways I listed property in a town it's very and I'm not gonna say too much because it hasn't closed yet but so it was a very hectic weekend the weekend before his surgery because we did two open houses and then I put another sale together and I'm gonna do inspection on that on Tuesday so it's been my real estate business sales which is what Jay and I do as independent contractors has nothing to do with my job with my company I work for a real estate company as a training person so I went very part-time in that position now my real estate sales is like getting very busy and that's okay so it's just been a crazy crazy crazy time frame super busy I just didn't have time for videos and I'll tell you when that week thing happen that kind of blew my mind so I also talked about the fact that I was gonna have my very second facial I I've made no secret about the fact that I went in March and I had some not not in another I can't remember the name of it it's like the Botox but not that brand or that name brand on my 11 because it couldn't stand my 11 so some of you that watch all my videos you know I've talked about that anyways I went to a nurse practitioner esthetician she's absolutely lovely beautiful beautiful kind-hearted sweetheart not anyone that believes in over injecting I just love her philosophy she's a nurse practitioner and she has a family practice etc so I felt super comfortable with her and she did my 11s and I absolutely loved the results now my nobody ever noticed much at all until I told them and it's such a minor change but I see it my husband says even when it wore off he never he said I don't see what you're talking about but anyways I see it so I was talking to her and I had set up an appointment to go back and have them because I did it much first and I could see my 11s they were pretty severe and all that stuff and they're still here right now because it hasn't been that long but so I made an appointment to go back and she had opened up a place in writing Mass as well and it is it is called on Haven aesthetics oh my gosh what a beautiful I'm gonna close some pictures and link the website but what a beautiful beautiful spot and her business I mean it's just set up really really nice I went in there before my injection for my 11th and 11th and I had a facial I had a facial by a lady named Theodora and oh my gosh she was absolutely she was absolutely fabulous she had studied for I think two years in Europe and she worked for a number of years in Europe and then came to the States and started working and she's an esthetician and just really you know really working on on I think helping women or anyone feel as good as they possibly can anyways I absolutely adore Theodora she did an amazing amazing facial on me absolutely loved it it was perfect and then Elizabeth who is a nurse practitioner she introduced me to her partner Crystal who is also a nurse and two of them have started the on Haven aesthetics and I tell you if you're in the Massachusetts area their pricing is amazing their service is amazing and they care they care they listen to what you want and they really care so I just got a little bit of a booster here and I probably will go back at the end of getting an appointment to anyways for the end of December to go back because then I'm going to be not around for a while at least up here so I wanted to you know make sure I'm going to link all of this information below but I can't I highly highly recommend anyone that's especially in the Massachusetts Southern New Hampshire area and if you're looking for someplace where you can really feel super comfortable you've got to check them out so I wanted to tell you about that because it only had two facials in my life my first one not all that great it was more of an uncomfortable experience but this one with Theodora I mean she made she just pampered me and it was awesome it was really really awesome I absolutely loved it so other things that I've been doing basically is I have I've been trying to find a bathing suit and bathing suits are really hard I mean I have my favorite bathing suit is this old green and it's starting to get so worn out you know in a bathing suit if you've had it too long it's just get so worn out I've always loved the straps on this in the back and you can see them the straps are just really really pretty but this is my favorite bathing suit it's a one-piece obviously I'm not going to get into anything whatever so I have been looking online and I've been like trying to find something similar but you know I know I'm not going to but just just trying to find a bathing suit you can't find a bathing suit right now and I need to have a bathing suit so I found this company called Cup She I saw it on Facebook and I'm like you know you you order from these companies and they're kind of like you know iffy and I was really worried about that so I kept reading their reviews and all that and so I said I'm going to go ahead and order so I ordered two vins it's ordered the screen one I do like the straps there's just a little bit of what I don't like sometimes the white shows here from the lining but I love the back the back is really pretty and it fits me awesome and I'll tell you the quality for an online one-piece bathing suit the quality has been really really good I really like it I ordered a second one this one I really like a lot but I might send it back because I feel the straps are so thin I'm almost worried that it isn't going to hold me up it holds me up so far and I like the color it's a purple again the quality of these the lining is coming out I don't know I don't want to pull the lining out because then I won't be able to return it if I want to I don't know if I'm going to return it I just yeah I probably won't the prices are amazing and for an online company I was like holy mackerel so I did buy that I also had this summer because I knew my green bathing suit was going to be history soon I had gone into Target and I don't shop in Target and I bought this bathing suit target I like it a lot but I think maybe I don't feel like it supports me I'm Chesty I really don't feel like it supports me on top as much as I really really need to be supported on top so I like bathing suits and I you know I like bathing suits I need to have a bathing suit because I'm going to be in warmer climate and I don't want to be skinny because I would look you know awful and I wear a lot of my hope my rips girth's Hawaii you know when Jay and I went to Hawaii the beginning of the year went to Maui and we stayed in Lahaina the western part of Lahaina in a very small Airbnb I'll link that here for those of you that haven't seen it and we stayed in this really nice Airbnb in a small small complex and I don't know if it's there or not but all I wore every day was I have several of these this is probably the latest one and the one I like the best because when you wear a solid color baby suit like my black one this rip skirt looks awesome so anyways I was looking for clothes to wear summer type clothes because of the fact that we're going to be in a little bit of a warmer climate it's also why I want Jay to hurry up and get better so yeah not easy to buy a bathing suits up here is certainly not easy to find ones that I like that fit me I oh I also got these little clips I should have I got four of them they're flip flops can you see them these are the clips you put on the back of the beach chair and it kind of anchors your towel to it I should I should have probably saw if I could get maybe football ones or something for Jay because I don't think Jay is going to want to flip a flip-flop on his beach chair but those and Jay and I celebrated our anniversary we went out to this is all prior to his surgery we went out to a place called the Sea Glass which is one of our favorite restaurants that we go to it's a special occasion it's an expensive restaurant but it overlooks the ocean and it's just it's just so pretty it's so so pretty such a pretty spot so we went out to the Sea Glass we had gotten a gift certificate we went out and celebrated our anniversary on clothes and pitches he a few to say and we just sat we had an ocean side table and it was it was just wonderfully relaxing my meal was wonderful Jay of course got his favorite oysters and he loves those things and they were good I can't remember what else he got but our food was perfect and then afterwards we went over to Salisbury Beach and we walked around and the Sun was just setting over Salisbury Beach over on the western part of it overlooking Salisbury Harbor and it was so pretty we took a bunch of pictures the backsplash was amazing absolutely amazing backsplash it was so gorgeous and the reason why we almost didn't even make that sunset is because after dinner they had the Sea Glass they have this really cool gift shop called Buy the Sea gift shop Buy the Sea and it is a lot of handmade items and just different kind of pieces you know from jewelry to pocketbooks to scarves to you know to all sorts of like charms and stuff like that my favorite key ring I bought there it has like a starfish on it I love that key ring it's so pretty anyways we went in there because we had gotten a gift certificate and they give you like $10 for each $50 you spend off so I had like two $10 gift certificates to use and I went in there and I ended up getting this necklace I just thought it was so pretty I don't know if you guys can see it I thought it was so so pretty I love it and you can you know make it shorter if you want or longer if you want to get this necklace and I got this little matching little beady bracelet and then I got these earrings I thought the earrings are so pretty I haven't worn them yet but they've got like a little bit of a pearl and a little it's like a flower I just thought they were really really pretty so I got those and I just you know I've I've been on a shopping spree I actually got two sundresses at Old Navy now I wouldn't even call them sundresses but they're sort of they're right on 100% right on the first one as you can see here is a very flowery sort of like a little bit of a longer short sleeve very light material so they'd be perfect and warmer climates and certainly perfect if you were going to be on a cruise or something like that and you just wanted to have like a dress thing to go over and the second one is black it doesn't look good on the hanger at all but on me where you can I can pull it down a little bit off the shoulder I can't wear off off the shoulders but I can pull it down and and a little bit and it looks really looks much better on than hanging but I bought those two I'm really excited about our future plans and I don't know I'm like oh gosh you know hurry up Jay and get better and come home and all that it's it's different being home alone you know it's different especially if we used to having the companionship and it makes you wonder it makes me think anyways you know I've gone through a lot of loss these last three years with my dad my brother and then my mother I'm so lucky that I have my children and I have my grandchildren and I have my husband I'm so lucky in that sense and it just made me think being here all the things that he does that I have no clue how to do you know like for example I've pumped gas maybe three times in my life I'm gonna have to pump gas and I know I can do it I just put my mind over it I gotta do things like change a water filter soon and I'm hoping they'll be home so he can give me the directions and just all those little little things that you know he does that I never think about I call them the blue jobs I do the pink he does the blue you know so now I'm gonna have to do I've been doing more blue jobs since he's been in the hospital obviously it makes me appreciate the fact that I do have my husband with me and I think of the women you know my age or even younger or older obviously who don't you know and you have to go in on your own it's like I've thought a lot about that I've thought a lot about that this week because it's like I need to know where things are how to do things how to access things just in case I'm not obviously anticipating anything but just in case I need to do that because then I will feel self-sufficient maybe self-sufficient you know that's what I'm thinking about anyways like I said I have no idea how long this has been I apologize I'm sitting here rattling and there's so much more I want to talk about because I bought these compression bags and you know some other stuff but I think I should end this here and just say if you actually stay to the end thank you so much I know I just going on and on and on and hopefully life will just get back to somewhat normal soon but thank you for those of you that have been reaching out to me especially Barbara Barbara you always reach out to me always and I so appreciate I so appreciate it I really do I went into my other email and I saw an email you sent and I said my god she even sent the email because I know who had left a message on another video as well and I haven't even paid attention to that email account in the last couple of weeks because it's just been so crazy hectic but anyways I just want to say thank you all and thank you Barbara always reaching out and Sheila oh my god my friend Sheila those of you that know Sheila simply Sheila aka life with Lily I want to say she's like the sister of my heart you know we're not really sisters but she's been every day every single day are you okay are you you know what's happening so even though she's far away I feel so blessed to have such a close friend she just you know I got up this morning and there was a message from her and I'm like oh you know she's worried about Jay how's he doing how are you doing and stuff like that so I'm a lucky lady and I know it so thank you all and I will end this because it's gonna be 40 minutes I should have timed it but thank you all so much and I hope you have a wonderful wonderful week and go patriots I hope they win today that'll make Jay happy bye guys


Dot to do a here back with another round of dragon ball fighters rank matches this time playing a team of all earthlings Oh the gojita intro. I'm gonna I kind of want to let it rock out. It's been a while since I've took taking it all in Still one of the bet probably the best intro now that I'm thinking about it No easily the best intro Hopefully Broly's can come out soon and show it up But now we're just gonna have to handle the earthling intro which got skipped which I'm not even gonna complain I understand it's an intro. We've been watching for the third overhead got me Where's the command grab show me the command grab dude nice jumping with the slide Here you go chasing me down. I Think we just played against this guy, but he might have had GT Goku out I don't know where he might have a third assist. I actually don't know or this might be he's exact same team I might have forgotten about the beam assist. I try to block this out. It wasn't our turn I thought for sure it was and now Yamcha is is gonna get proven just how wrong that was unfortunately Man, I feel like the earthling team was meant to be played together. It just has like Just has the right mix of us stuff to it DP like I forgot gojita had it. Oh now I remember it was a Bardock I think Not gonna go for the dragon ball surprisingly. I only got one bar get him out here, Jean That's what I deserve. I keep superdashing like the beam is it a one-button maneuver that can be done very quickly We're just gonna keep guard Shouldn't stop. We should keep guard canceling now that I'm truly thinking it is this TOD. Oh, yeah It is this is gonna be wild Oh, no drop. Oh, I can't imagine a worse feeling obviously goker GT dropped out But imagine going from the high of like, oh, yeah, TOD. Oh, I Try jump man. This this gojita is locking me down with those punches. It is just a menace to my health The two bars probably won't kill Might need to go for three He did not go for three so I'm gonna take a quick lunch break. Hey must be the lunch What even just happened then what even just happened there? Understandable except that don't understand All right, we're playing from a distinct disadvantage.  We're actually dead. So this is gonna have to be the TN show Never mind. We had it down heavy with criminal Name a more rare circumstance of a three mix. All right. What's the nose sparking to get out of it? You got your level three there, but what does that matter when I got to him? The most basic of options, but whatever we like it. We like the option all the same I should probably start doing to L more safer Wow, I am no fan of geometry, but that kind of sucked for me That sucked for me more than anything especially because TN supposed to be my anchor right now He's supposed to hold me down when times get tough and he is He's going to die He's not gonna die, but it does Distinctively take away our level three sparking plan. Okay, we might have to make the move for Vegeta I wish he didn't super dash right there. I was gonna super dash. I just took too long to do it I just took way too long to do it Sparking is gonna get run off the clock Hold on sparking hold on stay with me a little while longer I know you got to leave. I know you got a train to catch a heart out as some say We got it. We got a we got a game to win. Oh, no, we lost Two guard cancels one game You got to be kidding me. I play the newtch Play it play it play it well play well back dash on it. Go go go. He did it random DP Okay. Oh no. Oh We caught him. Yes, that was risky for me. I did not know the range All right, now we have one chance to get reset back to neutral. I'm not calling out any assist He tried to click a button. I think he wanted to go on the offensive. Oh, no, this might kill The I have two of the strongest level ones in the game It's my kill Tiet Earthling power. Oh my goodness the level one damage on this team and Utility with yonja. I think we got a game boys. That was intense though. We got we got really drawn back I think I want to use more um super dash this game Wow, that's a bold choice crazy daring today. Aren't we sir knock him down? There's the DP. I should have played it a little safer Dude, he's all right, this is it the super dash yonja sis the over what we've really changed up our game plan I think that might be what's throwing him off the most Where's the DP? No, no DG off the rock didn't work. Oh Oh No, no, he got me. Okay fair and square. He got me. I was playing a little reckless I was hoping to just overwhelm him honestly level one as you go level two. I probably go level three get gojita out of there Yep, that's about what that's about what I would do Probably exactly what I would do actually because both level three switch out of the corner I can't do it. I can't in good faith like gojita sit there with blue health Here I come that was nice. Don't even tell me it's not dude a little bit of sparking shenanigans I don't want to throw it to you one will work. I Did it anyway. I dude I try to fake confidence all the time. It did not work out for me there I I wanted to fake it so bad. I Couldn't do it though. I couldn't in good faith fake it for that long Whoa, whoa, whoa, whoa, whoa, whoa. That was a bad conscience. Oh my god trying to mix me. Oh Got me got me got got me good. We're down sparking by the way. I completely forgot about that Good, why did I go with heavy so many questions? No answers? Literally zero answers. Oh destructive Man, I really want to see more destroy. Oh, no sky Hello, no, not hello. Goodbye I'm sorry. I'm sorry. Yeah, I'm ready. I'm ready. I'm ready. Oh, thank you for waiting for me though, dude I found me. Oh my god. I'm I'm I'm lost dude. Oh I'm so gone. I'm not I can't even think about the game.  No, no slap me in the face. I just got out of the Skype call. I was sick Oh, we're definitely dead. Oh my god.  That's not funny. That's her. Oh my god Beautiful beautiful. We actually have a chance of winning now What am I doing? What am I doing? What am I doing? What am I doing? I forfeited the wind Well, the angle wasn't good. Why do I always click buttons when Goku GT has the angle on me? Oh, we're so dead We're so dead. We're so dead. We're so dead. We're so dead.  Spirit bomb is so good. All right to one. I gotta give it up. He did it like a gentleman boys So I honestly respect I got nothing but respect for my man He waited for me. He played outplayed me many spirit bombing Nothing but respect for this fine soldier, but I still got to do the I still got to do the thing All right, if he guard cancels me, I'm definitely pulling it out.  God thrilling is Amenis in this game and he's gonna continue to be a menace actually. No, he's not cuz I gotta go. Oh That was almost sick The down every would have been nice. I did not even attempt it though. I'll be honest I'm tech. No, we're getting taken for a ride probably even a spirit bomb ride. Nope and snap Don't ever snap yamcha in only if you express only snap yamcha in if you're ready for the yamcha player army army The union of yamcha players, dude. Oh That guard cancel he was definitely baiting it Definitely looking for that. Oh nice beam follow up to the key blast. He did. I've been super dashing I've been real super-dash itchy lately. I Definitely don't mind getting too heavy because it's just a reminder that you can't keep getting away with stuff like that I'm so slow. They really got to make that a little faster. I think I don't know I don't know. Maybe that's just me but Tien's grab is a little too slow It might be the same speed as other ones. It might just be like an animation thing. I just wish it was faster Oh God. Oh, I wanted to hit so bad. Oh Also, we got hit again level three. Yep. I can I can see these things come Do I use spark and give up Sparky the sparking battle with them? That's why I don't die here I think I do dude if everything goes right we can if everything goes right, which it just does not go right for your boy All right, hold it. Okay old Old That's not gonna kill. I don't know why I did it. I got it does he spark does he spark does he spark does he spark Does he spark does he spark does he spark does he spark? Damn it. Damn it.  I'm intense. I'm trying to end it right now He's got a level three again. Damn it. The Super Saiyan 3 just doesn't stop on this map May huh Structure disc the only one that can kill. Oh Nice and we get a little bit of blue health. We also get the yamches is that's a huge factor in this game Because now we have to really respect almost everything we do because it comes with a job How how did he call out the first stagger play? Was it too slow? Are his inhuman reflexes just that Godlike? And anyway, we got the hit so I'm gonna I'm gonna have passed that up now You just lost. Oh, no, we could have won. We could have won But I I accepted defeat too quickly. Oh, he's got four bars. I don't see how we can win this approach for the bar I tried I tried it. I tried a wolf angphis cheese probably not the best thing against somebody with Connects and that's too too.   I actually did say pivotal there. It doesn't make any sense. I went back to correct the record All right, come on. We always start off strong here though I feel like we started off strong in almost every match and then just slowly choked it away Whether it be through Skype calls or just something else entirely Was a clean mix up, dude What how did he catch me and we're dead? No, we're not dead, but damage look at the damage up. I will continue to be surprised Okay, thank you should have switched. I Wanted to try I just I regretted so much. Don't live in your regrets guys.  Superdash, I got to get my feet planted after those after I superdash and he blocks it and if I back up he likes a super What a way to lose a character. That's a way. That's a definite way. You can lose a character Not many people do and come at me, huh? Oh This ain't a great start ain't a great start and there's an unwinnable, but Why did he do that? Why did he do that? Oh, I can't read this guy Starting to get hot as well Starting to get hot under the collar. Oh Come on. Oh my god. This game is weird. This game is wild. Get him down This should kill. Yep. Yeah. Yeah, we're definitely killing here Nice TN does it himself. We don't lose much health from that because of sparking you were looking at the health bar I wonder what happened Okay, we're gonna have to waste the bar and then I think I do want the snap option. Oh, why did I click a button? I always expecting the DP now. I am terrified of dragon punches They scare me to death and then I just stopped playing offense. Oh, it's a mess It's a mess of a game. We got out our heads, but it's a game on the left. Okay, this should be a kill as well Maybe not but Now it definitely should be that hit, right? This won't kill.  Anyway, right as well snap, right? Yes, okay. Okay. Okay. We brought it back. We brought it back to any option. That's all we have lots of super dash Everybody to reflect if you see it. Oh, baby. Come here. Come to papa Come to papa No No Why do I do these things to myself? Oh, I didn't even exit cuz I figured why why waste it But whatever we win Not climactic at all, but whatever we got the way we take it. Thank you again to my opponent for the games Also, thank you for waiting Hopefully you guys enjoyed this one if you did and have a favorite moment Make sure to let me know down below in the comments as well as any teams you'd like to see me play Other than that while you're down there if you liked the video and enjoyed the channel Make sure to hit the like button and subscribe help support the channel out a ton It also keeps these videos coming to you on a daily basis because I can keep making them that way Speaking of videos there should be some more up on your screen So give any of those a watch if they catch your eye I have been dr. Doia, and I will see you in the next one


Welcome to module 50 of point set topology part one. So, let us continue the study of half starsness along with compactness and so on. Last time we gave some applications to functional analysis, normular spaces. In particular we showed that every normular space in which the unit sphere is compact then it is finite dimensional. Of course, we did many other things like we showed that the entire of R n no matter which norm you take is a cone over the unit sphere. When you change a norm the unit sphere changes, but the cone over the unit sphere is always homeomorphic to the underlying linear space R n. So, these are what we have seen. So, let us continue that as a consequence of the discussions last time we also get the following theorem that S and D denote the unit sphere and the unit disc respectively with any norm on R n. Remember if the R n, if the norm you have taken the help to norm then I have different notation. I have a special notation for this. This is the general notation for any R n. Then D R means S and D are respectively homeomorphic to the standard unit sphere and the standard disc. So, this is a consequence of our discussion last time. However, let us just go through the proof carefully. What is finally inordinate? Fix any arbitrary norm on R n. We know that it is continuous and non-managing on the standard unit sphere S n minus 1. The norm is a function into R right R plus actually. Restricted to S n minus 1 we already established that it is continuous. Now what I mean by continuous? Now I can take the standard R 2 norm on R n. With respect to that this function which is a norm on R n is also continuous. This is what we have seen. In particular on the sphere it is non-managing therefore 1 by norm x is also continuous. So, it follows that if I put lambda x equal to x by norm x then this lambda is from S n minus 1 to S is continuous. You see this you have to be careful here. This is the new norm. S n minus 1 is the standard norm. So, L 2 norm. L 2 norm is 1 does not mean that this norm is 1. So, I have divided by this so that I get into this capital S a ok. So, x going to x is continuous 1 by norm x is continuous the product will be continuous ok. Likewise, if we take mu x equal to x by L 2 norm now. So, the role will be reversed now domain and codomain then mu will be continuous from S to S n minus 1. What is easy to verify is that lambda and mu are inverses of each other ok. Therefore, each of them is what? A homeomorphism one is the inverse of lambda. Once that is the case you can take the cone over C s minus 1 which is d is homeomorphism the cone over C s ok. This is d n here this will be the d here. So, this also we have seen that is the thing that we have seen last time ok. The first one C S n minus 1 is d n. The second C S is the unit is with respect to the extra norm usual, unusual norm. This is the usual normal. So, once two spaces are homeomorphic their cones over them are also homeomorphic is one of the theorem that we saw last time. So, now let me take a genuine example here which you may have all be interested in. Maybe I will try to first stop and may have to stop this presentation and then do something. I wanted to show you a piece of string. Can you see that? What does it represent? I mean how do you represent this by a mathematical object? So, you can either say this is a closed interval. When you include the end points of the string or an open interval or a half open interval there is no other way to represent a half open interval or open interval or a closed interval. All of them are represented by string I have to tell me what it is ok. So, this is you can say it is 0 1. When you identify just 0 with 1 and nothing else what do you get? You will get some object like this which is a model for S 1 ok. Actually model S 1 is a model that this is now this is the object whatever it does not matter whether it is like this or like this and so on of homeomorphism this S 1. So, all this we know that you know a layman will understand that this is a circle. Now mathematically we want to rigorously say that identifying the end points of 0 and 1 gives you S 1 right. So, that is what I want to rigorously prove now. That was the that was my main interest here ok. So, let me go back to the slides ok. So, let us do this business namely let us prove that the quotient of the closed interval 0 1 by the identification namely the end points 0 and 1 are identified is actually S 1 ok. So, I will denote X by 0 1 ok. I do not have to take any other interval because they are all homeomorphic to 0 1 any closed interval is homeomorphic to 0 1 that we know already. So, define a relation 0 is related to 1, 1 is related to 0, X is related to X for every X. So, this is a equivalence relation ok. There is no other no other rule here ok. Now, let Q from X to X by R be the quotient map. This X by R which is quotient topology I want to show that is homeomorphic to S 1. Our task is to show that this homeomorphic to the unit circle ok. To get a homeomorphism we observe that whenever you have a quotient space what you do is you construct the function on X itself on the mother space itself ok. But now what we observe is X is compact therefore X by R is compact and S 1 is Hausdorff. Then one of the theorem that we have tells you that if you have a continuous bijection then it will be automatically a or homeomorphism. So, that is what we are going to exploit here ok. But now to construct a function from X by R to S 1 we appeal to this lemma which we proved long back. So, what was this? When you have a quotient map X to Y ok given any function f from X to Z that is a unique function f twiddle from Y to Z such that f twiddle composite Q is f filled on leaves this happens ok. Whenever two points are identified by Q the same thing should happen to the function that you are interested in that should also identify those two points. This is the condition. Of course, now our notations are like different X to X by R is same that is Q I want to get a homeomorphism here ok. To get a map here I should have a map here first of all such that g X is equal to g Y thus whenever X is not equal to Y implies only X and Y are you know the pairs 0 1 either X is 0, Y is 1 or Y is 0 and X is 1 that is an order pair is the same ok. Why I have put this one? Because finally I do not want any more identification here I want this one to be injective ok. I want first of all this function so it must have 0 and 1 should be gone to the same point here. So, g must have that point g of 0 must be g of 1 ok otherwise you know otherwise there is no equality I mean g X is not equal to Y ok then g X will be not equal to Y. So, such a map is readily available to us already this you do not have to hurt for it take g t equal to e power 2 pi i t restricted to 0 1 when t is 0 or t equal to 1 it is the unit of S 1 and everywhere inside it is injected. So, when you come down here you get a continuous function which is injective ok but g is already surjective because it is surjective also. So, this becomes theomomorphism because X is x pi r is compared and S 1 is also. Let us go to some other example now namely the projective space P n we have not much studied much of it, but we have we have studied its definition. So, I will recall the definition. So, projective space P n the real projective space called this is a complex version also as you may appreciate it is defined as a quotient of nonzero vectors in R n plus 1 by the diagonal action of R minus 0 namely R comma X naught, X 1, X n goes to R X naught, R X 1, R X n. So, it is also scalar multiplication you may say diagonal action of scalar multiplication ok. We have then claimed that this quotient map when you restrict it to S n namely unit vector is also a quotient map that time it was an exercise for you. Now this claim easily follows from our earlier theorem why because now S n is see S n is compact ok and and what we have shown is that it is a closed map ok. So, this will be also a quotient map now ok S n to P n it is easily seen that it is surjective, surjective continuous map, but now because S n is compact the function will be a closed map every say closed subset is compact as a compact and what the image of a compact as a compact a compact subset of a host or space is closed. So, that was the theorem ok. So, you can use that it follows that S n to P n is also a quotient map of course, it is not by action. So, it was very should be homeomorphism here antipodal points X n minus X are mapped to the same point ok. This quotient is easier to understand namely it is under the antipodal action X goes to minus X and both of X n minus L go to same point ok. So, in particular why I took this example now it will follow that P n is compact ok we did not have that one then P n is compact here. So, only thing we had to bodily verify is that P n is house doll which is not very difficult to verify the entire exercise here you do not have to worry you have to just tell it P n is house doll that much you have to verify then only this theorem can be applied ok. So, now I would like to take a different game here recall recall that we had defined an embedding long long back maybe of a topological space into another space Y. What is the meaning of an embedding? It means a continuous injective map such that when you restrict F from X to F X not the whole space Y this is a homeomorphism where F X is given a subspace topology from Y. So, that was the definition ok beyond definition we have not done much now at least we will have some examples here. So, now I specialize to the case wherein n is equal to 2. So, look at the projective space P2 of dimension 2 which is quotient of S2 by anti-borderly action ok. What I want to do is I want to explicitly write down a embedding of S2 inside R4 sorry embedding of P2 inside R4. S2 is embedded in R3, S2 is a subspace of R3 right by definition. But P2 I would like to embed inside R4 why because for some reason I am not able to embed it inside R3. Actually a deeper theorem in algebraic topology will tell you that you cannot embed P2 inside R3 ok notions such as orientability etc. I have to be studied to understand that ok. So, to get such a function what I should do I should construct a function from S2 to R4 such that F X is equal to F Y if and only X is plus minus Y then I will get a map F bar from P2 to R4 ok. If this is if it only if when you come here it will be already injective mapping. If F X equal to if it only if X is plus minus Y then the corresponding function F bar from P2 to R4 will be injective. Once again this is compact that is Hausdorff. So, F bar from P2 to F of P2 F bar of P2 that will be homeomorphism which means F bar is an embedding. So, task is to find a function F from S2 to R4 which has this property points are mapped to same point only if they are anti-border otherwise they are distant this is what I have to do ok. So, hunting around various examples it turns out to be a pleasant surprise that we can do it with a quadratic embedding namely you know when you whenever you have plus minus X going to same point. So, the natural thing is to look for quadratic function X square Y square X plus Y square X square plus Y square and so on the combinations. So, they will have this property right homogeneous quadratic and a hunt like that will is actually giving you that. So, naturally you can try X Y Z or 3 coordinates here. Remember I am restricted I am only interested in Z S2 ok not the whole of R whole of R3 this will make sense in the whole of R3 no problem because they are polynomial functions. Look at X Y X Z and Y Z these are obviously three easiest things of course I could have taken X square Y square Z square but there will be problem suppose I take X Y Y square Z square would it work. Finally, even X Y X Z Y Z also does not seem to work I have to take one more coordinate namely I pick Y square minus Z square ok. After that it is a matter of checking that when you pass down to P2 obviously. So, if you replace each X Y Z with minus X minus Y minus Z then this side RHS here remains unchanged ok do not just change X to minus X that is not the idea that is not required when you change minus it is minus X minus Y minus that. So, that is the antipodal point of X Y Z. So, that goes to say a point therefore, this will give you a function F bar from P2 to R4. Now, here to see that function is injective that is all ok. So, that part I am going to leave you as a pleasant exercise verify it ok. I will go to another interesting example once again this also something like a non-orient table surface P2 was once such this also cannot be embedded in R3, but similar efforts to P2 seem to work here also and we will get a embedding in R4 ok. So, first of all I have to explain what is this Klein bottle K ok. It is a quotient a double quotient 2 to 1 mapping is there of the torus H1 cross S1 ok what is the action? Action is important here by the diagonal action of a of minus 1 plus 1 Z2 ok.  So, this one is is the multiplicative inverse ok. First of all let us work out a geometric way of obtaining Klein bottle out of the above definition suppose I take this as definition ok this is not a standard definition, but now let us say using the well known geometric way of getting a Klein bottle out of the torus. The torus S1 cross S1 is defined as the quotient of a rectangle I cross I where in the four sides are identified in a particular way the 0 comma 1 cross 0 is identified with 0 comma 1 cross 1 so that is the opposite side right in the same oriented fashion. Similarly, 0 cross 0 comma 1 will be identified with 1 cross 0 comma 1 ok again the opposite side that is the identification here this is the square 1 cross 1. So, this bottom thing goes to the top thing here see the the double arrow and double arrow are indicating that is a triple arrow on vertical lines they are identified this is the identification the arrow shows tells you how the identifications are done for example here an element can look like t comma 0. So, where it will go it will be identified with t comma 1 similarly 0 comma s will be identified with 1 comma s here ok. So, this is the this is the once you identify this case the torus, but to get the Klein bottle I have to do some more identification because on the torus I have an action u going to u inverse and v going to minus v right. So, that is what I have identified here I have I have tried to put it here ok. So, what happens is if you look at this arrow the bottom line here this will get identified with the middle one dot dot dot dot 1 in the opposite direction these these elements will be there all the time there is no identification there ok, but what have further identification what happens is this will be identified that part ok this is getting identified this way anyway alright and this is getting identified anyway all these points here they will be identified corresponding points here something here will come here and so on in the opposite direction around the half of this one there will be self-identifications. So, there will be no identification sorry within the interior of this half rectangle everything above will be identified some point here. Therefore, the upper half part of this rectangle is unnecessary. So, I have cut it off and represented a half rectangle here of the original one now the identifications of this one will give you kind order. So, this is the geometric way ok to justify all these things rigorously you have to write down formula that is the only way ok. So, that is what we have to do ok.  Since, u and v are now representing elements of S1, S1 cross S1, but I am representing them on the plane only after identification name t going to equal 2 pi i t you will get an element of S1. So, what is the corresponding identification of this one in terms of real coordinates? So, I have to convert that and that amounts to the following thing t if you represent t the element u is represented by t means what u equal to e power 2 pi i t ok. So, this v is represented by S means v equal to e raise to 2 pi i S that is the meaning of this. So, when you get that u inverse will be represented by 1 minus t ok whereas, minus v will be represented by S plus half ok this is very easy you apply e power 2 pi i t what happens e power 2 pi i 1 minus t is just what is just the inverse of e power 2 pi i t and this is a minus of that because half of 2 pi i 2 pi i half will be pi i. So, e raise to pi i is multiplied by minus 1 ok. So, that is all I have done and the second part is see t this is in the in the t first half in the second half t comma S will be 1 minus t comma S minus half because plus half goes away out of that one. So, I have to take the half less than 1 less than 1 I have to take S minus 1 these two are the same effect, but the actual map will be S minus 2 because I have to be between 0 less than 1 less than half. If S is bigger than half S plus half will go out of 1 right. So, that is why I have to write it. The entire quotient map can be restricted to S equal to 0 1 cross 0 half you do not need a second part at all this part. Now, in the interior of S there are no identifications let us check what are the identifications on the boundary ok. So, this is what I have already done the two vertical sides have to be identified as indicated by the arrow. The lower horizontal arrow 0 comma 1 cross 0 gets identified with 0 1 cross half in the reverse direction why are the map t comma 0 going to 1 minus t comma half. I have to just understand what is happening here when you put when you put S equal to 0 and S equal to half ok. So, this is what happens thus the paper model of Klein bottle is given by the rectangle on the light hand side here in the picture which I have shown paper model means you have to indicate how you are going to identify the boundaries that is all ok. So, this is the explanation of how to construct a paper model of Klein bottle all right. But now I go back to my definition of the Klein bottle as a quotient of S 1 cross S 1 by the identification namely u comma v identified with u inverse comma minus v ok. Once again this time I am denoting this A B C D etcetera inside R 4 itself because S 1 is embedded in R 2. So, S 1 cross S 1 is embedded in R 4 from R 4 to R 4 I want to take a map such that whenever two points are identified they are going to same point if and only if. So, that is the same technique. So, what do I do take A B C D remember these are all A B A square plus B square is 1 C square plus D square is 1 that is the condition in R 4 for this to represent S 1 cross S 1 ok. Let it go to 2 plus A C square minus D square 2 C D into 2 plus A again then B C B D ok and take its interest in this restriction on S 1 cross S 1 this map is completely defined on the whole of R 4. But I am only interested in S 1 cross S 1 means put A square plus B square equal to 1 and C square plus B square equal to 1 that is all. So, once you put that you can think of A B C D S then sin theta cos theta cos theta sin theta and cos psi sin psi something like that also you can find ok. Theto action in this notation corresponds to A B C D right A B gets A minus B this involves C D goes to minus C minus D. So, it is negative. Now, it is a matter of straight forward verification to check that F factors down to define a continuous injective map F bar from K to R 4 which is just by the similar argument he is an embedded ok.  So, you put one more coordinate here to get an embedding of R 4 of the project of the plane border ok. So, this 2 plus A corresponds to you know the center of one of the circle is shifted to 2 comma 0 see and this 2 plus A 2 plus A and then this is cos 2 theta and that is sin 2 sin 2 theta. So, you just take instead of theta 2 theta. So, that that will give you when you help to give you map into P 2 that is a slight modification there ok yeah. So, so we have done something nice today let us stop here. Next time we will take 2 more properties which are very very important again the separation properties regularity and normality. Thank you.


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song on it. Or Pellet. Oh my god. That's a Daniel K. Tape as well. Yeah, I look so hot. Yeah, let's go. You just got this one, Dave. It was. What? Congratulations. Yeah, quote that.  Yes, guys. Close your throat. I'm begging you, please. 11, 1, 12, 2, 13, 3. Pass must go back into the area. Yes, yes, yes. One difference. The first exercise we're going to ask you to do is an RDL, which we will do eccentrically, because we haven't done those yet. Make sure you hit it. Slow down.