In today's video, we're going to talk about the top 10 most affectionate cat companions. Even though cat fanciers have been selecting and breeding cats specifically for their unique appearances, some cat breeds seem to be distinctly friendlier and more affectionate than others. A common argument used against cats is that they just aren't all that affectionate. People say dogs, on the other hand, are always absolute love bugs. Actually, the truth may lie somewhere in the middle. While some felines could care less about their owners, others absolutely love social interaction with their favorite humans. Here are the top 10 most affectionate cat breeds of all time. 1. Ragdoll. The ragdoll is considered to be a native kitty to the United States. It is said that they originated in California during the 1960s. The breed was created from a long haired, like gray cat named Josephine and a black cat. It's believed that Josephine may have been a hybrid of a Persian and potentially a Berman or Siamese. The elegant ragdoll is an easygoing kitty. They are calm, placid, and love their cuddles.  Your ragdoll will be the ultimate lap cat, and because of their varied docile persona, they make perfect pets for families with children or other pets. To boot, ragdoll cats are extremely friendly around children and other pets. You couldn't find a better furry friend to add to your family. 2. Siamese. Ever wonder who the royal feline of the cat world may be? The Siamese, of course. The Siamese cat has enjoyed this luxurious and royal status for centuries. This outgoing, chatty kitten is famous for its remarkable baby blue eyes, strikingly large ears, and sleek, muscular body. The Siamese are very affectionate and intelligent cats with distinct outgoing nature. They seek and enjoy the companionship of humans and as well as other cats. They will often strongly bond with their favorite human. These kitties are very vocal and will demand your attention with a loud, low-pitched voice.  Throughout history, this look has given them an edge that helped them gain popularity among cat lovers across the world. Their affectionate personalities didn't hurt either. 3. Sphinx. Sphinx cats are the utmost affectionate cats who love entertaining their families and treasure meeting new humans. These cats won't shy away or hesitate to demand your attention. Their sweet nature is accompanied by a high level of energy. They will always find a new way to entertain you. They are extremely playful and will find diverse activities around the house to keep entertained. When they're not on the go, they love cuddling and resting on your lap. Sphinx cats serve as excellent therapy pets due to their friendly, warm natured temperament and genuine love for meeting new people. They're known to cuddle with you beneath the covers at night and fix themselves on your lap during the day. 4. Raghamuffin. Raghamuffin is a variant of the ragdoll cat established as a separate breed in 1994. Like their related breed, they are notorious for their friendly and affectionate personalities and their thick, dense fur. All the colors except for color point are allowed. Raghamuffin is the kind of kitty that will follow you around everywhere and greet you at the door with a meow. They are social and outgoing and get along with most humans, as well as dogs and other cats. They are patient and intelligent, so you can teach them a number of cool tricks. They will gladly snuggle up against you while you're watching TV or reading a book, but they will also cherish any playful activity you can come up with. 5. Scottish Fold. Scottish Folds are captivating felines who carry a distinctive feature of forward folded ears and bear a reputation of being unusually loving companions. The folded ears of the breed are caused by an autosomal and complete dominant mutation. Folds come in all coat colors, patterns and lengths. Interestingly, they are notorious for sleeping in very peculiar positions. Scottish Folds are very social and friendly cats. They don't love being left alone, so if you're a working professional, this may not be the breed for you. But if you have time to love your cat unconditionally, you've found your match. Folds are moderately active and yet very intelligent and agile. They will gladly engage in games with you and follow you around from room to room. Hello. This video is sponsored by Beemix Pets. Are you looking for high quality cat collars at an affordable cost? Check out BeemixPets.com. Use coupon code Kittenlife to get 20% off. 6. Persian. The Persian's exceptionally beautiful and graceful coat, chubby cheeks, expressive eyes, and affectionate personality make them one of the world's most popular feline breeds of all time. Sometimes referred to as the glamorous puss of the cat world, the Persian is notorious for its calm nature and cuddly personality. Persians are typically not very athletic nor playful. They prefer a serene environment and long naps on a comfortable sofa.  These kitties enjoy the company of their favorite humans, as long as they treat them kindly and gently. 7. Tonkinese. A hybrid of the Siamese and Burmese, Tonkinese cats are characterized by their lively, social nature and pointed coat patterns of various colors. The personality of the Tonkinese is very similar to its ancestors. They're energetic, playful, and demand attention. The Tonkinese is vocal, but their voices are softer and sweeter than those of the Siamese. Like the Burmese, these felines often engage in dog-like behaviors such as fetching. They absolutely love attention. In fact, they will demand it. 8. Burmese. This is a compact and heavily built cat breed originating from Thailand. They have short, dense, and glossy coats that come in a variety of colors such as champagne, platinum, and sable. This cat is famous for being strong, elegant, and athletic. The Burmese make excellent family pets due to their loving, affectionate and playful nature. Due to their clingy personalities and puppy-like behavior, the Burmese are often considered to be the dogs of the cat world. These people-oriented cats often maintain their energetic and playful kitten-like personality in older years as well. Like their Siamese ancestors, Burmese are very vocal, affectionate, and sweet. They are not independent and don't like being left alone for long periods of time. They get along very well with kids and other pets, including dogs. 9. Berman. The Berman, or famously known as the sacred cat of Burma, is a strikingly gorgeous cat with a soft, silky coat, piercing blue eyes, and elegant, white gloves on its paws. The Berman is a docile and patient cat that doesn't cause a lot of trouble. As opposed to the close relative, the Siamese, these kitties are actually very quiet, and will often greet you with the softest, most adorable low meow. Remember, these friendly cats don't cope with solitude well and will easily get lonely if left alone for too long. If you choose the Berman for your new furry end, make sure to spoil it with extra hugs and snuggles. 10. Maine Coon. Tipping the scales at almost 20 lbs, the Maine Coon is said to be one of the heaviest cat breeds out there. This strong, athletic cat is gentle-mannered, friendly, and highly intelligent. Such a unique combination of characteristics led to their affectionate nickname the gentle giant. The Maine Coon is adored for its playful dog-like nature. They are loyal and obedient, but never clingy. They often learn many tricks, can be taught to walk on a leash, and are known to be one of the few water-loving cats. Due to their wonderful personalities, they get along very well with kids and other pets. Even though they probably won't fit perfectly in your lap due to their size, that won't stop them from stealing a few cuddles here and there. They are affectionate towards people of all ages and they love being involved in all family happenings, whether that means lounging on the couch or sitting at the table for dinner. Besides their affectionate personalities, Maine Coon cats are noted for their hunting skills and high intelligence. If you enjoyed this video, kindly press the like button. Also don't forget to subscribe with notifications on, so that you don't miss out on videos like this. Thank you for watching.


We are on kick. We are live Come join us if you want if not, that's cool. Just leave a like comment subscribe turn on your post notification bells Man, let's continue to grow the family from Chicago to the UK We also got the patreon. We double uploaded today and yesterday man. We all caught up We do start Sherlock tomorrow and we also Finish voting on the new show to replace misfits as well and We also got the discord which plays a big part in kick, you know, I'm a part-time streamer Full-time content creator Excuse me. Oh my god But yeah, man join in this though, I don't know what I just clicked That's let's not do that Uh The British language is proper rubbish in it.  What are you doing?  I saw the brick. I saw the blue bubbles on there. Who was it? Who's more important than me right now? I want to know who's number one in your life right now. Who is it? You were just looking at what better be pictures of me Better be my instagram. This is his whole comedic approach He like he tells jokes, but like he gets on the crowd as well. Yeah, show me if it was me Don't you scroll to another app? You won't be real Be present That's a fact. That's a fact.  LORna Lorna Say it like in America. I don't even a real name. I see why you missed Lorna What's your last name the rings?  Is that what that was just a whole crew here?  Mr. Rife. What are you people dude? Say learn sometimes.  She did it again She just said something that was so not a word aluminum Aluminium Aluminum man, it's not a word I See in Britain at this show He might be in Britain. I mean Britain. I mean Britain.  You're wrong. I'm thinking of the spelling of it in my head. There's no I after the M Nope, that's embarrassing you kind of I'm gonna do this right now.  Oh That's good. That's good jokes. Tell her leave a like comment subscribe turn your post. I'm gone I ain't know he was worldwide. That's tough. He was on wild and out too I don't know if y'all know what wild and out is with Nick Cannon. Anyway


Statistics and Excel, Standard Deviation and Variance Population Location Data. Get ready, taking a deep breath, holding it in for 10 seconds and looking forward to a smooth soothing Excel. First, a word from our sponsor. Yeah, actually we're sponsoring ourselves on this one because apparently the merchandisers, they don't want to be seen with us.  Like our CPA six-pack shirts, I must have for any pool or beach time, mixing money with muscle, always sure to attract attention. Even if you're not a CPA, you need this shirt, so you can like pull in that iconic CPA six-pack stomach muscle vibe, man.  As a CPA, I actually and unusually don't have tremendous abs. However, I was blessed with a whole lot of belly hair. Yeah, allowing me to sculpt the hair into a nice CPA six-pack-like shape, which is highly attractive. Yeah, maybe the shirt will help you generate some belly hair too. And if it does, make sure to let me know, maybe I'll try wearing it on my head. And yes, I know six-pack isn't spelled right, but three letters is more efficient than four, so I trimmed it down a bit, okay? It's an improvement. That's okay because we'll basically build this from a blank worksheet, but if you do have access, three tabs down below. Example, practice, blank. Example, in essence, answer key, practice tab, having pre-formatted cells so you can get right to the heart of the practice problem. The blank tab, having a blank worksheet except for our data set, so we can practice formatting the cells within Excel as we work through the practice problem. If you don't have this data set, you could type it in there. This is not a very long data set, or you can use some of our other methods to create your own data set. You can also take a look at Kaggle K-A-G-G-L-E dot com to find data sets to practice with as well. Let's go back to the first tab to get an idea of what we will be doing. In prior presentations, we've been talking about our statistical calculations in a bit more detail. Prior to that, we were mainly focused on the intuitive representations of the data in pictorial format, things like the histograms and the box and whiskers. So now we're concentrating in on the spread of data around the center point, around the mean, and we're using our standard deviation and variance calculations to do that. So now we just have a different data set over here that we can practice those similar calculations related to population information. So we have the location over here and then the population data that we will be working with. So let's just get rid of this. I'm going to just format our table and then we'll do the similar calculations we've seen in the past. I'll do them a little bit more quickly here because we have seen some of these in the past. So I'm going to format the entire worksheet by selecting the triangle, right-clicking on the worksheet, formatting the cells. I'm going to go into the numbers tab or go to the currency, negative numbers, bracketed and red, no dollar sign, and we don't need the decimals in this practice problem. Removing them, okay. I'm going to make the entire thing bold, home tab, font group, bold. That's too bold.  No, the whole thing is going to be overly bold, overly bold. And then I'm going to hold control, scroll up a bit and let's put our cursor in one of these items so we can go to the insert tab, tables, group and add the table. So we'll add a table dancing ants doing their magic mambo around this thing, mambo, mambo, mambo. And then I'll double click on the title to put a space between them so that I can wrap the text. I'm going to select the title, go to my home tab, and then I'm going to go to the alignment, wrap the text and center it. And then I can make these a little smaller. I can make this one a little smaller. And that looks good. So we have the name over here, I could sort by the name A to Z, it looks like it's sorted by, I could sort by then the results, the population from Z to A or from A to Z. Or I did that backwards, Z to A or A to Z, you get the idea. And then let's do our standard kind of statistical calculations that we've been working with, make it a skinny C so that we can see it better. We need a skinny C so we can see everything else because the C was in the way, it's too large, it's taken up too much time. So let's make us a YD and then we're going to say that we want the mean, we'll calculate the mean or the average. Average, we'll calculate the min number, we'll calculate Q1, we'll calculate the median, which is going to be Q2. And we'll calculate Q3. And we'll calculate the max. And then we'll calculate our focus here, the new calculations we have, standard deviation for the population. That stands for not really this population, but the population of whatever data we have as opposed to a sample. So we're going to do the population calculations and we'll look at the sample calculations later. So remember those are slightly different, but we'll just practice the population one. Variance for the pop. Okay, so then let's do our calculations function-wise equals the average. We're going to take in our data, double-click on the average and then just put in our cursor so we hit the drop down. Pick in the data and enter. The min equals the min, shift 9. And pick it up our data. And there's the smallest one, which we can see if I sort from A to Z. And then quartile 1. Quartile 1 is equal to Q, U, Tab. To pick up the quartile, select in the range and then I need one more argument. We get to the next argument by selecting a comma and then a 1 for Q1 or quartile 1. And then the median, which I could say is quartile 2, but I'm going to use the median calculation. And that's more common. It's easier to type, takes less arguments. We'll just take the whole thing. That's the one in the middle. Then we'll take quartile 3 equals quartile 3, Tab, selecting the data. Quartile 3, I need a comma for a second argument and a 3. We'll take the max or largest number in the data set equals the max control shift. I mean, sorry, shift 9, data, max. If I sort from Z to A, we could see that one. So there's the 30,348. Standard deviation equals the ST. And I'm picking the STD EV dot P for the population. The STD for the population, STD for the population. All right, selecting that one and picking it up. And then we want the variance equals the variance, which is the VAR for the population. And there's the calculations. So there we have it. All right, if I entered a histogram so we can see this pictorially, I can go to the insert tab up top, charts and graphs, histogram, putting in that histogram. We need a gram about our toe. And we call it a histogram. Okay, that's not what it is. But there it is. So now we've got our buckets of our data down below. We can see it's skewed to the right here for our data. Now let's calculate our variance and standard deviation a little bit more longhand in a formula type of basis or table basis, which might be giving us more insights than simply using the Excel functions. So you'll recall if I type in my variance, my formula, I'll just copy and paste the formula over here is that I'm going to lower the font a bit. You could type that in if you want, but I'm not going to retype it in because we have seen it in the past. We've typed it in there, but you could do it by going to the insert tab and go into the symbols equation. And I suggest using the ink equation so that you can just write it in there and it'll type it in there as you write it in there, which is nice. So let's make this home tab. I'll make it orange. So it's a different color. And then I'll pull this up here. I'll make this black and white on the header. There's our formula for the variance black, white. And then we have the standard deviation formula, which is similar, but has in essence another step in the series of steps, which is taking the square root, which is why it's represented by just a sigma instead of the sigma squared. And we can bring that down to 12 as well. 12 as well. And let's bring it on down and let's put some orange around that. Okay, so now let's kind of do this in our table form. This should be black and white. I need a header formatting before. Hold your horse. Hold your horse for crying out loud because it needs a hug. The horse needs a hug. You've got to hold your horse sometimes because it's really, you could tell it's got a long face. The horse has a long face that needs a hug. You got to hold your horse. All right, let's put some blue. Home tab font group. We're going to make this bordered and blue. And then let's just take our data set. I'm going to copy from A to B. Copy A and B. Selecting those, right-click and copy. Bring this on over to the right. And we're going to put this in column R. I'll just paste it normal. I'll make a skinny queue. Any questions? Skinny queue? We're short on the queues here, short on questions. All right, so then we're going to go up top and say that this is going to be... We're going to start doing our top bit here. So we're going to take each of these items minus mu or the average. So that's going to be the mean or the mean or the average equals the average. Selecting the average. And I'm going to pick my data set, recalculating the average again. There's our average. Everything compared to the mean. Every data compared to the mean. X of I up to X of N, which compared to the mean. So we're going to say, all right, this is going to be the difference. Which is equal to the population minus the mean. We have some positives, some negatives because that middle point is the middle point. And therefore we need to get rid of the negatives, not with a absolute value, but we're going to square them, squared, which gets rid of the negatives, but also squares it. So that's why we got the square root when we want to bring it back down to the standard deviation as opposed to the variance. But I'm going to take the prior cell and care it. It care it. That's to the power of taking it to the power of grace. No, the two squared to the power of squared. And then so that's the whole top part, but we got to sum it up now. We could do that by going to the bottom of our table. And we can we can then go into my table designs, table style options and put a total down here. And then over here, I could just keep the total as the name. That's good. Maybe here I'll do the average calculation so we can recalculate the average again, which we see right there. We can over here do the count because I don't need to sum it up or anything. I would like you to count the cells one, two, three, four and so on. And then here let's sum that up to give a double check that it should be zero. It is because we took the distance from the middle point of all of them, some of them being negative. And then we got this large number because we squared all of them, which took away the negative, but now we get this big number. Let's calculate then the variance. So this is the squared, squared difference from the mean, which is equal to this, that's kind of like the subtotal we have thus far, which is the top part of our variance formulas that has been squared here. And so we're going to pick that up. That's the numerator, I should say, if I could stop stuttering, if I could stop stuttering. And then we're going to pick the count. We'll say divide by that thing so I can hit a divide without it beeping at me. And then the count, which is represented by N in our equation. And this is going to be equal to the 78 counting the cells, which is counting the cells. And then that's going to be giving us the variance, which is represented by the sigma. Insert and symbol, cool symbol, Greek symbol here, Greek and Coptic. But I already have it in recent, so I'm going to pick it up down there, add it. And I'm going to hit enter and then go back into a type of two, shift left to select the two, right click on the two, format the two, and make it a subscript so that it does that, making it look like a squared thing. And then we're going to do the squaring, which is which I'm sorry, we're not going to square it. Divide now this equals this cell divided by the cell gives us the the variance or sigma squared. Let's put an underline home tab font group underline. And then we can also do the the standard deviation, which is also a sigma, but not squared. Insert symbols were in the Greek and Coptic. There's the sigma there. There it is down here, either one. Pick it. Pick the sigma instead of your nose. I don't know why I had to say that is I couldn't raise it anyways. This is going to be equal to the square root, the square root. You probably weren't picking your nose in the first place. I didn't mean to accuse you of picking your nose. I felt like saying that. So that comes out to 5134. Let's go ahead and make this border blue home tab font group blue and bordered. So there we have it. So we've calculated this a couple different ways. Again, just notice that when you do it this way, sometimes those separate steps can give you more intuition about what your what these numbers are actually doing. And they can give you sometimes the mid steps actually give you value in and of themselves. These numbers are often useful as well when we're comparing different sets of data, right? So then we can because then we're going to be comparing like the variance and the standard deviation compared to, you know, a similar possibly similar related variance and standard deviation. And so we'll take a look at that more in future presentations, but we can see we got the 5134 here, which ties out to the calculation 5134 here and the variance of the 2696. They're trying to dyslexify me there. The 26, the 26 variance here. Hold on a second. Something is wrong. Let's go back on over here. I think I picked up the wrong variance because if I go into here, I picked up the variance for S instead of P. Okay, so then that should be 2635388. See, they tried to sneak that by me by dyslexifying it with the 696 thing. But no, I totally saw who's I did it myself. I did it to myself. I don't I tried to trick myself by doing that, but no, so there, there it is.


So Chungyeongbing is one of my favorite Chinese dishes. Our mom, Shandong. That's why I said it. That's how you knew. One of the best slices of pizza in New York City. It's the best. It is the best. It is the best. See, this is one of your first times in New York, right? What do you think about the area? I just want to eat right now. What's going on, guys? As you may or may not know, we live in New York City. And the number one question that people usually ask you is, what neighborhood do you live in? Now, as far as building, we live in the One Manhattan Square, which is right behind us. And we actually made a whole video touring our apartment. But in this video, we want to tell you that we actually live at the intersection of three amazing neighborhoods, South Seaport, Chinatown, and the Lower East Side. Let's take a food tour of the three neighborhoods that are within walking distance of our building, South Seaport, LES, and Chinatown. Let's go. But you guys, we have arrived in the Seaport District. As you saw, it was a very short scooter ride. This is a historic district that was important in New York's history for 400 years. But really, since the 80s and 90s, it hasn't been popping until now. It is completely revamped. And some of the best restaurants in New York City are now opening in Seaport District. And of course, seafood. Let's check it out. All right, you guys, we're at the brand new Pier 17 building at the Seaport. Nell, it almost feels like bottom of the Venetian. They got all these brand new concepts from all these high-end chefs.  This is called a hot bay chicken. Of course, you got the S'morning Glory S'mores. They executed it just like the pictures. I'm going to go with the burger. OK, OK. I'm going to go for the fries, then. You got chili flakes on it, cheese, green onions. Mr. Dips. Oh, we're about to get a little messy. It really gives you those Americana throwback vibes. And I feel like that's what Mr. Dips was going for. Don't use no ordinary fries. They use waffle fries, very crunchy. What I really like is actually the chili flakes. Let's take one of these hot bay chicken tenders. I've never had chicken tenders look like this before. Not a lot of places that caramelize hot honey on chicken tenders.  Whoa! That's crazy, bro. Man, this was really good, but we're on the seafloor. I want to go get some seafood. This was really the foundation of New York. This is where all the ships came in. This is basically into the cobblestones over there. This is cobble fish. So we played off of that a little bit. There was a lot of water. The whole area was kind of decked for about 35 years. Nobody wanted to live down here. You have to look at some of these homes here, those 16th, 17th century homes. You can't even get your hands on it. Right now, it's one of the biggest, hottest places in New York City. Jack, a sea captain, a historian. Can we get some lobster rolls over here for these guys? Now, I know one thing in LA that's not that popular is lobster rolls, right? Lobster roll. You know what, now I realize too. Lobster rolls, they're always better on the pier. They have excellent execution here at Cobble Fish. This has got to be one of the best lobster rolls I've ever eaten, and it's mainly due to the bread. Yeah, you wouldn't think that. You wouldn't think that that would be the highlight. So good. All right, you guys, they adopted something from LA. They said they never had fish tacos in New York City, Piers, until now. Buy fish tacos. In a way, I like the grilled fish tacos more than the fried battle ones. For sure, it still gives you some of that like tasty umami vibe, but it tastes lighter, right? From cobble fish onto the cobblestone. That's what you guys run across from Pier 17. We are in the South Street Seaport Museum cobblestone district. Man, it's already a way different vibe. I actually never been here before, so I mean, this is all new to me. This is only like a five minute walk away from OMS, and even though OMS is very much in a neighborhood place, this actually has given you some tourist vibes. Let's see what cookies they got. Funny face bakery they sold out of the Drake cookie, Kanye cookie, 50 cent cookie. Who was that, no? The VP? Yeah! Shout out to the VP. I thought they did a pretty faithful reproduction here at Funny Face. Guys, it's a solid sugar cookie. These are fun, great gifts. You can get custom cookies. Oh, you can get a cookie that looks like your face. I'm telling you guys, man, the Seaport District is such a fascinating place. It almost feels like we're in Europe right now. This is a totally different vibe. I'm just saying, messy with PSG now, I'm watching all the games here. Ho-ho! Overall, the South Seaport is just a nice switch up from the rest of the area with a completely different vibe and crowd. For a second there, you might think you're in Nantucket wearing a Nautica Polo and Sperry Top Siders. Just a two-minute scooter ride away though is one of the oldest, most diverse and most trendy neighborhoods. If you've seen our videos, you're no stranger to it. The Lower East Side, aka the LES. I personally think one of the most unique things about the Lower East Side neighborhood is when you see people, you see kids, you see families. It's easy to feel like you lack that neighborhood vibe but here, it's like 3 p.m. right now so you know, people are getting out of school and it just really feels like an actual community. What I plan is arguably the most famous donut shop in all of New York City. And I got also a bagel with box cream cheese from one of the most famous bagel spots in the city. So we got bagels and then we got donuts. The circular ring of life. Wow, look at that cream cheese. Honestly, there's so much cream cheese. I wouldn't even usually put this much. This is crazy. I have to say that's one of the tastiest everything bagels I've had, period. All right, so here you have their classic wild blueberry bagel. This is one of their most famous ones. Here you have a yuzu bagel. They sold out. Here you have the black and white, another favorite. Here you got the Brooklyn blackout. Here we have the double chocolate sourdough donut. And then here you have the pumpkin seed donut. This is seasonal, also sourdough. I've never had a sourdough donut before and this is a seasonal flavor. Yours is the dark chocolate and here is the pumpkin seed. Bouncy, spongy, slight little sourness. Oh my God, delightfully surprised. I think sourdough donuts are the next wave. Look out for it.  Like we said on Grand Street, you can get all of your favorite New York classics, donuts, bagels, pizza. With Grand Street pizza open, now you're getting very high quality, super thin crust classic New York style pizza. David, that is the tomato garlic slice. Here we have a cheese slice and here we have a pep.  You've got options.  And we're in a very particular portion of the L.E.S. right now. This area is actually called Dimes Square, not officially, but that's just because Dimes was kind of the first hip restaurant in this area and it's very, very energetic down here. All right you guys, the food has arrived here with servos and these are Portuguese-influenced prawns, so we'll check them out. I hope there's peri-peri sauce. I'm just gonna eat it with the shell on. Very deep seafood flavor. Prawns were very, very tender. I like that dish a lot. I'm gonna go for this lamb burger with anchovies guys. I've never had fish on burgers. This is a brand new item here. I've never had Portuguese mackerel. Wow, surprisingly that anchovy adds a lot of umami, slight bit of that fermented funk to it, but it kind of all goes well with the lamb. I'm not usually eating mackerel in this context. They did a really good job flipping it on a tag. Mackerel, sometimes it has a very gamey taste to it as a fish, none here. And finishing off here at Servos, we got the manila clams and of course, garlic oil butter. What I really love about this area is that they actually block off the street so they close a part of canal during the weekend so that people can sit out in the street and it really makes it like this cool open-air scene where there's a lot of people skating. There's people having drinks, people on dates. To be honest, I would say it's very hipster. Dude, these manila clams are good. Guys, this is a very poppin' part of the LES. But of course, Andrew, the LES is known for cheap eats. We are in line at Scar's Pizza right on Orchard Street. It's right around the corner. It is so close to where we live. One of the best slices of pizza in New York City. It's the best. It is the best. It is the best. See? All right you guys, you gotta come to Scar's early. They're so popular. All they had left were square slices, Andrew. You have a pepperoni with garlic and I have a sausage with garlic. A lot of people consider this to be the best slice in New York City. A lot of herbs and a lot of details and a lot of textures. They make their crust crunchy, crispy. It's got a nice bite to it. It's not too fluffy. It strikes this very interesting bounce between artsy pizza and just tasty pizza. How lucky are you to have one of the best slice spots in the city right up the street from here? Here on Sunday to Sunday joining us, we got Crystal here from LA. I mean Crystal, this is one of your first times in New York, right? What do you think about the area? I just want to eat right now. And the key is that everything here is either vegan or pescetarian, right? Which you still are to this day. Yeah, since 2009. All right Crystal, Jiao, what are you going for first? Is it okay if we get sweet first? Go for that. Here we have their French toast. Here we have their avocado toast with a poached egg.  As a vegetarian or a pescetarian, what are you looking for from a good avocado toast? Cause I'm sure that's gotta be something you've had multiple times. Spices and pickled vegetable is elevated so much more. So much flavor, there's like sourness to it. There's pomegranate. It's just a great mixture of flavor. Yellow onions grinding something. Well, this is what's really cool about Sunday to Sunday is that one of the owners is Indian and one of the other partners, I believe is Filipino and they're kind of doing their own take on brunch food out here and they're right across from an Australian spot called Dudley's. And then one of my favorite delis slash bodegas, gourmet deli over on Broome Street. Here I have the Casio de Pepe popcorn. I thought it sounded interesting. It's got a little Parmesan cheese, a little salt and pepper. Gotta get your protein in somehow, right? I used to live in the Art District in downtown LA and this has a similar vibe. It's very hip, but it's more condensed. It's a great mixture of people. Sure, you have a lot of your hipsters, your art school students, a lot of transplants, but you have a lot of local people. All right, we'll be honest. We've lived in the LES three different times in three different homes. It's in close proximity to all the other great neighborhoods that we love, including this next one, Chinatown. So let me show you a few gems in Chinatown that we have never shown you on our channel ever before. In the heart of Chinatown is one of the hottest barber shops in this city, 12 Pell. That's a fresh skin fade. Here on Canal Street, you got Canal Street Market. It's a food hall that's trying to find new ideas and find new food concepts that wanna move in there. This one is Joe's steamed rice rolls churring fund. This is fully loaded, $13, but it's got everything you could possibly want in it. And Joe's is probably the most popular and widespread rice rolls chain in New York City. Buttery, silky. I could never get sick of this. Now I'm sure you guys know by now that there is a plethora of cheap Chinese food in Chinatown, but they actually have fine dining options as well. And we are outside of a gem in the neighborhood on East Broadway, something that you really wouldn't expect. This spot is called Hua Yuan. And trust me, with white tablecloths, the lighting, the flowers, it really feels like it's on another level. This looks like Shanghai. I didn't expect this to be in Chinatown. Looks nice. Actually for the price, I wanna tell everybody out there that they have the company. Yeah, thank you. You know, I was sitting in retire and I really wanna rest on business. People who are me, we open this restaurant and I do my job. Everybody is on top of number one, top of restaurant in New York. My job is, my job is people who enjoy their dinner here in Chinatown. If people wanna find you, what should they call you? Chef. Chef Tang. I give him my call. Chef Tang, Chef Tang. Oh, tossing the noodles ever so lightly and twirls it around the fork to make sure he gets a clean bundle of noodles. Guys, this restaurant has a history that dates back all the way to 1968. It's super famous. 1930s, 40s, 50s in New York City. These fancy Chinese restaurants were all the rage. All the rage back when all the rage was a phrase. Let's go in on their world-famous cold sesame noodles. I've got to say, guys, these cold sesame noodles mahjong liangmyeon, it was just a little bit more luxurious than other versions I've had. This is only $80 at a restaurant this fancy for the entire Beijing duck.  This is actually a great feature here at Huayun. I really recommend that people come and get this. I think it's a great thing to give it to friends. All right, David, we have our peaking duck all nicely sliced up. We have our thin pancakes. This is the more traditional way to do it. Huayun's peaking duck. Cheers. The drumstick meat from Huayun's peaking duck is incredible. I can see why Huayun won the Michelin Guide recommendation, honestly. The chunggou bing is one of my favorite Chinese dishes. Our mom's? Yes. That's why I said that. That's how you knew. We're not deep-fried, we're pin-fried. All right, you guys, we are looking at a dish called thungpoh rau. Yeah. All right, pick up one bone, open one pork right here. And I put one in the sauce. All right, all right. The texture of that pork belly, when he said it was like tofu, he really means it, man. That was some of the softest pork belly I've ever had. That was the fattiest, softest, and cleanest-tasting pork belly I've ever had. This is the chef's favorite dish. Be careful hot and tempered your heart. Spicy taste them all. Wow, that's really good. I can't say that. Not too many restaurants are going to make it to us. It's salty, it's spicy, I can taste the pepper. This may very well be the best mapo tofu I've ever had. Chef Tang, he's not here cloud chasing. He is here for the culture, he's here for the food, he's here for his family. Honestly, man, that's some of the best Chinese food I've ever had in Chinatown. All right, you guys, we have arrived at the iconic Zhou Xinghai in Chinatown over the past 25 years. This place has more from a Chinatown local spot to a New York staple. It's definitely a spot you can take anybody to. I've still been eating here, I still get takeout from here sometimes. Zhou's became famous for this gigantic, I would call this New York Xinghai style. These are gigantic Shaolong Bao's. Shaolong Bao. I have the crab one. You can see a little bit of the crab roe. It's more yellow on the inside. As you can see, the pork one, more of a basic flavor. I think the preference is up to you. We are looking at three classic Chinese New York style dishes. This is Yuxiang Qianzi. This is garlic pork eggplant, Gongbao Ji Ding, aka of course, Kung Pao Chicken, and Shanghai Chao Mian, aka Shanghai Chao Mian. I made a little rice bowl here. It might be my Chinese Southeast Asian side. Honestly, and we eat so much ultra-authentic Chinese food, sometimes it is nice to get the New York style. Zhou Xinghai, it's for everybody, man. All right, everybody, that's it for our neighborhood video. Please hit that like button, click subscribe. I hope you found this video useful and can appreciate lower Manhattan like we do. If you wanna watch our detailed apartment tour of the One Manhattan Square, please click on this video right here. Until next time, we out. Peace.


Hello, welcome to NPTEL NOC, Points at Apology Part 2. Today we will start studying Manifolds with Boundary. Last time we studied what are called as Manifolds, right? So this Manifold with Boundary, they are also going to Manifolds. Actually we are going to extend the definition of Manifolds to a larger class of topological spaces. So I will touch upon this usage of this terminology a little later. The first thing is that we are already familiar with the word boundary in a different context. Namely, we employed this whenever you have a topological space and a subset of that, the boundary of the subset A is defined as the closure of A minus the interior of A. This manclature for the boundary is totally dependent on the larger topological space X wherein the subset A is sitting inside. Now the term boundary will be used in a much subtle way in a technical sense and it is going to be an invariant of the homomorphism type and it does not depend upon where the manifold is sitting inside. So having said that much, let us come to brass tacks here. The important thing here is even to make this definition, we need the full force of Brauer's invariance of domain, namely remember this one, not the weaker form but the stronger form, namely if you have two non-empty open subsets of Rn which are abstractly as topological spaces, namely take the subspace topology from Rn. Suppose they are homomorphic to each other, if one of them is open then the other one is also open. This is a completely non-trivial statement, a profound statement and this is what we will need. We have used only a weaker version here, namely Rn and Rm cannot be homomorphic if m is not equal to n, that is an easy consequence of this one. So now we will need the full force of this. I will come to that again why we need it. So first of all, our model for manifold itself will change now. Earlier the model was Rn or equivalently the open disk inside Rn, the unit disk inside Rn because both those two are homomorphic to each other. Now we are taking the half space hn, namely the subspace of Rn such that the last coordinate of the points is bigger than or equal to 0. You can take any other coordinate because this is convenient one, only the last coordinate is bigger than or equal to 0. So this is like the half ray from 0 to infinity, half line, 0 to infinity open. When n equal to 1 this will be precisely 0 to infinity, 0 closed infinity open. So we are allowing that boundary point 0, the boundary in the older sense namely this point 0 will be a boundary point of the closed 0 to infinity inside R. That is the starting point, that is our model now. So in hn for example all points wherein the last coordinate is 0 that is subspace of that, that is going to be the boundary of this hn inside Rn. But once we have taken this one you can just forget about Rn and use this model hn to define our manifolds. So let us see how. Let us start with a topological space x, we will call it a manifold width boundary. So I am not defining manifold, I am not defining boundary, I am not defining in the word width here but I am defining manifold width boundary, this entire phrase as one single technical phrase. So a manifold width boundary if it is second countable house door and the local Euclidean part is modified here now. For each point x inside x we have a neighborhood ux and a homomorphism phi from ux to hn. Now you can take this as the whole of hn also no problem on to an open subset of hn that is more convenient that is all. We know that just like ux to Rn I could have taken in the case of ordinary manifolds right or ux to open subset of Rn or ux to just tell the open disc bn all that those three things are equivalent. Here also in the homomorphism could be ux to the whole of Rn or some open subset this open subset allows us little more freedom. Terms such as charts, coordinate function, coordinate neighborhood, atlas etc which we have defined in the definition 12.1 of manifolds remember that in the context of manifolds they all make sense exactly similarly except that now the coordinate functions phi of ux it is not necessarily an open subset of Rn but it is actually an open subset of hn. This is something funny here in fact every open subset of Rn has a copy an open subset of inside hn but not the vice versa. So Rn itself is homomorphic to the strict open upper half space here namely take all points such as last coordinate is strictly bigger than 0 right. So if you have an open subset of Rn that will be homomorphic to some open subset of hn strictly contained inside the interior of hn right. So this half space has more open sets than the full space anyway. So we can have this extra open subset here of hn which may meet the boundary which may meet the plane Rn-1 cross 0 that is the extra thing here that for this is clearly a generalization of the old definition manifolds okay. Let us denote by interior of x the set of all points x having a neighborhood ux homeomorphic to an open subset of interior of hn itself. You see there are two parts one may be just hn but this is a stricter condition interior of h homeomorphic interior of hn okay namely all xn the nth coordinate bigger than 0. So take such points then they will form automatically a manifold I mean a topological manifold in the old sense right. So why the interior is non-empty as soon as x is non-empty that is also clear because as soon as x is non-empty there is some open subset which is homeomorphic to a non-empty open subset of hn okay. If you delete the boundary part here that will be an open subset in the interior of hn which is obviously non-empty. So everything you have to see inside hn then you get this one okay the complement of interior of x in x is denoted by boundary of x. So this is a notation now I will read it as boundary of x and is called the boundary of x. So we had used this notation earlier for boundary of a where a is a subset of x if x is the topological space in which boundary of x is taken then this is empty. So this has no other meaning there right because the closure is whole of x interior whole of x. So boundary of a boundary of the whole space inside itself is empty all right. So here that is not the case x is the space it is not contained in anything okay this is a topological space on its own. Now boundary of x consists of those points which are not in the interior of x. So interior of x also has a different meaning here okay. So this is in the general topological case interior of x inside x would be whole of x that is not the case here. Here also it can happen that but then x will be a pure manifold not manifold with boundary okay. So I am talking about this one it may happen that boundary is empty just like in the case of manifolds okay which means that x is precisely manifold this is it still known. The points of boundary of x are characterized by the following property. So this is where you have to be you know you have to use that the browse invariance of domain. Namely there is a neighborhood ux of x in x such that we have homeomorphism view from ux question on to an open subset vx of question such that the nth coordinate of this point nth coordinate of vx this itself vanishes. So the Rn minus 1 cross 0 should be hit by this v okay take such a point such that vx is inside the Rn minus 1 cross 0 okay. So those are the boundary points. Now what do you mean by characterized let me tell you that okay so this is where I have to use the browse invariance of domain. So let us see start with such a point now suppose there is another neighborhood ux prime and another homeomorphism side from u prime x to hn similar to that one on to an open subset vx prime of hn such that the nth coordinate vx is now positive either it is positive or it is 0 right it is greater than equal to 0 always. So now it can this happen the question is the moment one this thing happens this other thing cannot happen that is the conclusion that is why this is a characterization. So let us see why this this cannot happen once you have another side like this you can restrict you know this psi to ux prime to a smaller neighborhood we may assume that psi nth coordinate of this psi x psi x of whatever this one is positive for all y inside the smaller subset by continuity of psi. If once it is positive in a smaller neighborhood it will be positive for all the points in that neighborhood that is all I am using and this ux prime is contained inside ux. This will imply that vx prime is an open subset of hn interior of hn now the smaller set and hence any subset of open subset of interior of hn is open inside rn because interior of hn is open inside rn. Now consider f equal to p component psi inverse from this open subset they are all inside our vx prime is inside where it is inside x right inside x. So psi inverse of p come back to f to this one so now what you get is this is inside hn which is a homeomorphism but by by Brauer's invariance of domain it follows that f of vx prime remember vx prime and f of this one these are both subsets of now rn this part is in hn this part is quickly inside interior of hn okay. So f of vx prime must be also open subset right and is contained inside hn but this contains a point vx which has its nth coordinate 0 so that is absurd because an open subset of rn contained inside hn cannot have a boundary point there in rn minus one the moment rn minus one hit if you take any open disk around that point it will flow out there will be nth coordinate will be negative also for some points. So it will go out of hn right so that is absurd okay so that is not possible. So the conclusion is that the definition of boundary of x as well as interior of x is now well defined thanks to Brauer's invariance of domain it follows that u hat which is just a notation take phi inverse of rn minus one cross 0 okay so this may be empty but right now I am assuming that the singleton x is already there so u hat will be a neighborhood of x this is now you know homeomorphic to some open subset here because phi inverse rn minus one cross 0 intersection with whatever neighborhood we have that will be an open subset of rn minus one cross 0. So I am taking phi inverse of that it is a neighborhood of x inside boundary of x because all these points in u hat are coming here in the rn minus one cross 0 so they are qualified to be inside boundary of x okay. So what I am talking starting with a phi like this okay ux and phi where x has gone to a point inside rn minus one cross 0 the point phi n of x is 0 so with that you take phi u hat equal to phi inverse that will be a neighborhood of x inside boundary of x and homeomorphic to by the same phi will give you homeomorphic to open subset of rn which just means that the boundary of x itself is a manifold pure manifold in the older definition of dimension one less namely n minus one okay this is if it is non-empty that is all otherwise of course empty is allowed okay and this boundary of x itself will not have any boundary because now it is pure manifold this is this is not half space now here it is rn minus one okay and easy consequence of this observation is that boundary of a manifold with boundary is a topological invariant what is the meaning of that if you have a homeomorphism f from x to y x is a manifold with boundary then y will be a manifold with boundary and f of the boundary of x will be equal to boundary of y you think about it this is not difficult at all okay so boundary always is mapped on to boundary of y by the homeomorphism so restricted boundary of x it will be homeomorphism okay now I come to again this nomenclature about manifolds somewhat apologetically strictly speaking our first definition of manifold should have been manifold without boundary right and the second one here should have been named manifold with or without boundary the only problem is right from beginning before defining the manifold I have to define what is the with boundary and without boundary so one does not like that one the second point is there is a standard convention of using the smallest words to represent the most commonly used concepts so we will be studying manifolds and quite often without boundary so that should be just called as smallest so we have called that one as manifolds okay so that is all explanation for why we are making this one in any case if there is an initial confusion about this what is a manifold what is a manifold with boundary what is a manifold without boundary having spent sufficiently enough time now explanation I hope this confusion if at all has disappeared now okay let us have some examples any closed dick in our n is a manifold with boundary so we have to we have to prove that one since any two of them are homomorphic to each other it suffices to show that the standard unit disc dn inside rn is an n manifold with boundary in fact boundary itself will be the n-1 dimension the sphere all x at norm x equal to 1 okay we will prove that let us look at first the case n equal to 1 then this closed disc is nothing but minus 1 to plus 1 the interval right you can write it as minus 1 to 1 open union minus 1 open comma 1 okay I have written it as two open subsets of this closure interval each of them is clearly homomorphic to 0 to infinity 0 closed infinity open and that is h1 so I have given a atlas there are two charts here okay so I have given an atlas over in the general case what I will do instead of you know subtracting this one and adding this one and so on there are two of them I will do this one dn minus the top point the north pole whatever 0 0 0 1 and dn minus the bottom part 0 0 0 minus 1 so this is similar to that but in n dimensional version okay and we claim that these two open subsets of dn are homomorphic to the entire of hn just like this one okay carbon copy of that one n dimensional version that is all so let us denote shorter notation n equal to 0 0 1 and this point will become minus n let us consider one of them u equal to dn minus this capital N okay this one the other one is homomorphic to this one by just taking the nth coordinate x going to all other coordinates same nth coordinate whether it is a reflection all right so it is enough to show that this is homomorphic to hn all right so we have ready made map here actually only thing is we have to see how to use it all right so first what I do I will identify this disk dn with a subspace of rn plus 1 indeed a subspace of sn itself see sn is sitting rn plus 1 this is inside rn so I need one more coordinate I will put it in the first one here or the 0th coordinate the rest of the coordinates are exactly as x1 x2 xn okay the 0th coordinate is 1 minus summation xi square and take the square root why I have done this if you take now squares of v all these that will be equal to 1 so I will be inside the unit sphere here in sn plus in rn plus 1 okay that is all clearly this is a continuous map is actually the graph of this function okay this function x1 x2 xn go into that to take this f is just like x comma this one so it is like a graph okay so you can rewrite it as y1 y2 yn plus 1 put fn okay equal to 0001 okay that is n prime that this n prime is a notation f of capital N is nothing but 0001 I do not I am not defining this one this n prime is just another notation it follows that f defines a homeomorphism of dn minus n the whole thing is defined the whole if you throw away that it will be a subset of b minus n prime where b set of all y is in sn the first coordinate y1 this part is greater than equal to 0 now this is a positive square so this is not the full sn it is only half the sphere right given by the first coordinate greater than equal to 0 so this is the preparation I have made now now what I do I take the stereographic projection stereography projection is defined from sn minus this n prime the north pole to entire of rn so that is a homeomorphism we have studied this one carefully restricted to the half disk it will go into the half disk h here half space h here namely y1 y2 yn plus 1 okay in the rn plus 1 the y1 coordinate is always greater than equal to 0 but the last coordinate yn plus 1 is 0 because stereographic projection so this is clearly homeomorphism to hn the only thing instead of last coordinate yn greater than equal to 0 I put y1 greater than equal to 0 so you have to interchange those two coordinates so look at these methods I mean these are important methods in just inside you know handling various subsets of rn that's all so more generally any convex polyhedron in rn being homeomorphic to dn is an n-manifold with boundary details are left to you as an exercise why I don't want to get into what is the definition of convex polyhedron and so on if you don't know you will have to learn it somewhere else that's all that is not central to the theme of this course note that the manifold boundary of dn and its topological boundary as a subset of r in coincide namely they are the unit sphere right in both the case indeed this explains why the term boundary is used in general topology you know the classically these manifolds were defined and studied even without definition perhaps even before the point set topology was conceived forget about the boundary of a subset and so on so boundary of a subset that that concept is a copy from examples like this okay unfortunately we are learning it the other way around for the same reason as for manifolds without boundary manifolds with boundary are also matrizable and hence para compact why because we have assumed second countable host dorm and whether they are inside you know homeomorphic to open subsets of rn or hm they will be locally compact okay so they are t3 second counter t3 space they are matrizable we should often use the word manifold to mean manifold without boundary the old definition okay so that is why this short term manifold often the results that we state for them are valid for manifolds without boundary as well though we cannot take them for granted there are some results which are not true at all and if they are true you have to prove them separately sometimes it requires quite a lot of effort as compared to manifolds without boundary okay whenever things are not true at all we will try to mention them separately okay so here is an easy consequence of existence of partition of unity because they are we have shown that they are host dorm and para compact okay we shall now obtain the so-called collar neighborhood theorem which is a very useful result on its own so that is an important result about manifolds with boundary let me make a definition first let x be a manifold boundary of a non-empty now okay otherwise the rest of the definition is actually true it does not hold at all so start with the manifold with boundary non-empty by a collar neighborhood or simply you can say just a collar of boundary of x inside x we mean an open set you okay open subset of x okay with a homeomorphism p from u to the boundary of x product with zero infinity this is a topological product boundary of subset subspace of x okay you must have a homeomorphism with one extra property namely for all x on the boundary phi of x should go to x comma 0 okay there are bound u is a neighborhood of boundary of x so boundary of x is sitting inside here right so for those points phi of that must be just x comma 0 so such a open subsets open neighborhood with a homeomorphism will be called a collar neighborhood now there is nothing special about the choice of this this half open interval 0 to infinity you could have taken 0 to epsilon because we know 0 epsilon is also homeomorphism infinity that is all indeed once u is a collar neighborhood as above you look at these subspaces namely phi inverse of boundary of x cross 0 to r take the inverse measure that will be another open subset which will contain boundary of x and it will be again homeomorphism boundary of x cross 0 r obviously which you can you know you can compose with another homeomorphism make it 0 infinity to satisfy this definition for h n the entire h n itself is a collar for boundary of h n obviously h n is a manifold with boundary right so what happens in the case of h n its boundary you know has a neighborhood which is the whole of h n that h n is homeomorph to r n minus 1 cross 0 mainly boundary of h n cross 0 cross r n minus 1 okay so this is the neighborhood the whole h n is a neighborhood similarly if you take the d n minus a concentric ball ball is center r center 0 radius r r smaller than 1 of course okay take a ball of radius r closure ball subtract it that is a collar neighborhood for boundary of d n that is namely the sphere s n minus 1 okay for an interval a to b okay closure interval a to b a to t union s to b is a collar neighborhood of boundary of a b what is boundary of a b where a b is an interval close interval little a comma little b right just the two elements okay that has a neighborhood like this a to t comma s to b where t and s are like this namely a less than t less than s less than b so that is the example now here is the final theorem so pay attention that x be a manifold with non-empty boundary and w contained in that x be a proper open subset in x such that boundary of w is contained inside w I do not want this w to be the whole of x okay then there exists a collar neighborhood u of boundary of x such that this u is contained in u bar contained inside w and the complement x minus u is homeomorphic to x again so throw away the collar neighborhood whatever left out is again homeomorphic to the manifold itself okay the collar is always like boundary cross an interval right so you you remove the open half open interval that will be again homeomorphic to x that is the that is the meaning of this one here and such neighborhoods can be chosen as small as to please inside an open set this w itself is a neighborhood okay then you can take u inside that just a u bar inside that all right so the proof is likely longer but not difficult if you understand what is going on the two examples three examples etc whatever I have given they are the guiding principles here let us look at what happens in that place of zero to say one okay open so zero is the interval zero is the boundary it has a neighborhood no matter how small you take zero to some epsilon which is homeomorphic to zero to infinity if you remove zero to epsilon whatever is left is again a closed interval proper closed interval not a sing routine so it will be again homeomorphic to the whole of zero one right okay similar to that that is what is happening but we have to work a little harder here we need to use partition of unity also because we are not assuming compactness if you assume compactness you can probably write down a much shorter proof over to you after after you learn this proof that will be left as an exercise to okay so idea here is a very very beautiful idea here what we do is okay we attach an external caller to x I am going to explain what the meaning of this one namely you know you are taking a larger space you are creating a space larger space containing x okay so the rest of the extra space is boundary of x cross in interval minus one plus one minus one zero so that is what I am doing so I am taking y as a quotient of disjoint union of x and boundary of x cross minus one zero if you take disjoint union that is not not much fun what I do I identify boundary of x cross zero this copy with the boundary of x here okay by identify x comma x belong to here with x comma zero here okay the rest of them is floating outside okay observe that y is also an n manifold which is boundary homeomorphic to boundary of x cross minus one this time so this portion becomes the boundary now the point the all the points x comma zero they have become interior points right the idea is to define a homeomorphism f from x to y which is identity outside this w prime and then take u equal to f inverse of boundary of x cross minus one to zero clearly then u bar will be contained inside w and x from here clear is a homeomorphism okay so what is this w prime I will explain this is a small subspace of w chosen right in the beginning like this you can choose an open subset w prime so that boundary of x is contained inside w prime contained inside the closure contained inside w so here I am just using regularity and this one is a closed subset closed subset contained in open subset in between you can introduce this another open subset right so now we begin with a countable partition of unity theta i how did you get countability because the whole space is second countable why where I am working on working on boundary of x which is the manifold so that for each i the support of theta i is compact so this is one of the remark which we made because because of the local compactness okay you can assume that the support of theta i is compact okay and is contained in a coordinate open subset ui the coordinate subset can be assumed to be such that their closures are relatively compact closures are compact right and if this is this support of theta i is contained inside this one it will be automatically being a closed subset will be compact also okay so so how do I do that start with a coordinate cover namely at last go to a locally finite cover because of para compactness after that go to a countable finite countable cover because second countability use the partition of unity which exists because of para compactness automatically you will get these ones are compact okay so these are coordinate neighborhoods i have phi i from ui cross 0 1 to vi these are coordinate neighborhoods for for points inside boundary of x okay now I am going inside the manifold x itself by using that these are all neighborhoods of this whole thing is a neighborhood of uh this w prime is a neighborhood of boundary of x and ui are compact ui bar are compact so I can extend this neighborhood to 0 comma 1 right phi i from ui cross 0 comes some see this is the first thing here because of the compactness you can extend it to a small neighborhood 0 to 1 to vi is vales theorem okay homeomorphism phi i to some rn is there now rn minus 1 is there now you can take an open subset in rn okay open subset vi these vi's are inside w prime of x everything is working inside of such that the phi i cross 0 0th coordinate x comma 0 is x inside vi prime for all x so these ui phi i's I am actually writing as a parameterization they are you know parameterizations for points inside the boundary but the parameterization is occurring for neighborhoods inside x itself okay so you choose such homeomorphism there is no problem now comes the construction so this is all preparation how to choose these things etc now put eta not equal to 0 I am going to do some uh what is it inductive construction so starting point no you do not do anything in or not equal to 0 eta k is the sum of theta i is up to k i rank to 1 to k remember these are just non-negative real value functions so it can just somewhat sum it up finite sum I am taken and for k greater than 1 that is a definition next put this z k is a space x comma t such that x is inside u k the u and u k's are open subsets of boundary effects x is inside u k the t varies between minus of eta k minus 1 to 1 after all all these eta k are taking values between 0 and 1 okay so I take minus of this less than equal to t less than equal to 1 up to there I so this is the interval strictly contained inside minus 1 to plus 1 this part is plus 1 on the positive side the negative side how much you can go at the most minus 1 okay so these are subsets of now u k cross minus 1 to plus 1 okay but much smaller so x must be only here now z k prime is all x t such that the same thing but use eta k here eta instead of eta k plus 1 so since eta k is slightly you know bigger because one more function is as been added here this will be slightly larger interval than this one only on u k so these are defined only on u k all right now take alpha k from z k to z k prime be the homomorphism which linearly stretches the segment singleton x cross this interval to this interval so linearly stretch it means what 1 goes to 1 eta k minus 1 x goes to minus eta k of x you know we have linear homeomorphism mapping any interval a b a to a prime b prime right so take those homeomorphisms for each x you do that automatically because you can write down the formula in terms of eta k minus 1 and eta k here so it will be automatically a homomorphism on the whole space okay so put y k see again I am defining these things inductively right put y k put x union this x comma t where eta k is taken up to 0 only okay so portion of z k is taken z k prime z k prime is taken only up to t less than equal to 0 okay that is your y k namely these are extra spaces you see that is what okay but take it with union text okay so they are all subspaces of y now that beta k from z k prime to y k be the embedding by given by beta k of xt is phi k of xt where t is positive remember this phi is ui cross 0 1 so the second coordinate is here non-negative so that is what you have to do so non-negative you take phi k when it is negative from 0 minus 1 to 0 you take it as xt itself okay identity map itself so that makes sense because they are also say we are working inside y now so it will go inside y k because this beta k is defined only up to minus eta k for t negative not less than equal to 0 you define this when t equal to 0 the two definitions coincide okay so we have got these embeddings of z k prime inside y k we define g k on the image of beta k intersection with y k minus 1 to y k by the formula g k of beta k of xt put beta k complete alpha k this alpha k is stretching k the smaller space to larger space okay so first stretch it and then take beta k every member here is this is the image so it is beta k of xt okay so take that xt stretch it and then again take beta k so it is like first you take beta k inverse instead of writing inverse I have just written g k of every element here is after all unique element like this beta k of xt if you write it as some y then this will be this xt will be beta k inverse of that so that is more interesting than writing so this write like this so g k is defined like that okay if xt is not in the support of theta k then eta k will be 0 right so if eta k is 0 what is this summation it will be eta k minus 1 so that point will be inside that k not in that k prime it will be here though I put this one okay for points wherein x is not in the support of this theta k eta k is eta k minus 1 therefore alpha k of xt will be just xt therefore g k of beta k of xt will be beta k of xt throughout outside for negative things it is just xt all both of them xt but now because of this one g k of beta k of xt will be just beta k of xt because this is identity so the stretching are extending one from the next one that is important okay now we define f k from y k minus 1 to y k see on subspace is this beta k so we have defined okay now for the whole full y k to be g k on image of beta k and identity outside image of beta k okay so this is very important here now because these things are just stretching things right so on this on whenever it is minus equal to plus 1 it is identity already there okay so on the press when it is identity so also when theta k is 0 for for points which are outside the support these two are the same also right so therefore what happens is that is an important the point is that once you are outside this image of beta k okay on the on the boundary of beta k itself which is 0 therefore on the outside of this one so you can just define it as identity so extend its identity then it will follow that f k is continuous indeed since each alpha k is a homeomorphism it easily follows that f k is also a homeomorphism finally there are infinitely many of them I am putting f equal to the compositions of these f i's in that correct order okay the reverse order so what is the meaning of this how do you explain that look at any point okay f 1 is defined f 2 is defined after the return stage it will become identity why because this f k once the point is outside theta k outside the support of theta k this f k will be just identity so you are taking only the composition up to here stretching stretching stretching along the these intervals vertical intervals x will be fixed x comma t goes to x comma t prime that is the kind of homeomorphism we have here okay so this makes sense because at each point only finitely many functions are are not identity after that it is identity so this is like infinite product when you take most of them equal to one infinite product of elements I mean real numbers that similar to that okay so note that compliment on the compliment of v which is p i union p i of u i cross 0 1 take the union and therefore outside w prime f is identity on v itself f makes sense since any point x belongs to boundary of x there are only finitely many i for which x belongs to u i and this is like local coordinate locally finiteness and so on okay and f k of x t is equal to x t if x is not in u k x is not in the support of theta k then already this f k is identity but these things may be they are all inside once it is beyond u k that is all theta k's are so theta on theta theta t are all 0 that is the whole idea indeed in a neighborhood of x all f k are identity except those k for which theta k is not 0 right so that is the whole idea so for this reason the map f is also a proper map proper map why it is outside you take any point here then the neighborhood it is the compact neighborhood and outside that it will be identity so this goes on like that okay so only at each you have to take some compact neighborhood of boundary of x okay then the rest of them you know these points where you can use this formula it will be identity so the changes are occurring only inside a compact neighborhood for each compact subset you will get a compact subset of for each compact subset of boundary of k you will get a compact neighborhood something 0 1 minus 1 to plus 1 inside that everything is working here outside that it will be identity okay so for this reason f is a proper mapping summation theta k is 1 it follows that f will be surjective summation theta k is 1 if you have anything less than 1 it will be hit in some k then y k to that y k plus 1 will contain that okay so it will be a surjective mapping okay stretching after all 1 will be also hit there from some point given any x some summation finitely many some this summation though it is infinite some given any x there will be finitely many coordinates theta i for which summation theta i is equal to 1 you look at maximum of this theta i at that point the point x comma 1 will be hit so that is why it is surjective since each f k is an embedding f is surjective also therefore f is a memory a proper injective okay surjective map that is why it is and it is an embedding for each embedding means what open map or closed map you can use there is an open mapping okay to see open mapping you have to restrict take an open open take a point take in our neighborhood compact neighborhood okay inside that only finitely many compositions are there they are all open the composite is open so all these study so this means that f is a memory okay yeah so that completes the proof that we have caller neighborhoods for the boundary of x the entire boundary of x whenever boundary of x is non-empty of course okay here is a easy exercise show that a manifold with boundary is connected if and only if its interior is connected this is a slightly tricky thing you have to think about this interior is connected of course interior is connected but what about the points in the boundary why the whole thing is connected that is what you have to show okay but that will make you think what exactly should I use here all right so next time we will do some more properties of manifolds mostly we will now deal with only manifolds that is manifolds without boundary thank you