Everybody forced here ripping 2122 ultimate hockey ultimate collection hockey here. Let's show the results. We'll scroll through good luck. Everybody find these late sports cars calm any time Steve see just go birds Go birds that I say go away Do we spam it again Soon enough the others are gonna show up like Sam and in Kyle and Jacob well now. I'm just time all out That guy's luminance has a pack promo as well. So the next five breaks with spots luminance plates and patches and Flawless Ryan be with spot luminance that might that we might have to do luminance before the The flawless boxes guys. Oh, yep. There's Kyle. Oh, oh, buddy. You're on thin ice already. Let me tell you All right, 21 22 ultimate Whoa, whoa, whoa, whoa Here we go y'all so how many four cards per box but Should get one auto, right? That's has one pack. Is it one auto in this? I think it's just hits and numbered cards Barl Let's go Red Sox not looking. Oh, oh foul foul We're doing a jangling jangling His name is jangling now like Bing Crosby singing or no Dean Martin Got a that's a nice one Beckman on the ultimate introductions rookie card for the wild there Like the base Rookie accents autograph is kind of the war for the wild to 99 Very nice on card auto the war Next we got some really nice gold ink Rookie's auto Nils Lung fist going out to the Rangers Very nice lung fist. There's Zach. I'm guessing Zach is another call In that household rookie jerseys of 499 is Dawson Mercer for the Devils No, but it sounds great because those two are amazing It's really good.  I can't think of it Ultimate signatures nice one here for the Knights as well a couple of nice vets William Carlson You have five cards in this pack by the way Something's extra. We'll take that though nice Carlson for the Knights there Jersey car that's that's gorgeous, dude displays Jason Robertson for the stars. That is a crazy looking design.  I'll be player worn Order just came in it's gonna be Jason G with a luminance spot luminance is under or is that five to go now? So you may end up doing some football before we get to flawless remember guys Next five breaks with spots so I'm a flawless basketball Luminance plates and patches even iMac baseball all have packed promos national VIP packed promos in them There's some gold packs going out and iMac baseball and flawless Basketball and then silver packs going out of luminance and plates So good Alex new hook ultimate introductions for the avalanche. I got to watch that. What's it? What's it called again? Life so that's gonna be hard to remember for some reason. Yeah, it's too simple So beautiful here blue autograph Wade Allison rookie auto for the Flyers That is on-card Allison eat Allison Rookie patch autograph for the Senators Igor Sokolov 14 of 99 three-color patch for the Sends and a Simon Benoit a 699 rookie jersey Ducks So good, man. Yeah, it's any Murphy any Murphy and Martin shorter like the best Oh more Lawrence. Yeah, either way Either way both hilarious Got flurry to 35 for the Hawks Ultimate signatures a rookie for the Panthers Cole Schwint Next it's a 49 Kyle Connor autograph for the Jets Kyle Connor gold ink They give him like no room to sign on this design right here Give him like the bottom right corner of the card You couldn't even sign your full signature if you wanted to and rookie Jersey up Tanner's you know to 699 for the mustard cats And they give him the thick gold ink.  Come on pack. Oh, dude Jokey Jokey Harju Jokey Harju to 15 on the Sabres card here 2020 rookies on the autograph Ryan McLeod for the Oilers Dude, that is awesome stick with me the 35 Clayton Keller coyotes stick relic That's fire game use stick relic coyotes Clayton Keller Look at that and rookie Jersey Flyers Cam York to 699 Ultimate is sick. I love a black diamond ultimate both. I love these on with a luminance spot again guys three silver packs on their luminance promo Five gold in each of the flawless breaks. We'll do some more tonight to find out where and when we'll do some more tonight. That's not good The one on the wire there is it is droney. I'd 149 Talbot for the wild Ducks with a nice rookie accents autograph the 99 Jamie Drysdale rookie ultimate auto is Jasper Weatherby for the sharks rookie Jersey Lucas Raymond for the wings to 499 and rookie Jersey for the Sabres are two rooster lineman to 699 And that player worn Jersey come on Bayo also come on offense. Can you just score some runs, please? Don't matter how much you give up being in a store non I Yeah, Casper Bjork vists ultimate 2021 rookie here for the pens Flames introductions auto Matthew Phillips Next up that's awesome to 49 for the Senators Tim Stutzler auto Beautiful hit there on the Stutzler. That is a second year students.  Just Give me a freaking boon here There we go. All right, there's a kale wears Macar to 149 or dude wears Macar to 149 For the abs of course ultimate signatures for the bolts rookie Gabriel Fortier. There he is. There's a bolt set Fortier autograph for the lightning Core is gonna get blooms job. I could see that actually that that does not totally surprise me But I don't know how I feel about that Carl Whoa, that's some fire actually that's some fire. That is four of six Jonathan Huber dough laundry tag patch auto Beautiful card that is so nice. That's gotta be game used right Yep used in an NHL game Huber dough laundry tag auto for the for the cats That is sick. That's going into a mag displays What a card and rookie Jersey is Salo to 699 for the Islanders Let's do a recap on this one actually recap this one. Okay Thane with a spot in luminance same with Andrew Luminance should be at two I think two or three Yeah, but that too guys flawless coming up got gold packs included the flawless Get that filled up all spots in the break eligible to get the gold back Give them away five of them random off five of them Again, it is also on sale. It's 400 or serial number spot monster upside Again higher risk higher reward in the serial number But the last time we filled them up there like for 2425 a spot They're 400 now big time on sale right relics of Salo Keller York just base there roost the line in Raymond York Clayton Keller stick relic Genoux Benoit a new hook base strewery Robertson Mercer in a Beckman base Auto is now of the war Hortier Matthew Phillips stutzel nice one sends Weatherbee Drysdale McLeod a Kyle Connor that is a cool Schwint Sokolov on the rookie patch auto think RPS case hit I think Maybe you can get more than one but Sokolov RPA for the sends got Wade Allison William Carlson. It's pretty nice in it And an Aho as well This auto so faded it's on cars is faded Pretty interesting used a weird must use a weird Writing tool and then a lung vest We also had the four of six game use laundry tag auto For the Panthers Jonathan Huberdo Now that's a pretty card right there. Thanks again everybody moving on to some basketball now


Scottish Fold Cat vs Persian Cat. What are the differences between these two cat breeds? There is nothing a Scottish Fold named for their folded ears likes better than to be with their people, participating in whatever they are doing. With the way their small ears fit like a cap over their rounded head, the Scottish Fold is often described as resembling an owl. The ears range in appearance from a single fold, bent forward about halfway up the ear, to a double fold, somewhat tighter, and the triple fold, lying tight to the head, which is desirable for show cats. Kittens are born with straight ears, which may or may not fold when they are about three weeks old. Wide open eyes gaze out at the world with a sweet expression. The medium sized body is also rounded, completed by a medium to long tail that sometimes ends in a rounded tip. A short haired fold has a dense, plush coat with a soft texture. The long haired variety has medium long to long fur with bridges, longer fur on the upper thighs, toe tufts, a plume tail, and tufts of fur on the ears. They may also have a rough around the neck. The Scottish Fold comes in a number of colors and patterns, including solid, tabby, tabby and white, bicolor, and particolor. I color depends on coat color. For instance, white and bicolor cats can have blue eyes or odd eyes, where each eye is a different color. The dignified and docile Persian cat is known for being quiet and sweet. The Persian has a distinctive appearance, a large, round head, large, round eyes, a short nose, full cheeks, and small ears with rounded tips. The head is supported by a short, thick neck and a deceptively sturdy, muscular body, a type known as Kavi. A Persian's legs are short, thick, and strong with large, round, firm paws. The tail is short but proportional to the length of the cat's body. A long, thick, shiny coat with a fine texture completes the Persian's look. It's long all over the body and includes an immense rough around the neck, a deep frill between the front legs, long ear, and toe tufts, and a full brush or tail. Persians can come in different looks. Some, known as peak face Persians, have an extremely flat face. Think twice about the breathing problems such a cat might have before deciding to get one. Doll face Persians are said to have a more old-fashioned appearance, with a face that is not as flat as the show Persian or the peak face Persian. Persians of all stripes stand out for their infinite variety of coat colors and patterns. Imagine, if you will, seven solid color divisions, white, blue, black, red, cream, chocolate, and lilac, plus silver and golden division colors of chinchilla and shaded silver or golden and blue chinchilla and blue shaded silver or golden, and then there are the shaded, smoke, tabby, calico, part of color and bi-color divisions. That's not counting the various pointed patterns of the Himalayan. Eye color is related to coat color. For instance, white Persians have deep blue or brilliant copper eyes, other solid colored Persians have brilliant copper eyes, silver and golden Persians have green or blue green eyes, and so on. 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. After their ears, the first thing you will notice about a Scottish fold is their habit of posing in odd positions, flat out on the floor like a little frog, sitting up for all the world as if they were a meerkat on a nature program, or lying on their back, paws up in the air. And although you might assume that their ears are less mobile than those of other cats, such as not the case. Scottish folds use those ears to communicate quite effectively, adding comments and a quiet, chirpy voice when necessary. This is a smart, moderately active cat. The Scottish fold enjoys teaser toys that test their agility and puzzle toys that challenge their intelligence. Their favorite activities include anything that involves human interaction. There is nothing a Scottish fold likes better than to be with their people, participating in whatever they are doing.  The last thing they want is to be left alone for hours on end, so they're not the best choice unless someone is home during the day or you can give them the company of another cat. Rest assured, however, that they will expect you to play with them when you get home from work or school or at least sit down so they can get in a little lap time or curl up next to you while you watch television. The dignified and docile Persian is known for being quiet and sweet. She is an ornament to any home where she can enjoy sitting in a lap, surely her rightful place, being petted by those who are discerning enough to recognize her superior qualities, and playing house with kind children will gently comb her hair, will her around in a baby buggy, then serve her tea at their parties. Persians are affectionate but discriminating. They reserve their attention for family members and those few guests whom they feel they can trust. Loud environments aren't a Persian style. They are sedate cats who prefer a serene home where little changes from day to day. With large, expressive eyes and a voice that has been described as soft, pleasant, and musical, Persians let their simple needs be known, regular meals, a little playtime with a catnip mouse or feather teaser, and lots of love, which they return tenfold. This is one cat who is unlikely to climb up your curtains, jump on your kitchen counters, or perch on top of your refrigerator. She is perfectly happy to rule her domain from the floor or more accessible pieces of furniture. When you are at work or are busy around the house, the Persian is content to adorn a chair, sofa, or bed until you are free to admire her and give her the attention she willingly receives but never demands. Come the Scottish Folds coat weekly to remove dead hair and distribute skin oils. A long-haired fold may need to be groomed a couple of times a week to ensure that tangles don't develop. The most important thing to understand about caring for a Persian is the need for daily grooming. That long, beautiful coat doesn't stay clean and tangle free on its own. It must be gently but thoroughly combed and brushed every day, and regular bathing, at least once a month, is a good idea. Excessive tearing can be a problem in this breed, so wipe the corners of the eyes clean daily to prevent under-eye stains from forming. Brush the teeth to prevent periodontal disease. Daily dental hygiene is best, but weekly brushing is better than nothing. Trim the nails every couple of weeks. Wipe the corners of the eyes with a soft, damp cloth to remove any discharge. Use a separate area of the cloth for each eye so you don't run the risk of spreading any infection. Check the ears weekly, especially if they are tightly folded. If they look dirty, wipe them out with a cotton ball or soft damp cloth moistened with a 50-50 mixture of cider vinegar and warm water. Avoid using cotton swabs, which can damage the interior of the ear. Keep their litter box spotlessly clean. Cats are very particular about bathroom hygiene, and a clean litter box will help to keep the coat clean, as well. It's a good idea to keep a Scottish fold and Persian cats as an indoor only cat to protect them from diseases spread by other cats, attacks by dogs or coyotes, and the other dangers that face cats who go outdoors, such as being hit by a car. Cats who go outdoors also run the risk of being stolen by someone who would like to have such a beautiful cat without paying for it. The friendly, laid-back Scottish fold is a perfect choice for families with children and cat-friendly dogs. They love the attention they receive from children who treat them politely and with respect, and they like to play and are capable of learning tricks. They're happy to live with cat-friendly dogs, too, thanks to their amiable disposition. Introduce pets slowly and in controlled circumstances to ensure that they learn to get along together. Persians aren't the best choice for a house full of boisterous children and dogs, but they have no objection to being the object of a gentle child's attention or to rubbing along with a friendly dog who doesn't chase them or otherwise cause them anxiety. 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.


I'm Amis the Basil Chapman as we do each and every Tuesday at 20 past the hour and don't forget folks Basil has an outstanding show here every trading day 10 to 11 Eastern standard time also has a great news there the opening call Now it's very easy to get his newsletter folks You can move to our website at TF and then you guys see a red of the featured content the opening call You just hit that button you hit subscribe you can get the opening call for one month $149 you get it for six months for 695 which is savings of $199 at 22% And you can get it for one year for 1195 which is a savings of $593 or 33% now they all come with a 30-day money back guarantee folks Okay, so bottom line you want to check it out just hit well a month six months a year Check it out Basil also has about approximately 10 to 12 archives on there That show you exactly how to ride that wave each and every day How's the Chapman? What's going on? Tom hope you had a long week a good long weekend. I did. I hope you had a great father's day. It was fun It was I know I love it. I love it So these three three-day weekends you can get used to them, you know Oh, man, I think we should have three-day weekends all the time man That's that's where it's at you get just as much worked on man Just depends on what's interesting is that the futures were open and Sunday night the futures ran up They ran down and they ran up very strongly and we followed through on Monday So even though it wasn't a market day I thought there was a really important session and On Sunday, I waited Saturday because I did a bunch of work. I need to go through and on Sunday I did my market overview video for my subscribers to my opening call And I said I like a lot of what's going on here Even if it's a short term the fact is that some kind of relief rally Should come early next week and the reason that I had looked at that was if you see this little tiny doji candle Yeah, this is the dating chart on the left of the down the middle is the weekly and the right is monthly This tiny little doji candle came about just as the unbalanced volume I don't usually talk about the MACD and stochastic older Used them all the time as being overboard oversold because the stochastic holding strongly above 95% is fantastic It's usually great. And if it's underneath 10% which is the exact opposite of being over 90% this is on the way down 10% That's usually a negative But I do use the unbalanced volume is very overboard and you can see right here on the what was that? That was a 27th of May the unbalanced volume made a high and then turned down quite sharply That's almost where we got that turn in the Dow at thirty three thousand two seventy two So a couple of things I like very much I like the fact that I went through all whoops I went through all the Dow stocks that he stocks and Why I didn't get any real buy signals many of them were extremely oversold and ready for at least a bounce That was number one number to the fact that I had this little Small doji candle which can either be a halfway market You can see this long one here was like a halfway market to the downside It could also be a reversal candle. So I like that I like the fact that the unbalanced volume Which when it gets oversold or overbought, that's the one thing they can turn with the price of whatever we're following so we went along on Friday and I needed the confirmation that we'd have a The futures would be up over the weekend and thank goodness. They were so this is a very good follow through but Until the Dow You can see this pink line here is a nine period moving average and way above it that and that's a thirty thousand eight seventy seven Way above it at thirty one thousand two forty nine is the fourteen period moving average So to get these to cross so that the pink turns green because it's above the fourteen will take Oh at least thirty one thousand five five hundred thirty one thousand six hundred So I like to go step by step. So we've got the first initial turnaround. That's important But it's still the days young we've got thirty five minutes to go anything can happen But so far this is good because I said to subscribers today that if after one thirty or two o'clock The Dow is holding strongly to the upside and the VIX is pulling back and the VIX is now at twenty nine sixty four That's the volatility index that will at least allow the Dow to close strongly because if there was a failure We would have had that by about two o'clock. So so far. This is good action So that's just one step. The other is that in looking at a number of the And I and as I say we went along on Friday already. So we're long via the diamonds, but also looking at the different Dow stocks We also took a position in IBM as a first. I can't even remember when last I ever looked at IBM as a long position But one of the reasons is look how it held look how the market The IBM was at 144 just where was it on the 6th of June Then the market took a huge dive and you look IBM only went down to 132 So my reasoning is that We've seen Microsoft Adobe and you can just go through the list of stocks from the year 2000 that just got hammered at almost 90% Or sometimes even more of a decline and then morph they they reinvigorated their company and and thought through the the process of what they need to do and Most of them went on to new all-time highs. So my thinking is that IBM in the AI and Information technology and the cloud There's a chance that they have finally got something right So I'm we we took a long position and it's at this particular point There's a lot of evidence on the upside that we will need to confirm that it's going to go much higher The fact that it held well on the downside We've seen stocks that have held well and when the market broke to the upside They didn't do very much in this case. It was really important to see what it did today And it's up $3 and 12 cents at 138 14 So I'd go one step at a time But I'm kind of putting it in the Dow category as kind of for me It's a kind of a visual icon because if IBM is able by this coming Friday or early next week to actually trade in the one 43 144 area I think that'll be a really good sign that that is you can't just look at a sector You have to look at the sector leadership and it would be surprising to everyone if IBM Suddenly became a leader in the tech So I like the action and so far is acting well And we're also we're in a another position that we just added that is in the It's kind of the former Nasdaq winners that really got beaten down So we have an ETF that kind of represents that and so far today's up about four and a half percent So when selective when when we get to how do you get to a a letter a? So you identify the low bar actually let me just show you right We don't know this the low or not. That's why that's why I'm asking Oh, that's it. That's exact. That's a very good point. So my my thesis always is we use different techniques on the way down I usually use the technical indicators. I use the magnetic. I use the stochastic. I use the unbalanced volume So does it have to get does it have to get to the 14 day of the 9 day to get to a day? Or is it just doesn't know it doesn't have to the other techniques that I use but most importantly on the way up I always use the alphabet a b c d e f d and d is where other things go That's a very good question. Right. So on the way down. I use other there I use the look. This is a trough B in the IBM I use that but it isn't as important because on the way up I'm always using the letters and yes simply shows there's a starting position and he go alphabet I got it. Okay, yeah Yes, so that's I mean that's basically it. So I we're looking at this is I think this Based on the oversold action that I'm looking at I think this can last a little longer than the usual just pop up and drop So I'm hoping for that because the technicals indicate we are very over oversold and folks It's very easy to get Bowser's newsletter come over to our website at TFN You see it right into featured content to hit that button and you are off the races Guys have a great one safe one and we look forward to show tomorrow. Thank you very much. Thank you


Hello, welcome to NPTEL NOC, an introductory course on point set topology part 2. So, as promised today we shall construct the ordinals. In this section, we shall construct the ordinals and study some of that topological properties. First of all, we have to make a definition and two well ordered sets x i less than equal to i i equal to 1 and 2 are said to be equivalent if there is a bijection f from x 1 to x 2 set theoretic function which preserves the order. What is the meaning of that a 1 less than equal to b a less than equal to b through this one first relation should imply f a is less than equal to b a second relation f a is less than equal to f b. See this order preserving which we have used earlier, there must be an order preserving bijection. Obviously, the inverse will be also ordered. An equivalence class of a well ordered set is called an ordinal order. You see you take the class the collection of all well ordered sets that is not a set. However, we can define the equivalence relation by this process. Nobody stops for that. Then you can verify that it is reflexive, asymmetric and transitive. So, you can look at equivalence class. This world class is very important now here. This class itself is not a set. That class is called an ordinal. So, in what follows you are going to construct a set of representatives for a class of ordinal, not all ordinal, but a large number of ordinals. So, we will construct that which itself has a well ordered one set of representatives and that will be a set. So, this is the whole idea and that will be a sufficiently large number of ordinal and that itself that set of collection itself will be a lot. The construction of this ordinal space. Of course, when you say when there is a well order then we can take the topology also then we can say ordinal space. The following construction works well with any uncountable set. Now, you must know that uncountable sets exist. So, that part is elementary or whatever you know set theory of cardinals. That part I am assuming that you know it or I will pretend that you know it that is all. So, I am not going to teach you the theory of cardinals here. So, be sure of that. So, indeed this is not the correct thing to do in the logical sequence, but since you are familiar with natural numbers in whatever quote unquote whatever way you know you can take the power set of the natural numbers and call it as x. That will do our job of understanding what are ordinals to begin with. Strictly speaking you should not do that even n will be constructed out of this thing. So, all that you have to know somehow that there is a uncountable set. Now, so fix an uncountable set x start with a well order. So, this is Jermelo's theorem that we have proved. Every set can be well order. Start with a well order next. Then you add one more point. So, once again I denoted by x infinity x star equal to x disjoint in x infinity and extend the order on a order on x to x star by declaring that all the elements of x are less than or equal to this in that way which will be strictly less than because they are different from infinity. That will be automatically a well order and a larger one. Clearly x star is also well order. That is what we have seen this extension. In fact below and above also we have already done that and we have used that one. Now let us put s equal to all points of x star such that l x is uncountable. Remember what is l x? It is a left raise. Take all those for l x is uncountable. Then s is non-empty. How do you prove that s is non-empty? Well you can take infinity prime here. x equal to infinity prime here. Then what is l x? l x is precisely x and we have started the x is uncountable. So, therefore this s contains infinity prime and so it is non-empty. That is the role of infinity prime here. If I do not put this one I will be hard to prove why this is non-empty. In fact it may not be true. That trick is there. Just put one extra thing. Then you have got a non-empty set. You are sure. Therefore once it is non-empty you can take intima mo base belong to x star because x star is the lord. Take the least element and call that as omega, large omega. So this makes sense because it is non-empty. That is all. So this omega will be automatically a member of this one. So it is s equal to infinity prime. We do not know whether this omega is less than infinity prime or for that matter it is equal to infinity prime. So this depends upon the x itself but we are not bothered about it. We have started with any uncountable set. Therefore we do not know what it is. If you take natural numbers and per set then maybe you can say that this has to be actually infinity. That is a different aspect. So we do not bother. This is an element of this x star. That is all we know. Also now let 0 denote the least element of x star. This time I am bold enough to use 0 itself. I better use 0 hat or 0 prime but that is too much of work. Whenever it is needed and later on I will do that. Right now let us just have this 0. 0 be the least element of x star. Always least element exists anyway because this is a lord. We then take the order topology open intervals constituting base as we have seen earlier on 0 close and omega open as well as 0 close omega close. Both of them are subsets of now x star. And they are themselves well ordered. So you can take this topology order topology on both 0 omega open and 0 omega close. Clearly this capital omega will be in the closure of that. That much is clear. Apart from the properties we have listed above under total order except for one thing well order we did that but other things were generally total order. They are all valid here also. Now we will have more interesting properties of this space this omega 0 omega. So basic properties of this ordinal topology. Here we are not I told you I am not interested in teaching you cardinality theory. I hope you know. So we assume that you are familiar with this. So start with omega the least element of the set of all x such that cardinality of this left ray lx is not finite. For example if I take 0 lx is empty. If I take one successor successor of 0 immediate successor of 0 then it will have only one element and so on. So that is where actually Piano said you know start if you have studied it otherwise you can you know surmise here itself. So cardinality of lx is not finite. You take all that set and take the least element. The least element is also an element of this set. So this check denotes a cardinality. Then for every x inside 0 omega that means what x is less than omega right. Omega is the least element. Less than omega it is not. So it must be the cardinality of the left ray 0 to x 0 to open x is finite. We add one more point it is also finite. It follows that if you take this check as a function from 0 omega this interval to the natural number namely cardinality is now finite. It is a natural number. Take anything here it will at least have the 0 right lx will have at least 0. So it is non-empty. So natural number will be positive. So it is all 1, 2, 3 and so on you will get. You get a function here it is an order preserving bijection. You see it is an order preserving bijection. Here we have the order that is whatever we have started with. Here we are pretend we know the order of natural number. These order okay both of them are totally ordered sets. If you add both of them are now well ordered sets also. This is a order preserving bijection okay. So in our definition this equivalence class will itself denote an ordinal and that ordinal will be omega itself. That is the whole idea here okay. Indeed if you do not know what n is then 0 omega as above you can think of that as n okay and then construct the algebra out of that by using the ordinal the successor theory. That is not I am not going to do that. So here are few terms I want to recall. Every member of 0 omega, 0 close omega open is called a finite ordinal okay. Omega is called the first limit ordinal. There are other limit ordinal obviously okay. There are many more. This is called first limit ordinal. This is some name you can say things which are not finite. Amongst them omega is the least one. That is what what you are is the first one okay. So you can say not finite. Not finite ordinal omega is the least one. So that is called the first limit ordinal. Now cardinality of 0 omega is that of set of natural number because there is a bijection like that. Clearly elements of x, elements x of 0 omega close can be broadly classified by the cardinality of Lx okay. How many elements are there before that? Look at the cardinality. That is how many means okay. So it may be finite. It may be countably infinite like natural number 0 omega or it may be uncountable. I do not want to go on further classifying these what kind of countability is there. That is a you know that is for purely further logic which we are not interested in right in this space okay. We are only interested in broad classification. All elements x less than omega are finite. Omega is uncountable and all other elements are countably infinite. All finite elements bigger than 0 are immediate successor of the previous element. Omega is not an immediate successor. So all these things elementary things are intelligent. Note that there is a finer classification of ordinals and a very vast literature on them okay. You can study them elsewhere. We are not going to that. The elements of 0 omega close which are not immediate successor are called limit ordinals. I am just I am just repeating this one. Here we only define first limit ordinals but all of them which are not immediate successor are called limit ordinals okay. There are of course limit immediate successor for every element. So omega is not an immediate successor. Omega plus 1 is there. That will be immediate successor. Omega plus 1 plus 1 will be there and so on right. So again there will be another one which is not an immediate successor and then again after that only immediate successor will be there and so on. This is a wonderful space okay. Never ending. So that is why we have put this capital omega here. So that is the and this is the maximum element among all of them. There are several equivalent as well as slightly varying definitions of limit ordinals. Different names are also there. One of which we have chosen can be justified partially as follows. Why limit ordinals? Just ordinary name okay. So I am going to tell you that. Take any element in the open interval 0 omega. Suppose it is not a successor. I mean 0 is not a successor anyway. We do not know that omega will prove that soon. Immediate successor. Then there exists a strictly monotonically increasing sequence xn. Strictly monotonically increasing very important xn which converges to x. Every immediate successor has this property. Every element is not an immediate successor has this property. Because it is a limit in this then we are calling it limit ordinals. That is the justification. Soon we will see that it is not a very good justification either okay. First enumerate the countable set Lx, y1, y2 and so on. A countable set means just it can be enumerated. It can be put in the 1-1 correspondence with natural number okay. But these y1, y2 are not in the order in which you know there is an order in Lx already we have chosen. That order we do not know. This may not be the order okay. That does not matter. Just take a just an enumeration. Start with x0 equal to 0. I am constructing this sequence now. Start with x0 equal to 0. You can start with no problem. Having chosen xn, I want to define xn plus 1 inductively okay. So how do I do that? Once xn is there, look at a equal to maximum of xn and yn plus 1 here in this list. Go up to yn plus 1 okay. Take the maximum of them. Then clearly this a okay is still less than x because all these Lx are less than x okay. And this x is not equal to a plus 1 right because x is not an immediate successor. So if you even add one more you are not hitting the x.   That will be less than x. So what is this? This is maximum of these two. So I have given you what is xn plus 1 here okay. So why I can just do one more, one more and so on because x is never reached right. I can take one more, take x0 and one, two, three and so on. But you will never know whether you will reach x. So if you look at this list and I am going to exhaust this list by this method. Because once I am told x0 I am not interested in y1. I will go to y2. Bigger the bigger of the two I will take. Then having chosen something I will go to not y3, y4. I will go to something which is bigger than that one you see. Maximum of the two. So I will go to somewhere. This way I will be keeping on exhausting elements of lx. So that is the whole idea. That will imply that xn tends to x. So here is the caution. Do not be carried away by the above theorem. Omega is a limit ordinal. But it is not a limit of any sequence in 0 omega. If capital omega is not a successor. Yet there is no sequence which converges to this omega. The sequence being inside 0 omega. Of course you put omega, omega, omega that will converge. So we shall soon see why this is true. Observe that omega is a limit of the net. If you think of this 0 omega as the domain of a net. Because it is a directed set right. This is totally ordered set anyway. So 0 omega as the domain of a net. What is a net? Just the inclusion map.  Then this omega will be obviously a limit point of this. It is a limit of this set. So in this one single paragraph. I have both justified as well as cautioned you all. So I am giving you another justification for calling this one as limit ordinal. This time I cannot take a sequence but again take a net. That net has this as a limit. It is easy to see that the ordered topology on 0 omega is discreet. It is just like natural numbers. We have already seen that right. So ordered topology there is just discreet. However if you include omega it is not discreet. If you include omega something nice can be seen here. For singleton omega it is not open. Okay discreet means it is not open no. So to see that singleton omega you have to write it as maybe intersection of two left one left and right right right. That is the only way you can say this thing is open. So you cannot do that. But this one is a g delta set. So it is a countable intersection of open sets. Okay. What are they? Start with any n go to omega closed. That is an open subset in 0 omega. As n keeps increasing intersection over all n we will be left with only singleton omega. Okay. Indeed there is another way of looking at is 0 omega. There is a ordered reversing homeomorphism 0 omega to the space 1 by n n belonging to natural number include with 0. Send each x to 1 by n where n is its cardinality of Lx. Then this omega itself we send it to 0. So that will be a continuous function. That will be a bijection. It is an order reversing homeomorphism. You can take this also as the order of set. Okay. Order topology then you will have this. Okay. 0 is a limit point. So that is the way you have to think of this one. This omega is a limit point of 0 omega open. A subset A of 0 capital omega open is bounded in 0 omega. Bounded in 0 omega is very important here. Because everything is bounded in 0 omega if you include omega you know included. Okay. This is bounded above if and only if it is countable. If bounded below is obvious. So we are only interested in bounded means bounded above. Okay. Everything is bounded below in a well order set. Alright. So if it only if it is countable. Only countable sets and all countable subsets are bounded. Inside 0 omega itself. So this is the crucial thing I said to wait a minute. Wait a minute. Okay. Slowly we are building it up. So why? So in particular if this is the case then it happens that every compact subset of 0 omega is countable. Because compact subsets we have seen has to be bounded. That we have seen last time. Okay. So bounded subsets are countable is what we have to prove. Of course every subset A is bounded below. So we are only interested in things which are bounded whether it is bounded above or not. So let A be a countable subset of 0 omega open. To see that it is bounded above considered a subset of 0 omega it is bounded above. Right. What we have to show? Take the smallest bound namely least upper bound and show that it is inside omega open. Okay. That is what we have to show. See that it is bounded above whatever means inside 0 omega consider a subset of 0 it is bounded above and hence has a supremum S equal to sup of A. We claim that this S is less than omega. S is not omega. Then we are done. Got it? So how do we do that? Look at this set B union of all the left rays where X range is over A. So A is some scattered countable set. Right. So for each point you fill up all the things which are below that. So it is like a saturating this set. Everything below let them come inside. So that is B. So this is a very big set now. Okay. So for all X inside A you take LX and take the union. Then each LX is countable by the very definition. Okay. Of 0 omega. Okay.  Therefore B is countable. Okay. But now if S is this omega. Okay. Then it follows that B must be the entire 0 omega because all of them will be there. Right. If S is supremum take any X that will they will satisfy this property. This less than that one. So B will be the whole of 0 omega. But 0 omega is uncountable. Right. So if we started with set X is uncountable then we constructed this omega. So set 0 omega is uncountable. In particular this implies that no sequence in 0 omega converges to omega. See because supremum will be inside omega. Inside 0 omega. Sequence is what countable set first of all. Right. So supremum. Supremum will be if at all if it is increasing you can take subsequence is increasing that it will be supremum. So 0 omega open. Okay. Inside that itself it will be the supremum will be there. So it cannot convert to this omega itself. Alright. So this fact is useful. So this is an extra thing I am telling already I have proved that if A is countable then it is bounded right bounded inside 0 omega. Now let us prove the converse. Suppose A is bounded in 0 omega. Okay. If X is in 0 omega. Okay. Is an upper bound for A that means bounded means what? Within 0 omega must have an upper bound. Then this A is contained inside LX plus 1. Right. In the way LX plus 1 because everything is less than X. Because since X plus 1 must be inside 0 omega because X is inside omega. Okay. But LX plus 1 is countable that is definition of 0 omega. Alright. So A is countable over. Okay. So now we come to little more serious business here. If X and Y and R they need to interlace increasing sequences in 0 omega. What is the meaning of interlace? You take X1 that will be less than equal to X2 or Y1 or X2 does not matter. X1 less than equal to Y1. X2 will be bigger than Y1. And then X2 will be less than equal to Y2 and so on. Okay. So they come interlaced. Interlacing can occur in different ways, slightly different ways also. Okay. This is one. If you change the labels there you will get the same thing. That is all. So I can define like this. Then they have the same supremum. The increasing sequences in 0 omega they are countable so they are bounded. This much we have seen. The supremum is the same. So this is similar to what happens in the real number. So I will leave it to you. Okay. 0 omega claw is first countable. Now we are talking about topology now. First countable at all points except omega. So how do we show that? Say X belongs to 0 omega open. If X is 0 then singleton 0 itself is open.  Singleton 0 is open. So that itself is a base. So it is a countable base. Otherwise look at open interval Y to X plus 1. Okay. Where Y is less than X. Look at all this. Now X is fixed. I am looking at all Y less than X. Okay. Then take interval Y to X plus 1. So it will cover X. These are open intervals. So this will form a countable set because you can't have any more than countable elements Y less than X. But this is a local base. Every open set must contain one of them. Okay. It may contain X plus 1, plus 1, plus 1, so on. But it can up to X plus 1 and we have to be there. And below it may be, you don't know how far you have to go. So you have to keep coming smaller and smaller. So there is always this. Alright. It's a local base at X. See, this X itself may not be a successor. That's why I have to do this trick. If X is a successor, it will be like 0. Okay. Lower part you don't have to take. And upper part you can take X plus 1. So singleton that will be an open set. So all immediate successors are singleton sensors are open. Now to see that there is no countable local base at omega. See, I said accept at that point. So why there is no countable local base at omega? Let us suppose there is one such. Then it follows that there is a countable set U. May Y n. I can denote them by Y n and inside natural number. Contains a 0 omega such that the half closed interval is Y n to omega closed because beyond that we don't have anything. This must be a local base. Each opens of set must contain such an open set. This part. So these themselves must be local base. Now you take supremum of this Y n because a countable set now, supremum of this set now. We have seen in 5 that S must be inside 0 omega. Now if you look at the open neutral S2 omega, no element will be there at all. None of them will be contained at this point. Because S is supremum. So this subset is not, this subset is an open neighborhood of omega which does not contain any member of U. Now that is a contradiction. This is more S1 actually. It is intersection of all of them. When you take supremum, you take intersection of Y n omega. This will be precisely S omega. So 0 omega, omega open is not separable. This is the next thing. For countability we have seen but separability is not possible. And hence of course the larger space is also not separable. Because separability is after all hereditary. So we shall prove this one is not separable. Automatically you follow that 0 omega close is not separable. So why this is true? Take any countable set. What happens? You will have supremum. If you take that supremum plus 1 that will be still inside omega. So that will not contain any of this. It will not be dense at all. So plus 1 is single 10 open. So it will not intersect this one at all. That is all. So these are the easy consequences. That 5 is the key here. Bounded subset is countable. And if it is not lived it is countable. Something is bounded inside 0 omega. So now it is countable. Well, so let us stop here. Next time we will see some more serious topology of this ordinal. Thank you.


Kraft presents the Great Gilder Sleeves. Yeah! Kraft Cheese Company makers of Parquet Marchion and a complete line of famous quality food products presents Harold Perry as the Great Gilder Sleeve. Kraft brings you the Great Gilder Sleeve every week at this time, written by John Whedon and Sam Moore. We'll hear from the Great Gilder Sleeve in just a moment. But first, let me tell you about the grand cheese food that gets such a royal welcome from shoppers every time a new supplier arrives at their favorite food store. Yes, it's your old favorite, Pabstet, the delicious golden cheese food of a hundred uses. Pabstet, an economical, nutritious cheese food, is such a wonderful help in preparing wartime meals we just wish there was a whole lot more of it to go around. Pabstet, you see, makes those smooth, appetizing cheese sauces that add such nourishing goodness to macaroni or noodles, cooked vegetables and fish, prepared in dozens of tempting ways. And Pabstet also spreads slices and toasts to perfection. So naturally it's in big demand too for sandwiches and for crackers with dessert. So ask for Pabstet and buy it when you can. Look for the round, flat package, the delicious golden cheese food, Pabstet. Now let's join our friend the Great Gilder Sleeve. The ups and downs of his campaign for mayor, plus the complications of his passion for Miss Eve Goodwin, have reduced him to a bulky shadow. Yesterday afternoon he inadvertently announced his engagement to Miss Goodwin at an Arbor Day rally. And now as he drops in at Peevy's drug store he's in a mood to accept congratulations. Love is the sweetest Well, thanks, Commissioner Gilder Sleeve. Oh, hi, Floyd. Who's taking care of the barbershop? Closed up. Shave them a quarter for the day, Commissioner. Have a soda? No, Floyd, you have one. I'm buying today. Where's Peevy? Out back there, making up some of my special hair tonic. Guaranteed to grow hair in a pool ball.  I use it myself. Very good.  Well, if you're buying, I'll have a cigar. What's the occasion, Miss Gilder Sleeve? Haven't you heard? I'm engaged to be married. Well, congratulations. Say, didn't you just get out of one? What do you mean? Well, wouldn't you engage to that Southern party with the... Well, that was some time ago, Floyd. This is different. This is the real thing. Well, if you're satisfied, why should I kick? Who's the... Miss Eve Goodwin. The school principal? Never would have thought of her. Why not? I don't know. I had a teacher once that I wondered about, but a principal. She's a very lovely woman, Floyd. Oh, sure. I wasn't casting any asparagus on the lady. Asparagus? Five gallons on the last little Christmas. Hello, Mr. Gilder Sleeve. Hello, Peevy. Have a cigar. Well, I rarely use tobacco, Mr. Gilder Sleeve. Is this some kind of an occasion? Sure is, Peevy. The commission has gone out and got himself engaged again. So are I here? Yes, Peevy. I'm engaged to the nicest woman in Summerfield. Come on, have a cigar. Well, as I say, I seldom smoke cigars, but just to be a good fellow, I'll ring one up.  I'll get out the box of Floyd and I can smoke. Yes, sir. Your, uh, regular brand? No, let's have the three-for-a-half model. Three-for-a-half at the end. Thanks, commissioner. I think you should be very happy, Mr. Gilder Sleeve. Miss Goodwin always seemed like a nice, wholesome, substantial woman. Wholesome?  You haven't got a spark of romance in your whole system. Well, I wouldn't say that, huh? Well, don't call my fiance wholesome. Oh, no offense. I'm very happy if you've met Miss Wright. Was it, uh, love at first sight, would you say? Instantaneous. You can always... Well, Judge Hooker, come on in. Join the party, Judge. Thank you. Gilder Sleeve's given out cigars for his engagement. Yeah, that's right, Judge. Have a cigar? I've already offered my congratulations, but I'll accept the cigar. I wish you a long life of happiness, and I think you used a lucky girl. Oh, thank you, Horace. The commissioner claims it was love at first sight, Judge. There's no such thing as love at first sight, Floyd. Love is a matter of habit. Habit? Love is not like brushing your teeth, Horace. Well, I think there's something in what he says. Man, Floyd's a married man, Gilder. He ought to know. All right, Floyd, what's your theory? Well, I'll tell you the way it happened with me. I was going with the wife for about six months, I guess. You know, she seemed like a good kid. Her family knew my family.  But to me, she was strictly a good time. Nothing serious, few laughs, little neck and once in a while. Then one night I took her to a dance someplace, and when we were going home after, there was some moonlight or something. Before I knew it, I asked her to marry me. I didn't have any more intention to marry that girl than... Well, as soon as the words were out of my mouth, I could have bit my tongue off. That's very interesting, Floyd, but what's your theory about love? My theory is that it's an accident, and there ought to be a law against it. You're just cynical. You can't convince me. Sure, me either, Mr. Gallus. What do you think, Pee-Vee? Do you think love is just a habit? I think love is a miracle. Oh? What do you mean by that? Well, I'll tell you. The minute I laid eyes on Mrs. Pee-Vee, I was swept off my feet. There you are, fellas. Tell us how it happened, Pee-Vee. Well, we met at a picnic. There were several other... Oh, I'd never fall for a girl at a picnic. They make you do all the work. Such, Floyd. Like the fire, boil the water... Let Pee-Vee tell his story, Floyd. Burry the garbage. A lot of work at a picnic. Go ahead, Pee-Vee. Well, Mrs. Pee-Vee was the only stranger in the group at this picnic, so four or five of us young fellas were meeting her for the first time. As soon as I saw her, I knew she was the girl for me. It was like an electric current, huh? Well, no, I went... Yes, by George Edward. It was like an electric current, and it sure filled me full of ginger. You know how young fellas are. I commenced to show off, climbing trees, jumping across the creek. Had on a brand-new pair of trousers, but I just didn't give a hang. That's it, Pee-Vee. That's the miracle. Oh, no, that's not the miracle. The miracle was that I was the only fellow there that fell in love with her, and I was the only one she married. That was the miracle. I don't believe it, Pee-Vee. Now, wait a minute, Floyd. What is it you don't believe, Floyd? I don't believe it was any miracle. Did you clean the pots after the picnic? Well, yes, I believe I did. You buried the garbage? Well, yes, I recall that very distinctly. Well, there goes your miracle. A woman always picks out the man that's got his nose to the grindstone, and boy, does she keep it there. You'll find out, Mr. Gilda Sleeve. I still hope it will turn out well. Do you think it will? This is the first time in my life I've ever made a cake. Now, honey, just take it easy. Take it easy. Don't get yourself in the uproar. Just do like I tell you. Put it in the oven now. One side, Leroy. Oh, sorry. Leroy, for goodness sake. I said I was sorry. You have to hang around the kitchen.  I don't feel like it. I don't know why it is. Every time I bake a cake, Leroy has to be under foot. One side now. I know what Leroy is hanging around for. Here it is, Leroy. You can lick the bowl now. No thanks, Bertie. What? No thanks. Leroy, honey, what's the matter? Nothing. I just feel like punching somebody in the nose, that's all. Well, what on earth? What does he have to go and get married for? Why can't she leave him alone? Oh, why can't he? Oh, Lady Goodwood. They were doing all right till she came along. Now, Leroy, you had not a feel like that. Well, I do. Who wants to marry an old principal? Well, it looks like your uncle does. Besides, she's not old. She's not even as old as Uncle Moore. What's wrong with marrying a principal, anyway? It's like marrying a cop. Very fair to Miss Goodwood. Was she fair to me? Made me clean up the whole playground when I didn't even throw it. Whitey threw it. I don't know what you're talking about. And anyway... Ask Whitey. He admits it. He admits it. Anyway, I like her. You would. I don't know that I think she's just the person for Uncle Moore, but I like her. What do you think, Birdie? You don't want to marry her, do you, Birdie? Well, you have to look at it this way. There's all kinds of women. Now, Miss Ransom, she's one kind of woman. Miss Goodwood, she's a little different. It's up to the man. Mr. Gillsleeves knows what he's doing. I hope. Well, gosh, do we have to go through this every spring? This year is Miss Good... early. Yes, Uncle Moore. Did I interrupt something? Oh, no, we were just talking. Yeah, just talking. I've been teaching Miss Marjorie how to bake a cake. Yeah, she's been teaching me how to bake a cake. Yeah, baking a cake. What's for dinner, Birdie? Besides cake. Oh, something you like. Good. Well, um...  Fine, fine. Birdie. Yes, sir? Birdie, I suppose the children have told you that I've decided to take the fatal plunge. I'm taking it again, as it were. Oh, yes, sir. I was just getting around to congratulate you. Many happy returns, Miss Goodwood. Well, thank you, Birdie. Thank you. You approve of this step, I trust. Oh, everything you want to do is all right with me. You know that, Miss Goodwood. Yes, I know that, Birdie. But I'd like to know what you really think of it. Oh, I think it's fine, Miss Goodwood. I think it's fine if you do. Oh. Well, thank you. Of course, I don't know the lady, except in the one time she came here to dinner. Yes, I must have her dinner again soon. Perhaps tomorrow night, so we can all get acquainted. After all, we're going to see a lot of each other. I can't help noticing that you children haven't committed yourselves as yet. If Marjorie, what do you think of all this? Oh, I think it's fine, Uncle Moore. I really do. I hope you'll be very happy, and I think you will. Well, I think we should be. After all, Miss Goodwood and I have many things in common, which should... Lee Roy, what do you think? Oh, I think it's fine. Fine, I think it's fine. You do, eh? Yeah, I think it's fine. That's fine. Front door, I'll go. Come on, Marjorie. Let's get out of here so Birdie can get dinner. I've got to watch my cake. Oh, I'll watch it, honey. You run along. Oh, thanks, Birdie. But I'll bake it, remember? Don't worry, you'll get the credit. Uh-oh, I suppose she's heard. Yeah, I'm glad to come over, Mrs.... Pretty polite all of a sudden. What's the little devil up to now? Oh! Hello, Marjorie. Drug vaulting, I'm so happy for you. Happy? Hmm, I just heard the news just this morning. Of course, I had a hit from Judge Hooker. Oh, I was going to call you, Leela. Oh, I understand. Gracious. You're much too busy to have any time for me. But I just couldn't wait to tell you how happy I am for both of you. I called Eve up right away. You did? Oh, we have the nicest child. I think she's a dear Throckmorton. She'll refine. Oh, yes, Eve's very refined. I'm such a lady. I think you two should have a very interesting life together. Well, we plan to, of course. I bet the children are just thrilled about it. Aren't you, Leera? Yeah, it's great. Don't you think it's wonderful? I'm so excited about it.  Well, that's very nice of you, Leela. But after all, who has a better right? I mean, think of all the parties that were given for us when we were engaged. I think I ought to do something in return, and I'd love to. I love parties anyway. Well, I think I ought to speak to Eve first. She isn't much the party type. Oh, don't worry about Eve. I've already spoken to her. You have? Mm-hmm. I invited her for tomorrow evening, and she said she'd love to come. Oh, Eve is wonderful. I've invited the judge, too. I thought we'd just make it, you know, kind of intimate as long as we're all such old friends and all. What about the doctor? Uh, Dr. Hograve? I had a town again. You know these doctors. Sometimes I think all they really care about is microbes. Hi-ho! Who wants to be a microbe? Leela by George, sometimes I... sometimes I don't know. Well, then, Dr. Motton. Wait, I'll go to the door with you. Oh, thank you. I, uh, I just want to say, Leela. Yes, Dr. Motton. Well, I think you've been a darn good sport about this whole thing, and I want you to know it. Oh, gracious, Dr. Motton. I couldn't be happier for you, really. I love giving a party. After all, it isn't every day you get yourself engaged. Not quite. See you tomorrow. Yeah, see you tomorrow. Tomorrow. I wonder, it could be one of my bad days. This leave will be with us again in just a few seconds. A few packed lunches for school-going children and hard-working grown-ups. I surprised them some day this week with the appetizing cheddar cheese flavor of Pabstead. This delicious golden cheese food makes wonderfully good sandwiches. And, of course, Pabstead is also one of America's special mealtime favorites. It melts with luscious smoothness into a golden cheese sauce that's simply grand with macaroni, vegetables, eggs, or fish. And another main dish treats for a hungry family. Pabstead also slices perfectly to be served with desserts. Pabstead, in fact, has won a reputation for their different uses. And it's wonderfully nourishing, too. Pabstead helps provide food energy, milk protein, milk minerals, rival-flavin, and important vitamin A. So for delicious nourishing sandwiches and other mealtime treats, buy Pabstead whenever you can. Look for the round, flat packets, the delicious golden cheese food, Pabstead. Now let's return to the great Gilder Sleeve who finds himself facing a delicate domestic problem. For the first time since the announcement the good one is coming to the house. And Gilder Sleeve senses that Leroy is going to require a little coaching. The coaching is going on in Gilder Sleeve's bedroom as he cleans up for dinner. Wait a minute. Wait a minute. Yeah? Yeah. Come back here. I want to talk to you. What about? Well, I'd just like to talk to you, Leroy. Sit down on the bed there. Okay. Shoot. Well, uh, what I wanted to say, Leroy, was, uh, was that the necktie I gave you? Yeah, what's the matter with it? What I was going to say, my boy. Yeah? Did you wash your neck? Sure. Come here. By George, you did, didn't you? She's not going to get anything on me. Who? Miss Goodwin. Now that's just what I want to talk to you about, young man. That attitude. You must realize that because Miss Goodwin is the principal of your school, she has certain duties to perform. It's her duty to be strict with you. God, somebody's got to be strict with you.  That's all right. I don't like her. Uh, let me put it this way, my boy. When Miss Goodwin's out of school, she's a different person. A very different person. And you must learn to think of her that way. You must learn to think of her as well as somebody who's very dear to me. Will you do that? I'll try. That's the boy. I know you will. You know, Leroy, you and I ought to get to know each other better. I've always been intending to spend more time with you.  I know how you feel, my boy. We've always been a pretty tight little corporation here. You and Marjorie and I. But I feel that the time has come now to expand a little bit. We've got to take a new partner into the firm. Why? Well, I don't know that I can explain why. Perhaps when you're older, you'll understand. When I'm older. That's right. When you're older, you'll understand a lot of things. When I'm older, everything's going to happen when I'm older. For corn sake, I might as well go to bed and stay there till I'm 21. I know how you feel, my boy, but there are certain things. Well, you just have to grow up to them. What things? When did you grow up to them? I haven't got time to explain it all now, Leroy. Miss Goodwin will be here any minute. When she comes, I want you to be polite to her.  Well, I'll be polite. Don't worry. You're not going to get anything on me. I'll be polite to her. You'll be more than that. You'll be nice to her.   Okay. Now stick in your shirt tail. Let's go downstairs. Why didn't they tell me you were here? I've only been here a few minutes. May I? Careful, lipstick.  Well, your day is over, thank goodness. Well, no, I wouldn't say that. I've just been having such a nice talk with Marjorie. You have a very tactful niece, Brock Morton. I have a very lucky uncle. Your uncle has very good taste. Luck had nothing to do with it, did it, Eve? It was sheer determination. Yours or mine? Where's Leroy? Oh, he's upstairs. He'll be right down. Well, let's sit down, shall we? Dinner won't be ready for a few minutes. I'm looking forward to Mrs. Ransom's party this evening. I think it should be such fun. Yes, I hope so. Marjorie was telling me she baked her first cake yesterday. I think that's wonderful. Well, it wasn't wonderful, but it wasn't bad. Was it, Uncle Moynihan? It was food for the gods, my dear. You must teach me how sometimes to cook. Oh, I'd love to. It's very simple, really. You just take a cup of flour and... Marjorie, not right now, my dear. That was very simple. Oh, here's Leroy. Come on in, my boy, and meet your future... Say hello to Miss Goodwin. Hello, Leroy. How do you do, Miss Goodwin? Well, won't you come over here and sit beside me? I'm quite comfortable here, thank you. I might cloud you. Huh? There's plenty of room. Leroy, do as Miss Goodwin tells you. Now, Dr. Ackmorton, that's no way to talk to a boy. Let him do as he pleases. Did you have a good time this afternoon, Leroy? Yes, Miss Goodwin. What did you play? Baseball? No, Miss Goodwin. You didn't play baseball. Well... Tell Miss Goodwin what you did play. Yes, sir. We played Commando. Oh, that sounds like fun. How do you play it? Tell her how you play it, Leroy. Yes, sir. Well, do you have foxholes? Do we have foxholes? Boy, honestly, we have foxholes. Go on, Leroy. Miss Goodwin asked you a question. Answer it. Just a minute, Dr. Ackmorton. Leroy? Yes, Miss Goodwin? Come here a minute. Yes? I don't know that this is the place for it, Leroy, but I owe you an apology. What for? I found out who threw that paperwad in assembly. It wasn't you. Who was it? Paul Marx. They're still cold. Am I? Well, don't tell me who it was. It wasn't you and I apologize. Miss Goodwin, I'll tell you something. Yes? I didn't actually throw it, but I watered it up. Well, I'll forgive you, Leroy, on one condition. What's that? When we're out of school, don't call me Miss Goodwin. Oh, I'm sorry. Well, call me Eve or anything you like, but not Miss Goodwin. I like to get out of school once in a while, too, you know. Okay, I'll do it. If you'll do something for me. What's that? Don't call me Leroy. The name is Leroy. No, young man. It's a deal. Oh, here's Birdie. Is dinner announced, Birdie? Come and get it. Birdie's a jewel, but she's no Emily post. Shall we, Leroy? I get to sit next to... Happy little family. How do you mean, Throckmorton? Well, the way you handle the children. So tactful. They're very nice children. Well, it's a big load off my mind. Now, if things go well at Leela's... Well, if you're worried about it, perhaps we should have stayed at your house and been cozy. How about it? Shall we tell Leela we couldn't make it? Oh, Throckmorton, I was joking. We can't possibly back out now. Come on. All right. How about a quick kiss? Just one before she comes? Lipstick. I don't mind. I like the taste. Well, I'm my Throckmorton, dear. Well, the happy couple. Here they are, George. Well, congratulations, Throckmorton. My profoundest sympathy, Eve. May all your troubles be little ones. Don't worry, Eve, the first hundred years of the hardest. I'll take care of it. I'll take care of it. I'll take care of it. This is the hardest. Oh, brother, this is going to be worse than I thought. I want to tell you that's the finest homemade punch I've ever tasted. It's simply delicious. Great. Oh, now y'all are just trying to turn my head. But my mother always told me a girl should be able to do one thing well. Well, I'm sure you have other accomplishments. Well, cook a little.  On my way downtown this afternoon, I... You might... You might just as well realize Eve the Throckmorton simply loves tea.  Throckmorton, you pig. Now, Leela, please. Oh, this is fun, Throckmorton. Tell me some more things about him, Leela.  Leela. Throckmorton thought she wanted to wrestle. My, how he's changed. Eve, please. But, darling, I like to know things about your pads. Then let me tell them to you. Eve, have you ever seen that snapshot of Throckmorton, the one that was taken when he was a three-year-old baby? Now, Leela, I forbid it. He was all stomach. Throckmorton, I'd love to see the picture. So would I. You stay out of this, Hooker. Oh, please. We're just having fun. Yes, that's how it is, Throckmorton. Eve, tell me, does he ever... Does he ever invariably...? Why, George, I don't have to stand for this. Eve, has he ever sung Speak to Me of Love? Leela is nothing sacred to you. Well, it can't be terribly sacred, Throckmorton. You've sung it to me.  Oh, Throckmorton, sing it for us now. No. Oh, come on. No, I don't want to sing it. Oh, come on, Throckmorton, sing it for us. Eve. He wants to be coached, girl. I do not. Throckmorton, won't you sing it for us? Please, pretty please. I will not. All right, ladies, I will sing it for you. What? Oh, what? The music, please. The music, please. I'll play it for you, George. Speak to me of love, and say what I'm longing for. No. Come on. Women. Floyd was right. Well, sir, Throckmorton, Pete Gillerslee, has learned his lesson. What do you want? Darling, don't you see? Don't you understand what Leela was doing? You were just as bad as she was. She was trying to make you look silly. She was trying to make me lose my temper. Who wouldn't lose his temper? I'm not blaming you, darling.  Would you forgive me? Well, I won't go back there. We don't have to. Isn't there anything else you'd like to do? Yeah. TV was right after all. Love is a miracle. Dear diary, this will have to be a brief entry, because here it is two o'clock in the morning. How did it ever get to be so late? And I thought this was going to be one of my bad days. Good night, everybody. Music heard on this program was directed by Clark Sweet. 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