Fax Uncountable Company with airSlate SignNow

Fax uncountable company

the last thing that we did was show that that every infinite subset of a comfortable set itself company and the infinite subset of a countable set is comfortable it has to be infinite because you don't want to have some finite subset you know what you said that's where we stopped that's where we stopped and so a couple more things that we wanted that we want to say first is that if you have a countable set and you let see let B be some finite product some finite Cartesian product of the A's okay in other words it will be vectors right vectors where each component is a name then let's cost in this event then piece of enemies comes and this confusion from the Cartesian beef a Cartesian product of capital sets please so you've got a bunch of you haven't you know Cartesian said I'm saying you accountable said you take a Cartesian product of these guys you make that goes out of them and then that set of vectors it's gonna be comfortable okay yeah it's mine yes not a real question just yes it feels like we're getting none of this accountability is like what we would normally mean when we say accountable so what are we trying to get up when we see it the guy isn't the definition of accountable yes but what do you mean it's not you mean it's not what do you mean in English when you say comfortable finding find it okay yeah no that's not moving yeah we mean I know fine either or has a one-to-one correspondence with the natural okay okay okay can anybody what's the what's the what's the idea of the proof let's let's just do it let's just do one case for example let's so in the case N equals two okay so in other words we're looking at things that look like you know a 180 like why is that why is that accountable why is that comfortable you know that you know that the set a is comfortable right you know that the Sunday is helping anyone remember remember the Christmas or yesterday just imagine prefix Bruin this is the right thing assisted that tangle no this is not right are you thinking of the counter is diagonal puzzles no this is not it but the idea is the idea is simple you to say well look if you look at the set may I cross date okay so they sent a icon saying you fix one of the numbers you know fix one of the elements cross it like in other words we're looking at everything that looks like so let's look at say a 1 cos a so a 1 comma you just fixed pick some element okay anywhere now this is a countable set right because the first element is fixed and that the number of elements is the same as you know it's the number the number of elements is the same the cardinality of this is the same thing as 8 by 10 a was comfortable right okay so you know that this is a comfortable scent right is everyone okay with that this is a comfortable set because this this thing for any a one hey this thing has a wonder one correspondence with a right just use the second element okay and then B - well b2 is just the union of all a union of all these guys right so B 2 is just the Union over all elements of so it's a countable Union it's a countable union of comfortable sense the end so simple thing are you looking kind of dull is it because you're not it anybody anybody in on understanding anybody confused anybody willing to say that they confused okay so so and then you know if you want to do you know so the higher guys then you can basically do it by induction by yourself in the end at the end by you'd say well you know you've proven it for the for the you know first you and assume that B and minus one is countable and then just use the same argument just to bootstrap your way up to get to the end okay so just by induction so general case the case of the end and then zoom true assume the end - what is comfortable I'm going to use exactly the same argument to show that then the end is awesome right and then those two things together would give you and give you the inductive proof that's so there was one of the questions in your that people sent in the other day was is Q he is cute comfortable right and the answer is yes it is right and one way you can see it is by using this as they know it in the book one way you can see it is by using this stamp right so you say well look consider Q cross Q consider Q cross Q I'm saying bunch of sq consider is equal Z right that's countable Q repeat you and then we can identify you a season of Z kasi right because every element of Q is of this one P or Q where P and Q are integers right so just use for every element of Q just it's not that we cannot from that to P comma Q okay so you're gonna get a subset you'll get a subset of Z cuisine please don't use your phone you know please unplug unplug yourselves pay attention okay um yeah so so what's what's what do I say at this point you've got this thing he was identified on the subset of Z cross C and then me oh yes just point to the theorem and that's the end right so Q is a subset of a is an infinite subset it's you know it's infinite it's q q is an infinite subset of a countable set and so it is it is itself comfortable yes so when we're trying to prove a set to be countable and we do find and other sets that the set that we're interested in the subset of on the other side which way of the mapping which way should the mapping be you want this to be so so for this case we improve q is a subset of a bit uncomfortable of Z processing yes that's well which way of the mapping should officially be have function from Q that maps to Z cross Z or the other way around or either won't be fine well you you're trying you're identifying Q with a subset of Z cross Z right so this Q is you're gonna think of q Q and do you not the map I mean you know you have a you know map taking you have a one-to-one you know that taking Q you have a you have a one-to-one map taking Q into into Z cross C right no no what you know you don't need it to be right because all you need is to have Q as a subset of Z cross C machine not as all as the crossing in fact it won't be all of Z plus C because you're not going to hit through the guys with 0 in the second second entry yeah generally use like what you mean when you see C crosses e it's just when I say is a Class C all I mean is like just like when we talk about bar to our to you know being come our cross our class our it's just pairs of pairs of elements yeah just pairs of balances that's all we mean when we say this cross yes so are we assuming that the set of integers are terrible yes yes because right can you make can you make a map from from the integers to the to the natural numbers I mean can you make a one-to-one map going from what other words can you put the integers into a unique list and that's that's clearly missing 0 1 negative 1 right so if you can put them into a list where there's no repetition so yeah it state clearly clearly comfortable okay any any other questions any questions okay so okay so what we see is that you know if we take these Cartesian products then a Cartesian product of countable sets is still going to be accountable but then you could ask you know what happens if you take an intimate Cartesian product so we've been doing it for a finite we said you know provided confusion ponder and what about an infinite Cartesian product in other words what if you take you know basically just sequences of elements in a countable set and the simplest one for the simplest one so what we do is we'll say well I'm a be the set 0 comma 1 so yeah only two choices it's 0 or 1 and then I'm sorry let me not call it a consider decide this at 0 1 and let a be just all sequences set of sequences these elements are either 0 or 1 I guess zero okay okay so this is the same thing as thinking of this set cross itself infinite infinitely may be infinite but infinitely many times you've got my infinity long vectors where the entries are 0 1 and the claim is that this is uncomfortable okay this is you know sort of a famous argument it's not really useful anywhere else but this is the famous argument see - well no it captures dying off the process so what's the inputs the idea can somebody tell me tempters diagnosis for in you you seem like you you can link layout very cool that but that's okay the diagonal is it's like how you somebody helped her in somebody there I'm aligning what's do you think please what's in it JB JB JB what's what's the country style thanks so the proof is suppose it is comfortable suppose so all that you've got it thank suppose a is comfortable so you've got a list of sequences you can make you can make a list of sequences you can you can make a list that one's gonna hit anybody man you want to do it as do you want to say oh I only remember the beginning of the proof but sure suppose I tell ya right now is go through yes except go through each element yes so anyone has anyone in some sequence you know saying it's you know suppose that entries are anyone what so then you go like for like if a1 1 is equal to 0 of M you set a1 equal to 0 or something and then like if a2 a2 2 equals to 0 you something equal to 0 something like that yes yes yes so you okay so if a is countable the point is this but if a is countable then right then there exists a list of all right a list listing here of all such sequences right then we have this there's a complete list and it had it can be you know is it come it can be written down the first elements a second forty fifty right okay but in that case you know lay them all out like this lay them all out like this and construct a new sequence construct a sequence my where i j is is not equal to Beijing as I only discovered be can construct the sequence be be in b1 b2 where Jake is different from the diagonal of Jason okay so like if this if this guy is zero then we're gonna choose B 1 to B 1 if this guy's is 1 which is big V 2 to V 0 and some and so forth right then the sequence B difference is not equal to any right because it differs from a differs from each of those other sequences at the I term when the difference from the first guy at the first time the difference from the second sequence of the second term difference from the first sequence at the third term and so on and so forth right so that means that B is not on the list right but we said that we have a complete list of all sequences and we've just created somebody who's not on the list so that's a contradiction and so what if the ad JJ doesn't exist like if you have a sequence of the zeros and ones that is 0 1 1 and it's the 5th element whether it's the food sequence that you're dealing with it doesn't these are infinite sequences it's like and these are these are infinite sequence right because yeah we're not when you consider this Lake this Cartesian product you're not allowed to say I have a vector but I think might want some of the vectors are short okay they're all they're all the same there's another question meat okay okay so that is that is the proof that this is an uncountable set it's also a proof if you think about the binary representation of numbers so if you instead of using decimals you have presented numbers using you know in using binary it shows you that the numbers between 0 the numbers between 0 and 1 are uncomfortable also so I give you a look at the real numbers between 0 & 1 they they correspond to you know some subset of this and this this thing is uncomfortable actually they they themselves with okay so that's enough about uncomfortable and uncomfortable question okay okay so actually the way this whole board is entering a new section okay so the next topic is this tape on metrics cases okay so metric spaces and the point of this this idea is that a lot of the concepts that we're going to deal with and a lot of the results are we gonna get in the near future don't really depend on the fact that we're working with the real numbers they don't really depend on the fact that we working with the complex numbers they would hold in a more generalized setting one called metric spaces okay so we'll be tough we'll have a space and then we'll have a notion of distance okay and you know that's that's that's all easy but we do need to say what we mean by distance thanks so what what what does that way counts as a distance you know for us you know in in our are our standard distance you know the distance between two points and we usually take as this by the absolute value of the difference right in RN right the differential change in vectors is the normal normal laughter displayed so the distance like the length the length of the vector you know that the length of the vector that goes from one and Y to the other right these are standard distances but a lot of light like I said a lot of what we're about to do doesn't depend on on these specific notions of distance but just on certain properties of distance okay so we generalized the notion of distance to something called a metric I have tried to change the name of this course from principles of real analysis to principles of just analysis but I've never been able to pull it off I protest the name of this course it's incorrect because we're doing we don't care about the real numbers like we don't care about about we don't care about the girl members we're in metric spaces okay and people always I always get objections I don't know why shouldn't you change that we should you tell me the name of this course is principles of mathematical analysis you start off with a set as a set and then you have this function being from Xbox hence into our sum function if for all for all points as the points in X well as the points and X the dpq is bigger than zero and whenever the P is zero to fifty PQ is the same as DQ p + 3 B P Q is always less than equal to b PR + d RP for some I'm thinking are cute cute for some Third Point are if all these things happen then we call be a metric we call this function any metric so metric is just a distance and we call pair X comma B okay so you know like I just said and it's the standard examples of the metric space are you putting in space with the EQ defined as the magnitude like in standard your standard magnitude right and weight and you see that the magnitude satisfies this plate the length right the distance between two in any two points is bigger than zero it's going to be zero if this two points are the same if and only if those two solutions same the distance two points it doesn't matter what order you take them and then the distance from one point to another is smaller than the distance from that point to them to some intermediary point plus the distance from that intermediate point to the until the last time the triangle inequality right which we which we saw earlier absolutely for the video city okay so that's that's one example of metric another sort of cute example is the taxi cab taxi cab metric are are to leave your comment so that is where the distance between two points is the length not the duck but not collected the diagonal with the length of the sums of these signs here okay so you're in a taxicab and the taxi cab cannot do go diagonally I can only go horizontally and vertically okay so how far is this building from that building well you know you have to go this way and then that way so that's this is this is the distance this is the taxicab metric okay can you see it's easy to see that you know that that it's an easy exercise to see that that that notion of distance and light so this would be the distance between Y is X 1 X 2 like if this point x1y1 and at this point this point is excellent x1 minus y1 x2 minus x1 x2 why are you sanitation because I called this guy X and I call this guy why sorry yeah yeah yeah yeah that's that's that's a trouble because I called this guy why and physically messed my ice cream myself I was right okay okay okay okay um okay so metric right space metric metric space okay some basic definitions from pretty accordion space or Euclidean space you know what these guys are right open open interval r1 in our end you have a generalization of that so say you have a bunch of numbers a line less than B I wear one pink I see a bunch of basically what we're sending with my selves up for is a bunch of intervals so you've got a bunch of intervals and you take the Cartesian product of those guys right this thing's set of X K in R K where X is bigger than a big this thing is called K so it's just like the it's just a box right it's just a parallelepiped now you've taken we take an a1 b1 a2 b2 impossible to prosper we gave you taken the product of all these open intervals sweet or less than these guys are our numbers right X is in R K it's the guy whose components his entries on x1 through the next king right so always thank you it's like if I say 1 comma 2 costs costs 1 comma 3 right then all that means is that going taking this interval when you're crossing against this interval and you get this open box you get this box okay right - first the first entry has to be in the first first interval the second entry has to be in the second interval okay so yeah don't get don't get confused these guys the X I are not in R K X isn't recognized okay okay very good notion third notion you have some X in our King and you have some real number of R so if I ever put say something like this you know art is greater than zero and it's I'm talking about a real number if some real number are we define select be our X denote all points in R K such that X minus P in absolute value is less than water and just be open the open ball the open ball and subject and X with radius it's just in a patient right it's all points whose distance from X is less than R right so doesn't contain doesn't include the surface does not include the surface of the ball it was point next if radius R you don't include M include the boundary of it if we want to include the boundary then we usually do it with a bar over it guys and and these are basically just examples of what's gonna follow so say yes now we're gonna give a bunch of definitions that you know are sort of the basic vocabulary of topology they will seem a little bit strange but you'll get more and more comfortable with them at time in time but at the beginning there they're always a little bit confusing you may have noticed in mine reply to the questions of day four I I made a reference to Hitler learns topology can anyone look at his Midlands topology no okay some of some of you did okay good so it's you know to the authority from some German movie in the German movie I think you know things are going badly for Hitler and he calls all these people and him like give him this this lecture away you know I I don't know the story he's clearly upset but corporate people have taken this movie and dubbed it over as Hitler lecturing a bunch of topologists and he's so he's been spending his time trying to learn points and topology that gave thought to get and he's upset with with hobby how the technologists have done it you know at one point you know he says like is this you know it's not even clear to me that yeah it could be you know you can even have a set that's open and closed and then one of the people says my Kiera there is such a sense it's called Copeland okay okay so um welcome welcome to that world here we go okay okay so leaving leaving Euclidean space behind you just have a metric space you have a notion of distance so the analog of this ball is gonna be what's called a neighborhood so some point in the space and again you have some positive real number light and some RP being all the points in the space is distance from this income is less than honor right and it's the same thing as a ball right this is called my neighborhood neighborhood with radius of a neighborhood wins okay so for example if you are here in r2 you're here at r2 and I say what's the neighborhood of radius 1 with respect to the taxicab metric ok what's the neighborhood of radius 1 with respect to the taxicab district so simply I can go this far and I can go this far I can go this far and I can go this far right but I can also go like 1/2 here like half distance this way and half up right I can go you know because this way and so what is it it's gonna be it's gonna be this set here and this would be that this would be the neighborhood at that point with respect to that that metric right okay so just get used to the idea that new metric is not Euclidean the trick definition too you have some subset and subset of X and P and element of X if every neighborhood at P contains a point Q naught equal to P such that Q is empty let me say P is a limit point a limit point of e so you have a set I mean let me introduce some convenient notation if if I put a star on neighborhood then what I mean by that is the neighborhood but excluding the center the punctures the pumps where I'm going to call that the functioning since the it's the it's their ball but the center is removed sorry yeah so that if I put a star on it then I mean the punctured name it's a neighbor that but with a little hole in the mouth so if I want to write I can write this more cleanly I hope that every for every are bigger than zero every punctured neighborhood of P intersects P is not empty tonight but every that every radius every punctured neighborhood of P intersecting is not in that case P is a P as a limit point because it's the same thing every neighborhood of P contains some point not at the center like contains some point not at the center that is it me in other words every punctured neighborhood right it's gonna it's gonna intersect somewhere this is the same okay so for example right if if we are so simple the father site look at the open ball in in our to Euclidean Euclidean space that's standard standard metric right and I I take a point inside the ball well that would be a limit point right because every neighborhood every day with the doctrine of that point it contains some some element of e that's not the center okay so there's a set of obvious obvious limit limit points right one less less obvious would be what something anything on the boundary right cuz that means well any neighborhood contains a point not that's not the center that is a neat okay so this is beside the boundary although he's not part of the set itself is one of the limit points right it's not part of the set right because the set doesn't contain its boundary right but every neighborhood of it intersects the intersects Cosette in a point not the center you know so the only time that something wouldn't be a limit point is if it was to find too much way that it was like super isolated for me that's right so what is it one would mean to not be a limit point that would mean that there is some there is some neighborhood that whose puncture does not intersect me okay there's some neighborhood whose punctured does not intersect D so for example let's go on like a one-dimensional example suppose I look at the set one over m and and is like this okay so here is one here at zero except that one I've got a half I've got a third fourth and so on and so forth right is is is this a limit point is one a limit point of this set you can be in the set and still be a limit point doesn't matter like smaller than half that's right so you say well look I can find a neighborhood and the punctured version of this the concrete version of this doesn't intersect the center right so in fact none of these points are limit points right none of these guys that I drew up here none of the elements of the site are actually limit points but this set has a limit point light let me point is it what's what is wet where is a limit point for the second 0 right 0 is the only point it's not remember the set but you see that no matter what no matter what neighborhood you take but if you maybe puncture it of course it don't matter it'll it will intersect it still it always intersects the set mutation we like fine is defined to be the limit points set of little points okay any questions okay any quite many questions okay okay position of a city them people need to take breaks so yeah if P is indeed but P is not is not a limit then we say which makes sense because it means that there's some neighborhood where the punctured neighborhood doesn't intersect the sent right in other words you in the middle are the only element of the scent of your scent in that function in that neighborhood right right so this term isolated makes sense it means that being isolated being not being a limit point but being part of the set means that if you shrink down your neighborhoods at some point you're gonna be only only one from your set in that in that neighborhood that there's always isolated points no no it's just we just we just sing these sorts of points are called isolated ones they may be they may or may not laughter yeah yeah there's no exclamation points here everybody's a nice answer okay and before and if every limit point you say if every limit point of B is ending everything is closed so so this for example is not an example of a closed set like because it doesn't contain its boundary and the guys on its boundary are limit points so this is the thing which of course is called the open bar is not closed well okay so those are probably the ones that you know are more tricky in the definition of getting up to the notion of a closed set there's a bunch of definitions I that get us to the notion of open okay these I think are sort of easier so again yeah this so they exist a neighborhood if there exists some radius such that and our key is contained in V then we say and we call G an interior point an interior point of view so what this is saying is that you have some set and you have some point and there is some neighborhood that is centered at the point and it's contained entirely in the center right it's some neighborhood that is in the set and contained Center that you point then we call this guy in the material he is called an interior interior ok again yeah this would not be an interior point this would not be an interior point not on interior because I can I can't find any gayborhood I can't find any neighborhood that's contained entirely inside the set i no matter what i do i can't find a neighborhood that's contained in tiling suneson so not is that an ordinary isn't okay number six if every point of E is interior we say P is open easel so for example the open ball is an open set because any point in it whatever point I have inside there well I can find I can find a neighborhood that's contained entirely inside the center this is a pseudo standard definition when we say super see what we mean is the complement of the set so it's all the points in the space where X is not in okay there's a definition of something called a perfect set I'm just gonna skip that that's not that's not important number nine if there exists a point and a then some positive number such that the distance from any point in the set is less than M from Q then we say found it so all this is saying and pictorially all these things that there's some point Q and there is some videos M like there's some radius em and you're set lives inside of that book commencing write your sentence inside of some ball some finite ball thank every point every point P in the set is distance the distance from those points from Q others points from Q is less than n okay so then we call it a bounded set like I mean this look involved is is bounded right because I mean it lives inside some other some other larger ball and it's open it's an open set and you know valued basically needs not heading off towards infinity with respect to the metro so yeah in the in the standard Euclidean econometrics if this guy is open and it's also bounded because those notions are contradictory and in the last last nation if every so you have some set fee that lives inside you know this do the metric space X if every point of X is either in E or a good point B then we say B is dense he is dense enough so your number earlier we talked about this result that Q was dense in our way and it's in this sense that when we showed earlier gives it gives this result like hi there every point in every point in R is either irrational or it's a limit point of the rationals right because if you're if you're an irrational number you know you can take a take an interval around you and then by the density theme that we proved before you know that there's some rational number in in that in that interval remember like the imperious thing said that you take some arbitrary number take some arbitrary number and you take there's something a little bit bigger than it then you've guaranteed a rational number you guarantee the rational number between those two guys right what what's it saying this is saying that in every in every neighborhood of you know a of every of every real number you know there is their irrational their vegetable points right so what it says is that every every real number is a limit point of the rational of the set of rational numbers so the rationals events as let's that's why why we caught attendance okay okay so these are the the basic definitions anybody anybody have questions let's go okay so um okay even exercise the result that every neighborhood is in a metric space every name sure let's go on to something more interesting if P is a given point then every neighborhood a P contains infinitely many points every neighborhood is going to contain infinitely many points of so remember what a limit point is a limit point you have this point every punctured neighborhood intersects the set every punctured neighborhood intersects the set right so P being a little point means that every giving any any arm and punctured P intersects intersects the site right so if I take a radius of one start off with saying I know that there's somebody in here not at the center who is an element of eat okay why do I know that there are infinitely many dyes from me in there I know that there's this one die here you could do this sort of constructively by induction you could say well what you think you'd say well so I'm gonna take a second neighborhood right I'm gonna take a second neighborhood whose radius is smaller than the distance from key to e1ke one's not in there any one's not in there right P being a limit point means that this must be somebody somebody in there who's put is from me right so call that guy you two right and so when it's a point that you can do this by induction right so you can construct it let R be one then they exist some one that lies in and our one punctured pee in a semi and let our to be the distance from A to B 1 then there exists me to not equate e 1 and R 2 and so on and so forth properly you should do this by induction or if you wanted to do it by contradiction suppose some neighborhood contains only finally members only finitely anyone sitting through this point P and there's some neighborhood and there's only finally many elements of V in there then you get a contradiction like because you just say let um let R be the minimum of the distances from being taken to be di and but take the take the smallest take the smallest distance and take that as the radius of a new neighborhood but then there would be nobody from he in there and that's a contradiction because this he was supposed to be a limit point being a limit point means that every punctured neighborhood intersects E or we've just created a neighborhood that doesn't intersect okay let's let's try this phasing one more episode which is kind of nice okay it's only if the complement is closed so please so suppose let's go for it spoons ears open in other words it consists only of interior points and we want to show that a compliment is first so I want to show the complement is closed in other words that it contains all its so um let let X be a little point well if your limit point of e complement then every punctured neighborhood is going to intersect B complement then for all radii the function neighborhood around X intersects B complement right every punctured neighborhood is going to intersect the complement say here intersects the complement but that means that the neighborhood is not contained in that neighborhood those neighborhoods all of those neighborhoods are not contained to me because they intersect the compliment they intersect the couplet so they're not contained and eat right but that means that X is not interior right but he was open so every point of being has to be interior right he was open so X cannot be because only he being open as only interior points everybody everybody all night you got a few not all right I'm gonna go on there's only one winner one more minute so slide backwards and let's say that the compliment is closed you want to show every point of E is material so same accent let's choose a point energy we know that X cannot be a limit point we compliment like the compliment is closed so it contains all its winning points closed so X can't Galen okay if it's not a limit point that means if not being a limit point means that there's some neighborhood of it right there exists some neighborhood of it where the puncture neighbor that where the puncture of that neighborhood intersects the complement is right that's empty so that means that this thing he is hated is contained in China insanity right because that thing it doesn't intersect B complement so it must be inside a fee right well that right X was in he and the punctured neighborhood around X isn't me so that tells you that okay so this gives us another way of thinking about being what it means to be open or closed right closed if you don't like the limit definition limit point definition you can just think well what closed means is that my compliment isn't open sense it's a compliment all the points in a compliment are interior to that compliment but you will need to know you know you're gonna need this limit point idea don't think you can get away from okay that's it for today