 Welcome to module 13 of point set topology course. We shall now introduce a number of notions. They all arise in the study of metric spaces as well, but they are useful also in the abstract topological setup and most of the time we will have our examples from metric spaces. Throughout this section, X is a topological space, so I may not, every time mention it, for example this X will not be used for any other thing in this section, in this lecture. The topology comes from a metric, then I will specially mention that X d is a metric space. Take a point X in X and a subset. We say A is an open neighborhood, quite often lazily we will write it as O n B d. So A is an open neighborhood of X in X. If first of all X must be inside A and A must be open, that is why it is O. O n B d means open, neighborhood means that X is inside A first of all. In general, A is called neighborhood without that O. If X is inside A of course, that is part of there. But you have an open subset V such that X is inside V contained inside A. Of course, this containment notation is equality allowed here. So there is again a lazy notation. But it may happen that quite often that V is smaller than A and A itself is not open. Even then we call it a neighborhood. Then we do not call it open neighborhood. That is all, that is the difference. A point, let us now define interior of A. So interior of A is called union of all open sets contained in A. So it is denoted by interior of A or A on the top. But I will have to read it as interior of A only. So this just means that if I take a point X inside interior of A, that is already a point of A and there is a neighborhood, open neighborhood because all open sets are there. It is union of open sets. It belongs to one of the open sets contained inside A. And therefore, therefore what? It is union of all open sets contained in A. So interior of A itself is open. Not only that, obviously it will be union of all those sets. It will be the largest open set contained inside A. That is another term, largest open set contained inside A. So to begin with A itself may not be open. So that is why interior of A is defined. That is a subset of A that is open. But if A itself is open, obviously then being the largest open set contained inside A, interior of A will be equal to A. Of course, Conor is also true right. If interior of A equal to A, then A is open because just now I proved that interior of A is open. So we will meet these things again and again. So it is better to by heart them. So it does not matter if I repeat once or twice. Let X belong to X. It is said to be a closure point of A now. So everything is happening in the head X. If every neighborhood of X intersects A. So here I have put every open neighborhood of X intersects A. If every open neighborhood of X intersects, then every neighborhood intersects. If every neighborhood intersects, open neighborhoods being neighborhoods, they will also intersect. So these two are equivalent. We can take whichever one we like, every open neighborhood of x intersects a fine. The set of all closure points see when you take the closure points we are no longer inside any point of x you can this verifies this condition it will be inside which will you call the closure point. Look at all the closure points we will have a notation for that it is denoted by a bar on the top is a very popular notation but some people may not use that one they may use closure a and then some other people may use a suffix c but I will never use that notation ok. CLA I may use for closure a bar also I may use but a you know suffix top at the top a raised to c that small c that I will never use because many people use that for the complementation. So, that is a very confusing notation. So, I will not use that one. So, this descriptive notation CL of a is very easy to remember it is closure of a. Now, another concept the boundary of a is nothing but all the closure points of a a bar it throw away the interior. Interior is already a subset of a. A bar is larger than a perhaps or may be equal to I do not know but interior of a bar interior of a definitely is contained inside a bar ok. So, you throw it throw that away what you get is boundary of a. A point x is called a limit point of a if for every open neighborhood u of x u minus x intersection a is nonempty. In the definition of closure point we did not have this u minus x u intersection a is nonempty was a condition. So, u minus x intersection a is nonempty is a stronger condition ok. If u minus x intersection a is nonempty u intersection a is also nonempty for every a for every u. Therefore, a limit point will be automatically a closure point but not vice versa always ok. A limit point is a closure point that is another notion here that is called accumulation point of a ok. Sorry another notion another name. So, there is another name accumulation point is also used by some people. The set of all limit points of a let us have a notation for that L a. So, in geometry this L will be denoting length and so on. So, do not confuse it for any length here this is the limit points of a. But then there is another notation here these are not very standard different people use different notation, but not too many. So, 50 percent of we may be using this and that and so on. So, its name is derived set of a some people use this a prime for this notation L a. What is L a set of limit points what is limit point every neighborhood of x has the property u minus x intersection a is nonempty. Let us have a couple of definitions here because all of them are related to these open sets right all of them are interdependent. So, it is better to study all of them together. So, let us have some more definitions. A subset a of x is called a closed subset this I am repeating I have already told you what is closed subset if x minus a is open in x yes the complement ok. We shall use the notation this a c for the complement if there is no confusion ok. Usually this is a much notation you can take complements of a in different subsets containing it right instead of x. So, then for all of them if you use a c then it will be complicated whereas, this will be b minus suppose a is contained inside b b minus a makes sense. In fact, b minus a makes sense even if a is not contained inside b all that you have to do is throw away all the points of a if they are already not there then you do not have to throw them. If they are there means what you take intersection a intersection b then throw away a intersection b from b that is b minus a that notation is a set theoretic notation I am not introducing that I am just recalling that. So, a set a subset of x is dense in x if the closure is the whole space. So, every point of x is a closure point in that case a is called a dense set exactly opposite something very strong stronger than just negation of this one of course these are not this is not negation it is not dense is not the same thing as another one I am going to define no inheritance no inheritance is very stronger if a is we say a is no inheritance if interior of the closure of a is empty set that is first you take closure a bar and then put a circle there that is interior. Interior of the closure is empty the closure of a you have take and then look at the interior interior is empty means what take a set a interior is empty means what no non-empty open set is contained inside a because interior a is a union of all of them. So, that is the no inheritance a subset a of x is said to be isolated set if for each a inside a we have an open neighborhood of a such that if you take u intersection a it is just singleton a see u is a neighborhood of u is a neighborhood of a and a is already inside a so u intersection a contains a it should not have any other element it is singleton a if you can find for each a if you can find such a open set u then the set a is called isolated in x okay this is like each point is isolated it is only one point in that neighborhood that is meaning of isolated if each of them is a whole set is isolated a subset which is isolated and closed also that will be called as discrete subset this is just a terminology some people may not put this closeness and say isolated is same thing as discrete or some people may interchange them also okay wherever I have called isolated them you call discrete wherever I have called discrete they may say isolated this can happen in you know in some by with some authors but this is my definition I am going to state okay one general remark is all these terminologies occur right before the topology as such the definition of topology that we have introduced was adopted okay in the works of Weistra's but the ideas the terminologies took some time of course and then there were different terminologies by different authors so that is why even today there are slightly different terminologies by different authors but many things have come to the central theme many things have you know come and other things have fallen out only few things have survived okay so many of these notions were there some people were using different terms and each time you know Weistra's was not using any of any of these terms but you every time you will explain the whole condition so it was taking a very tortuous thing you know reading those papers etc is very difficult so let me first give you the simplest example discrete in discrete are two simple the next simple example is the sierpinski space that we have produced right sierpinski space consists of just two elements and three open sets what are the open sets open set empty set and x must be there so you put one of the things namely either singleton 0 or singleton 1 so put singleton 0 okay check that singleton 0 is dense in x this is what I have said but I am I am not sure of myself so I have to check so what is the meaning of this tell me there is only one other point right namely 1 I am not sure that that 1 is in the closure of this 0 so what does that mean I must take an open set containing 1 but by this definition x is the only set which contains one which contains 0 also therefore that open set intersects this one that is a condition to be in the closure therefore 0 closure will contain 1 also of course it contains 0 therefore it is a whole of x is that easy to verify that singleton 0 closure is this whole space that is the meaning of it dense okay now you can do more generally whatever I have introduced namely take any sierpinski point in a typical space you remember what is sierpinski point sierpinski point was the point where in the only open set containing that point is the whole space therefore it will be inside the closure of every other point take y to be any other point x will be inside the closure of that point okay this does not mean that y bar is the whole of x of course singleton x the sierpinski point would be there okay so that is similar to this but when I say more generally I am not concluding this one namely the closure of y is the whole space closure of y contains x okay now let us come to I as I told you for examples are from R2 or R3 and so on so let me take in between let us take a subspace of R2 all right so I am taking open interval 0 sorry closed interval 0 1 open half open interval cross the same thing closed closed 1 1 0 is closed 1 is open okay so namely all x y such that 0 is less than or x y less than 1 both x and y less than 1 and bigger than equal to 0 here equality here strictly inequality okay but what is the topology now I am taking it is the usual Euclidean topology on R2 so that is a metric topology also right to try to work out the whatever we want to do without the concept of the metric we have defined the topology what is an open set etc we use just use try to use them that's all note that this a is not an open set in R2 why to say them thing is not open you have to produce one element such that no neighborhood of that element is contained inside the given set right for example I can take 0 chroma 0 that's a point of a every neighborhood of 0 comma 0 first of all will contain an open ball of some positive radius no open ball of positive radius will be contained in this is just for the quadrant part right inside inside R cross R this thing is just contained inside the first quadrant right right so this set is not open in R okay you can do the same thing with all the points on the x part x axis part or y axis part okay if y is 0 or x is 0 those points are not in the interior therefore the interior does not contain those two sides you have to throw away those sides x positive y positive take all those points are in the interior why because as soon as x is positive y is positive okay so you can take the minimum of them and then take a radius take a ball around that point x comma y of that radius that will not intersect the x axis and y axis part where it will be contained inside complete contain inside okay so you want to go you don't want to go out of one also so you should also take care of that all right so this is actually an open set because it's a interior okay it's a largest open set contained inside A okay let me let me show you the picture here so in this picture I have taken 0 1 cross 0 1 that is why I have put these two dot dot dot here okay what does it mean 1 cross i is not contained in the set similarly this one i cross 1 is not contained inside that set okay everything is here but these strong lines here full lines they are contained inside that is a set if you take a just now I explained that to take a point 0 0 here then any neighborhood of that will contain a ball around that right so it won't be contained inside once I am inside here I can just look at the distance is various distances to the boundary take the minimum of all of them then that ball will be that this could be contained inside it's open this therefore every point other than these solid lines here they are inside what inside the interior therefore the interior is precisely go to that one that's what okay next thing is check that 2 2 is not a closure point of A this is just right 2 2 is somewhere far away here okay that will not be in the closure of A why I am telling that of course points of this interior here they are in the closure even this boundary dot dot dot line will be also in the closure why because if I take a neighborhood of this point some ball like this this half the ball will lie inside here like this here also so this entire dot dot dot dot dot dot dot here that will be in the closure but if I take some far away point okay far away means what anything which is not in this not in this I cross I 0 1 cross 0 1 both closed so one moment here is here you can choose its distance from there to here take the minimum least of them that open side will be not intersect this one at all one open side does not intersect this one means it is not in the closure okay so closure is precisely equal to 0 1 class okay next let us look at the boundary of A the boundary of A was defined by defined as A bar minus A interior we have computed both A bar and A interior therefore what is boundary of A all those points here lying on these sides the four sides here so this can be defined as this y equal to 0 this is x equal to 0 this is x equal to 1 that is equal to y equal to 1 so all these four conditions can be put in one single condition x y into 1 minus x into 1 minus y equal to 0 if this four product is 0 one of them must be 0 one of them is 0 this is 0 y is 0 or 1 minus x is 0 is x is 1 1 minus y is 0 which consists of four sides of square okay so why this example I want to give you tell you is that this concerns with the layman's idea of a boundary of a piece of land so here the piece of land was a square and the sides of the square are the boundary points so that is the kind of word boundary you must have already used in your standard okay boundary of a triangle is a ferry the length is a perimeter right the boundary is the the geometric object there it is the empty triangle when you say triangle you may be taking all the points inside also right so that is what it is if take a think of this an area of a triangle then the perimeter perimeter is what total length of the boundary okay so that is the same thing here also for nice things but now we have defined it for arbitrary surfaces so you should have some properties which are inherent in this pictures and so on but some other properties which may be strange which are not depicted by the pictures so you have to be careful because finally it is just the logic here of sets nothing more topology is always that strange it produces so many ideas so many you know visual things are reflected there and so many strange idea that is that is the power of set theory okay let us make a few more points here one by one in this case I have computed a bar okay we computed interior of a we computed the boundary of a also one more thing I want to say what is the limit points of k the LA here LA will be equal to the entire of k bar it is not always true of course if that is the case then there is no need to define LA so let us verify that so why that is true what is the meaning of LA take any point you must produce a neighborhood such that from the neighborhood even if you throw away that point it should intersect the set and that is very clear inside here it is clear even at this point this point if you throw away that point but you keep the keep the days half the disk will be inside this one so it will intersect right so it is very easy to see that every point in the closure is a limit point here for this set okay that is every point in the closure is a limit point or an accumulation point another world larger world second thing is if you look at the boundary of a that is no air dense in R2 see I start with the set a namely this time boundary of a that is my set when it is no air dense its interior must be empty okay not that you must take the closure first and then take the interior but here I don't have to take the closure because boundary of a closure is the same thing as boundary of air okay so if you take this set and look at its interior I want to say that the interior is empty there is no point which is in the interior what does that mean that's very easy to verify okay take a point on the boundary this boundary okay take a point here here now I must produce an open set that means must produce an open ball at least open ball contained inside that line no open ball will be contained in the in the boundary of the square it will flow inside or outside in fact both sides that's the beauty to take any point here take a ball here it will intersect both the inside part A I said complement of A right so that is the characteristic of boundary of A we'll come to that one later on right now it has no interior no open ball around that is contained in here so if no open set is contained in here how to prove no open ball is contained in here that is enough because if there is one open set non-empty and there will be a ball also open ball so it's in no open ball is contained here it's very clear right therefore this boundary of the square is no air dense similarly if you take a triangle the boundary of the triangle will be no air dense all right if you take the open part zero one cross zero one that's a subset of A that is dense in A bar okay now I am not taking dense inside X okay I am just saying dense inside A bar therefore I am thinking of this A bar itself as a space all that I have to do is what I have to do take any point of A bar show that it is in the closure of this one zero one cross zero one that means take any point take a neighborhood it should intersect it and that is precisely what we have been showing all the time these two lines are not necessary okay when you take the closure it they will automatically come so take a point here okay in A bar anywhere here here here if you take a neighborhood if you take an open ball it will intersect the interior right so these two lines these two these and along with this line also finally these four the four lines are in the closure of this zero one cross zero one so much easier to see that if you take the inside R if you take open interval zero one both zero and one are in the closure and when you take them you write a closed interval okay so that's easy to see I have taken middle thing so that that will be also explained and something more r3 r4 etc also it's possible that's why I'm taking now we come to little more little more serious examples look at the set of all rational numbers inside R let us check these points one by one this rational number set is neither open nor closed why why it is not open if it were open for each point you must get a interval okay because an open ball is nothing but an inter open interval an open interval must contain inside q but you know that rational numbers do not contain any interval any open interval okay of course if you want to take empty set it is there but we are interested in non-empty nor it is closed the same thing as saying that compliment of q inside R is not open the same reason all the rational number or take throw away them what you get all irrational number they also don't contain any non-empty open interval okay so neither q nor its compliment is open however if you look at the closure of R closure of c closure of q that is over of R okay so every point in R is a limit point of rational number take any point take an interval there will be always some rational number that's all i'm using here okay every open interval non-empty open interval will intersect q therefore q bar is the whole of R but in terms of our terminology this just means that q is dense in R okay let us look at the limit points of q that is also over of R just now it said that one take a any point take a neighborhood it can open interval throw away that point even then there will be many rational numbers there okay therefore every point is a limit point not only just q everywhere take any point R irrational irrational and so on okay take an interval around that point throw away that point still you will have two open intervals there right so they will intersect q so the next one is if f is a closed subset of R we take a closed subset of R okay q is not closed neither component of q is closed by that now take a closed subset of R and suppose it is contained in a q like you know you can just take a single point which is rational or one two three four five six all the integer there are lots of closed subsets inside q you take any closed subset inside q or the same thing you can do in irrational numbers also pi you know 2 pi 3 pi any anything like that okay so take a closed subset of q or R minus q then that set is no air dense so I am giving examples of every concept here at least one example to begin with that is no air dense no air dense means what if f is already closed so I don't have to take the closure okay first of all and then what I have to take then you have to take the interior so interior means what f should contain some open set right but f is contained inside contained inside q so q doesn't contain any open set open interval therefore f doesn't contain same thing is f is contained inside the whole of R minus q irrational number irrational numbers also don't contain any of the set so it is a consequence of these observations the last one namely every closed subset of q or R minus q is no air dense okay note that union of finitely many closed sets is a closed set also intersection of any family of closed sets is a closed set these are two axioms dual to au and fi right fi says what finite intersection of open sets is open so you take demorgan law finite union of closed sets is closed so that's demorgan law so what it amounts to saying that the t axiom au axiom fi axiom could have been replaced by three other axioms which you may call t prime au prime and fi primes which are in terms of closed sets okay so our definition of our foundation definition namely topology definition of topology could have been in terms of closed sets and then we would have defined open sets as complements of closed sets that would be exactly parallel to whatever we have done so there is no no surprise no no change at all in theory okay inside a metric space every singleton is a closed set why because the complement is open take any point other than the given point so let us say given point is x naught take any the other point then the distance between that y and x is positive so it is epsilon or so delta whatever to take delta by two and take the ball around y which is of diameter which is of length which is of radius delta by two that will not intersect x that will not contain x right that means show that the complement of a single point is open therefore the single point is closed okay every finite subset for a metric space is actually discrete this is the whatever I just proved actually proves this one because I have taken finite set what you can do is look at all the various distances between xi and yi xi's belong into the set okay it will be finitely many they are all positive so you take the minimum okay now you take a ball around that each of these points with that radius playing minimum that will not contain any other point at any point you can do that therefore it will become an isolated set but for being a finite it's also closed it's also closed so it is discrete set however if you take infinite sets they may not be closed they may not be discrete okay so let us we have examples very large set like a cross we have taken what a that a is definitely not an isolated set neither it's discrete okay so that is very easy to see but even more complicated sets can be may violate this condition so here is a very simple example we just violate being finite namely look at all one one by two one by three one by four and so on okay inverse of the positive integers there is an infinite set okay this is an isolated set because you take any say one by m then I have to take something between one by m plus one and one by m minus one take the open interval between these two that will contain one by m and it will not contain any other things right therefore it's an isolated set however it is not a closer set that's very easy to see that because zero is in the closure of this set look at zero take any interval around zero it will have these some one by n's after some large and if any summation large all the one by n's will be there so in any case every open interval around zero will intersect this set so zero is a limit point zero is in the closure therefore if you have an extra point which is not in a and it's in the closure a cannot be closed okay all right more interesting examples will be seen namely like cantor sets cantor set is uncountable set it is no inheritance not like a cross a not like zero one cross zero one okay yet you know it is uncountable yet it is no inheritance there are many interesting properties of the cantor set we will discuss that when it comes to but right now we want to study the closer sets integer etc more appropriately so let us stop here today