 Now, let us define some more properties of sets. So, there are three different concepts that we will need, open sets, closed sets and compact sets. What is an open set? So, let S be a subset of Rn and define this B of X, R as the set of Y in Rn. So, here X in belongs, X is a point in Rn and R is just a real number, X is just a positive number. B of X, R is defined as all the Y's such that they are, such that they are at distance strictly less than R from X. So, again my axis, I am just drawing two axis, here is my point X and if I take my R here defines a radius B of X, R is then all these points, all of these points that lie in a ball of radius R around X. So, this is called a ball of radius R around X. Now, because we are keeping this strictly less, what we are excluding is those points that are at exactly distance R. So, the shell of this ball is actually excluded. So, if you want this shell here, this one is, this is excluded from this set. Now, we say that S is an open set for all X in S. So, take any point in S, then around it, you should be able to find some ball such that it lies completely in S. For all X in S, there exists a radius R such that, look at this ball of radius R around centered at X, this ball is completely in S. We say S is an open set, if for all X in S, there exists an R such that the ball centered at X of radius R lies completely in S. So, what does such a set look like? So, just imagine the set like this. So, what is, if I take a point here, if I take a point X here, so the challenge is that you have to, for every point X in the set, I should be able to find a fit a ball around it such that it lies completely in the set. So, let us look at a point like this. Around this, I can fit a ball very easily, I take a ball like this, it fits very easily or inside the set, I take this point, this ball also fit, there is a ball around this that fits completely in it. The difficulty starts arising at this point. So, if this point which is here on the edge of the set, now what can I, around this can I find a ball such that it lies completely in the set S. So, whatever ball I choose, however small I make the ball, I will have a problem because some of that, some part of that ball is going to spill outside the set. It is just the way the set is. No matter how small I make this, how tiny I make the radius, some part of the ball is going to spill outside the set and this condition here, this condition is going to get violated. So, sets like this, sets like this which have a shell or what we call a boundary, those kind of sets are not open. Now, there are other kinds of sets, now a set does not necessarily have to look like this, like a contiguous body. For example, a set can be just isolate, bunch of isolated points. This is also a set. Now, is this set open? No, this set is not open. This set is not open simply because you take any point around, at any point in the set and try to draw this ball around it. It has several other points apart from the points in the set. So, the ball has to be completely within the set and that is not possible here. So, a set that is open, you should imagine it as a set that is contiguous and has no shell. Now, that does not mean a set, it does not mean that it should have a certain overall shape. For example, here I have drawn it almost like an ellipse. A set could have a shape like this also, some strange shape like this. And if I exclude the shell around this, that would be an open set. As an example, take the ball itself, the ball itself, the ball of radius r around x, that itself is an open set. Because every point I take inside the ball, I can find an even smaller ball that lies completely inside this particular ball. And I can keep doing this. Now, complimentary definition to this is that of a closed set. A set is set to be closed if it is compliment, a set is set to be closed if it is compliment. So, the way you should think of the closed set is precisely this. It is basically you look at everything that is not in this set and check if that is an open set. Then the set that you started with is a closed set. A set is set to be compact. Now, here remember I am talking of sets in Rn. There is a much more general definition outside of Rn. But for Rn, this is equivalent to all definitions. A set is said to be compact. So, let me be more precise. Here a set in Rn is said to be compact if it is closed. So, a set is, so this sort of, so a compact set is a set that can be enclosed within some large ball. And it has, it is such that its compliment is open. The compliment is open means the compliment does not have a shell. If the compliment does not have a shell, that means the set itself has the shell. Now, the set need not, can be neither closed nor open. These are not the only categories of set. Of course, it can be neither closed nor open. Say for example, but whether a set is open or closed depends on thus the ambient dimension in which we are looking at, looking at the set. So, when I am in the definition of an open set, I am looking for, I am asking that there should be around every point x, there should be a ball of radius R. There should be an R such that this ball, the ball Bx, R belongs to the set. Now, this B of x, R is a ball defined in Rn. So, the set is asking for a ball in Rn to be fit inside the set. So, I will give you an example. For instance, let us look at R2 and look at this square excluding its boundary, excluding the boundary. This shell is not included. Then is this an open set? This is an open set. No problem. Around every point I can find a ball that lies completely in the set. Now, let us look at R2 itself and look at this segment here. Says it starts, which lies along the horizontal axis, starts from some point A and ends at some point B. I have and let us suppose this is an open segment, you can draw this better. So, it starts from some point A ends at some point B, but A and B are themselves not included. Now, is this set an open set? So, what is the definition of open set in demand? So, if I want to think of this as a subset of R2, I want to think of this as a subset of R2, then I need to be able to fit two-dimensional balls around every point in it, such that the entire ball lies in the set. But two-dimensional balls will always spill outside the set. So, as a subset of R2, so if I express this set using two coordinates and coordinate for the horizontal axis as well as one for the vertical axis, then as a subset of R2, this set is not open. However, geometrically the same same object can be immersed also in R1 and the entire set can be expressed using only one coordinate. In that case, if I then that case, the picture looks different, then I do not have two dimensions, I am only in one dimension and I am looking at something like this, A and B excluded. And now around every point, what I need to fit is one-dimensional ball. A one-dimensional ball is simply an open segment and a one-dimensional ball can be always accommodated around every point. So, whether a set is open or not depends on what ambient space you have immersed it in, the technical word is what ambient space you have embedded it in. So, it depends on the ambient. Being open or closed again has no relation to how many pieces the set is made of. For example, I can take an open ball like this and another open ball and their union is this and this. Let S be the union of these two, this region and this region together. If I take this as my definition of my set, this is an open set. It is made up of these two pieces, that is fine, but individually if you look at the pieces, they are themselves are part of some sort of contigu- you know intuitively like a contiguous mass. So, this actually brings me to another point which is what sort of operations preserve openness and closeness. So, if you take the union, union of any number of open sets is open. You take any collection of open sets, take their union that the resulting set is always open. So, by negation of this or by complementation, you also get that the intersection of any number of closed sets is closed. The intersection of finitely many open sets is open and union of finitely many closed sets is closed. So, can someone tell me is the intersection of any number of compact sets compact. So, the intersection of any collection of sets, let us take this as one set S1, take another set S2. The intersection of these two is the common region between. So, if I take the intersection of any number of sets, it is the common area that belongs to all of them. If I have another set S3 here and another set S4, S4 like this, etc. I can put together many of these sets. The common area that belongs to all of them. So, here in this case, the common area is turning out to be I think just this red region here. So, the common area of all of them is the intersection. So, what is the claim made here? Here is that the intersection of any number of closed sets is closed. Is the intersection of two bounded sets bounded or any number of bounded sets bounded? Why is it bounded? It is part of any one of them for sure. It is part of each of them. Since each of them is bounded, that intersection is also going to be bounded. So, you are looking at this tiny region which belongs to all of them. So, to claim that this tiny region can be put inside a ball, all I need to say is that any one of them can be put inside the ball, which is what I have been told anyway. So, intersection of any number of compact sets will be bounded. And this statement here was telling you that intersection of any number of closed sets is closed. A closed sets are already, so compact sets are already closed. So, the intersection is also going to be closed and we know that the intersection is going to be bounded. So, the intersection is compact. So, intersection number of compact sets is the intersection of finitely many open sets is open. So, when you take, if you take S1 and S2 as open sets in this particular figure, I have S3 and S4 here also. So, let me just draw this separately. So, here is your S1, here is S2. S1 and S2 have both, in both in their definitions, I have excluded the shell around. So, even in the, even in the intersection, the shell, these shells are not present. So, so let us, let us, so the question here is, why is the intersection of infinitely many open sets not open? So, let me look, let us look at the, so let me write it here. Why is the intersection infinitely many open sets? Technically, we should say not necessarily open, not necessarily open. Why is it so, so can we give an example? So, let us, let us look at, let us look at this example in the real line itself, here is my, here is 0, here is minus 1 by n, here is plus 1 by n. I look at this interval that starts from minus 1 by n to plus 1 by n excluding the endpoint. This is an open interval, it is an open set in R, all right. Now, so I define Sn as this, minus 1 by n plus 1 by n. Now, what is the intersection of all these essence in starting from say 1 and going up till infinity. The intersection of all of these is, has only one element in it, which is, which is the point 0, right. So, the intersection of all of these is, is this set, it is a singleton set having, and this set is not open, because any ball around 0 will have points that are positive as well as negative, not just 0, any ball of positive radius. This is, all of these were open. Let me ask you another question, again about compact sets. So, we saw, we saw intersection of any number of compact sets is compact. What about the union of finitely many compact sets? Union of finitely many compact sets, each, each compact set is, is both closed and bounded. So, they are, all of them are individually closed. Union of, they are finitely many of them. Union of finitely many of closed set is a closed set. So, their union is closed, thanks to, thanks to this statement. So, the union is definitely closed. Now, is the union bounded, right? Now, why is the union bounded? Let us look again, look, look at this geometrically. Suppose, here are my axis, here is one set I have S1, here is some other set S2, here is a third set S3, here is a fourth set S4. We are looking at the union of all of these, right? So, what is the region I am looking at? The region is this, use a different color. The region is everything that is there in S3 and also S4 and also S2 and also S1. This is my set. Now, is this bounded? Each of them were individually bounded, right? So, S1 could be fit inside a big ball of radius r, r1 suppose. S2 could be fit inside a ball of radius r2, S3 could fit inside a ball of radius r3, S4 could fit inside a ball of radius r4, right? So, each Si can be fit inside a ball of radius centered at the origin and of radius ri, suppose, right? Now, if all, now all of them collectively can be put inside the largest of such balls, of these balls. So, these are r1, r2, r3, r4, they are concentric balls, they take the largest of them, maybe make the ball a little bit larger and that will definitely fit all of them, right? So, the union Si is certainly contained in the maximum of this, just to be safe if you want to add a 1, you can add a 1, right? The union of all of these are contained in all. So, the union of, and this maximum here is a maximum of a few real numbers, it is a maximum of some finitely many real numbers, whatever it will be a finite number, right? So, consequently, this ball is also of finite radius. So, the union is, we just checked is closed and by this argument, the union is now also bounded. So, the union is both closed and bounded. So, consequently, the union of finitely many compact sets, union of finitely many compact sets is compact.