AI Assistant
Transcript
00:01:580Paolo Guiotto: Okay, we are ready to start.
00:06:60Paolo Guiotto: Good morning!
00:09:410Paolo Guiotto: So, it seems that, I am having some trouble, with,
00:14:580Paolo Guiotto: transform it into a PDF these slides. I don't know what is going on, but let's hope that… I think it's a software issue, one note
00:25:960Paolo Guiotto: Is the software… let's see what happens in the next days. I tried it even now, but it doesn't work.
00:33:870Paolo Guiotto: Okay, today we are, the topic of this class is, topological…
00:47:380Paolo Guiotto: Concept.
00:55:640Paolo Guiotto: Now, topology… Let's say that is a mathematical word.
01:02:110Paolo Guiotto: That stands for the geometry… a qualitative geometry of sets.
01:09:120Paolo Guiotto: So… In fact, here we are going to introduce
01:14:250Paolo Guiotto: A number of three-fold properties of sets that will be very important for the future.
01:22:340Paolo Guiotto: All these properties start from a definition, which is the definition of closed the… Bold.
01:36:900Paolo Guiotto: So, a closed ball is the following, so given,
01:42:50Paolo Guiotto: a point, X in, space RD.
01:48:10Paolo Guiotto: A number R positive, R positive, or 0, which is… which will be called the radius.
01:55:150Paolo Guiotto: We define this set, B, X… R, with this notation.
02:03:480Paolo Guiotto: As the set of points, the vectors, Y in RT,
02:08:639Paolo Guiotto: Such that the distance between Y and X, which is the norm of the difference, Y minus X, huh?
02:18:670Paolo Guiotto: is less or equal than R.
02:21:780Paolo Guiotto: But this is distance.
02:25:730Paolo Guiotto: Between… Why?
02:31:290Paolo Guiotto: And… X.
02:35:400Paolo Guiotto: So, in other words, if, let's see… remark.
02:44:840Paolo Guiotto: If the dimension is 1, So… RD… is R1.
02:53:210Paolo Guiotto: R1 is the set of vectors made by one single component, so we may identify with the real numbers.
03:03:650Paolo Guiotto: What is a ball? So, we will not use arrows in this case. A ball centered at X with the radius R is the set of Y real.
03:15:960Paolo Guiotto: such that the distance, which boils down to the modulus, modulus of Y minus X is less or equal than R. Well, this is an interval centered at X,
03:28:700Paolo Guiotto: So it is precisely the interval x minus R, X plus R.
03:35:30Paolo Guiotto: So, in the real line, This is the real line, R1.
03:41:300Paolo Guiotto: you take a pointer X, the ball is just an interval centrally, that points X,
03:48:680Paolo Guiotto: we say… we still use the name Radius for R.
03:52:550Paolo Guiotto: So we go from X minus R to X plus R. So that's… that red set is the ball centered at X radius R, closed, because… because in this case, there are the end points, no? The two end points of the interval.
04:10:300Paolo Guiotto: In Dimension 2,
04:14:260Paolo Guiotto: Which is basically the case we can keep in mind for the… to help the intuition. So, RD is now R2, is the… identified with the Cartesian plane.
04:27:960Paolo Guiotto: Now, Ebola… centered at vector X with the radius r.
04:35:560Paolo Guiotto: Now, we are… here, we may imagine that, a picture of the situation. Of course, I will always draw figures as if they are in the first quarter, but that's not the case. So this time, let's put the vector here.
04:50:470Paolo Guiotto: So the vector will be identified with a pair, let's say, A and B,
04:55:600Paolo Guiotto: So, this is… the ball is the set of vectors y, I'm just copying the definition, of R2, such that the norm of Y minus X is less or equal than R.
05:14:390Paolo Guiotto: Well, maybe it is better if I…
05:17:560Paolo Guiotto: Write this a little bit below.
05:20:910Paolo Guiotto: Otherwise, The ball is the set of Ys in R2.
05:27:630Paolo Guiotto: Such that the norm of Y minus X, so the distance between Y and X, is less or equal than
05:35:410Paolo Guiotto: Now, this is the Euclidean distance, so if we write a little bit in a more concrete way with the coordinates, let's say that the coordinates of point Y are UV,
05:48:80Paolo Guiotto: So these are points UV in F2, you see that I'm not using letters X and Y, because I'm using X and Y for the vectors. So you have to imagine that we called here the axis, the U-axis and the V-axis. Letters are letters, you can call as you like.
06:05:870Paolo Guiotto: Okay, so this is the set of pairs UV in R2, such that… what about that norm? So, it is the norm of UV minus X, we said it is the point AB, less or equal than R. Now, if you write the Euclidean norm, this is the root of
06:25:680Paolo Guiotto: Now, this is the… you know that the difference between the two vectors is the vector with components, the difference. So we get U minus A squared plus V minus B squared less or equal than R.
06:41:720Paolo Guiotto: that we can.
06:46:200Paolo Guiotto: You know, as I love this thing.
06:49:680Paolo Guiotto: Okay, now it will sound. I have to remove these sounds and these things.
06:55:260Paolo Guiotto: Okay, so if you take the square, you can see that this is the set of points UV of R2,
07:04:320Paolo Guiotto: Such that u minus A squared plus V minus B squared is less or equal than R squared.
07:13:820Paolo Guiotto: And you probably know that this is the interior of a disc centered at point… or circle, if you prefer, centered at point AB with the radius R. So if I go back to the figure, let's say that the radius R is this one.
07:32:390Paolo Guiotto: So, my set is, Something like this.
07:38:180Paolo Guiotto: And it is everything included in this disk.
07:42:40Paolo Guiotto: Including, also, the, let's say the… vehicle.
07:47:120Paolo Guiotto: Surrounding this disc. So, including these points here, okay?
07:52:440Paolo Guiotto: Because those are points exactly a distance equal to I.
07:56:950Paolo Guiotto: So we can say that in dimension 1, a ball is an interval, is a closed…
08:03:50Paolo Guiotto: and bounded interval, centered at point X. In dimension 2, a ball is a disk, centered at some point, which is the center X of the disk. In dimension 3, I'm not doing the figure, but as you can imagine, this will become a true ball, a three-dimensional sphere.
08:22:270Paolo Guiotto: In Dimension 4, we do not have an intuition, physical intuition about that, but we may imagine it will be more or less the same thing.
08:32:470Paolo Guiotto: However, for practical purposes, it will be sufficient that you keep in mind this figure, okay, if you needed to have a figure of that.
08:44:730Paolo Guiotto: Okay, so with closed ball, we introduced… we will introduce a number of definitions, but let's first introduce also the definition of open ball. So this is called closed
09:00:960Paolo Guiotto: close the… Ball.
09:05:630Paolo Guiotto: centred… That's X… with the… radius.
09:18:450Paolo Guiotto: Now, the open ball is similar, it's basically the same thing, but without the boundary. Now, we will give a precise definition to this word, boundary, but the boundary for the moment means that continuous red line, the circle, no?
09:35:140Paolo Guiotto: So, the open bowl is the following definition.
09:40:340Paolo Guiotto: So, B, XR. As you see, I changed the direction of this parenthesis.
09:47:640Paolo Guiotto: Because in dimension 1, what happens? The closed ball, as you have seen, it is a closed interval, and in dimension 1, the open ball will be an open interval. So this is the set of vectors, Y, of RD,
10:05:40Paolo Guiotto: Such that the distance, of…
10:09:240Paolo Guiotto: X of Y to X is strictly less than R.
10:14:520Paolo Guiotto: So, the intuition suggests that in dimension 1, we are talking about
10:21:80Paolo Guiotto: on R1, we are talking about an open interval from X minus R to X plus R, with the two endpoints not included. And in dimension 2,
10:37:880Paolo Guiotto: We are talking about an open disk, so this means that we have a center as above AB, we have a radius, R, so this will be a disk without the edge.
10:52:50Paolo Guiotto: Of the disk.
10:54:180Paolo Guiotto: Okay, so whatever is in the interior, but without the circle at the bottom. That's why we… we normally use this dashed line to emphasize that those points are not included, okay?
11:08:60Paolo Guiotto: And in dimension 3, it will be an open like an orange without the skin, or an apple without the skin, something like this.
11:15:920Paolo Guiotto: Okay. Now… What we do with this definition is to introduce, first of all, some
11:25:750Paolo Guiotto: characteristics of points with respect to sets. So, these are the first, let's say, topological definitions. So, let S be a subset of our data.
11:41:970Paolo Guiotto: We say that…
11:47:300Paolo Guiotto: A pointer X.
11:50:50Paolo Guiotto: of RD.
11:53:410Paolo Guiotto: Jeez.
11:54:710Paolo Guiotto: Well, these are basically two definitions in one.
11:57:930Paolo Guiotto: As you will see, a point cannot be the two properties I'm going to state.
12:03:650Paolo Guiotto: So, I… we will say that the point axe in RD is an interior point.
12:16:810Paolo Guiotto: S, huh?
12:19:650Paolo Guiotto: If… well, let's do a figure first to understand what we're going to say.
12:25:260Paolo Guiotto: So now, imagine that this is our set S.
12:29:860Paolo Guiotto: A point is in the interior of S if this point belongs to S, but it stays into S with an interior closed ball around him.
12:41:450Paolo Guiotto: Maybe the radius of the ball is smaller, but it doesn't matter. If you can find a ball, centered at that point X anteriorly contained into S,
12:52:920Paolo Guiotto: you say that you are in the interior, because the idea is that if I am in S, and I am surrounded interiorly by S, I am inside, well inside, I mean, not just on the… on the boundary, okay?
13:06:690Paolo Guiotto: So, if… how do we write this? If, the property is not every ball is contended into S, but if there exists at least one ball.
13:17:980Paolo Guiotto: So if there exists basically a radius, because the center will be X, so if there exists a radius positive, such that the ball, centered, the closed ball centered at that X into radius R, is contained into S.
13:34:310Paolo Guiotto: So this is the definition of interior point, huh?
13:37:440Paolo Guiotto: And the set of all the interior points D… Seth.
13:44:940Paolo Guiotto: off.
13:45:970Paolo Guiotto: all… V… material.
13:52:50Paolo Guiotto: points.
13:53:900Paolo Guiotto: of S, is called… Jasper.
14:00:230Paolo Guiotto: interior.
14:03:960Paolo Guiotto: of S, and it is denoted, notation, We just write int… OVES.
14:14:340Paolo Guiotto: So the point… the set… this is a set… is a set of points which are interiorly well contented into the set S.
14:25:460Paolo Guiotto: Okay, now, as I told you, this is a sort of double definition, because now we introduce a second property that points may have. So we say that, still, we have a set of RV, and we say that the point X of RV is…
14:44:260Paolo Guiotto: A boundary point.
14:50:990Paolo Guiotto: Point.
14:53:920Paolo Guiotto: 4.
14:57:240Paolo Guiotto: S.
14:58:470Paolo Guiotto: So, you know, what is the boundary of a country, no? It's the place where you, you transit from a country to another, no?
15:07:70Paolo Guiotto: well, normally the boundary is not under known of the two countries, but let's say the idea here is that a boundary point is a point, not necessarily a point of S, I will return on this.
15:23:260Paolo Guiotto: But with this characteristic, that you are surrounded by both points of S and points of the outside of S. Outside of S means S complementary.
15:34:920Paolo Guiotto: Okay, so imagine that S is your state, which is this one, so how do you write these characteristics? So you say that no matter how you take a ball here.
15:46:980Paolo Guiotto: And this ball, in your intuition, can be very small, so it means you are very close to point X.
15:54:760Paolo Guiotto: In that ball, there will always be both points of S and points of S complementary, so points which are in S and points which are not in S.
16:07:570Paolo Guiotto: Because this is saying, no matter how I get close to that point, I always find a citizen of S and a citizen of the other state. That's a boundary point, okay? Now, let's write this… let's translate this into a property. So, if…
16:25:900Paolo Guiotto: We say that no matter we take this ball, so this means for every radius r positive.
16:31:890Paolo Guiotto: If we take the ball centered at X with radius R, this ball has points in common with both S and S complementary.
16:42:800Paolo Guiotto: How do we say this?
16:44:300Paolo Guiotto: Well, we say that this intersection with the S is non-empty.
16:50:330Paolo Guiotto: So, this means there are points of S which falls into this ball, and at the same time, also the intersection between the ball and the complementary of S is non-empty.
17:07:660Paolo Guiotto: Be careful, because I cannot just write, like, a unit intersection B in the section S in the section S complementary, because S in the section S complementary is empty, so there is no point that belongs in the tree, now simultaneously. We cannot be in S, in S complementary, and in both.
17:28:190Paolo Guiotto: In the real world, you can have two citizenships, but, like, imagine that you can hold only one citizenship, you cannot have two, no? That's the idea, no? So, the quantitative property is, for every arm, whatever is the board. So, in our intuition…
17:46:500Paolo Guiotto: R should be small, because this says more difficult to find points, no? If I find points in a small bowl, I will find in a large bowl, no? Because there are those points already there.
17:59:660Paolo Guiotto: But if I find a point in a small ball, it is not for sure that I will find points in a smaller ball.
18:06:480Paolo Guiotto: That's why I need to say for every R of these things, no matter how I am close to X, I will always find points of S, points of S continental.
18:17:990Paolo Guiotto: That's what we call a boundary point. The set of all the boundary points is called the boundary of S.
18:26:710Paolo Guiotto: So, D… Seth.
18:31:200Paolo Guiotto: off.
18:32:430Paolo Guiotto: Foundry.
18:36:970Paolo Guiotto: Points.
18:39:130Paolo Guiotto: Jeez.
18:40:160Paolo Guiotto: Called up.
18:44:140Paolo Guiotto: boundary.
18:48:900Paolo Guiotto: of S, and we will denote this by this notation.
18:58:780Paolo Guiotto: Notation. We will use this symbol.
19:02:350Paolo Guiotto: this, sort of, reverse D.
19:08:760Paolo Guiotto: Okay, let's see on some simple examples these two definitions, okay?
19:14:650Paolo Guiotto: Soft.
19:23:570Paolo Guiotto: How do they work? And let's see also some remarks, because,
19:28:970Paolo Guiotto: there are certain, apparently obvious things which are completely false, okay? So, we have to be careful about these two definitions. So, examples, imagine that, let's start to make things easy in dimension 1. So, we are on
19:47:250Paolo Guiotto: R1.
19:49:940Paolo Guiotto: So, let's take a simple set. For example, the simplest possible set is a set made of one single point, okay? Let's see what happens.
19:59:760Paolo Guiotto: On R1, we take S, a set made by a single point X, okay? So this is the real line, and the set is made of just X alone.
20:13:670Paolo Guiotto: Now, can you say if that X is an interior pointer?
20:19:10Paolo Guiotto: to S is interior to us.
20:21:850Paolo Guiotto: No, because you have to imagine that there is a ball centered at point X. This means an interval, like X minus R, X plus R, interiorly contended into the red point. It's impossible.
20:35:410Paolo Guiotto: for positive radius. Now, you see.
20:38:110Paolo Guiotto: interior point, if you have a positive radius, means a true ball, not a degenerate ball. So, it is clear that this…
20:49:980Paolo Guiotto: S… pause.
20:52:340Paolo Guiotto: not… interior.
20:57:620Paolo Guiotto: points.
20:59:670Paolo Guiotto: Well…
21:01:10Paolo Guiotto: Of course, we can always, we should always provide a proof, a justification of what we say. In this case, proofs can be difficult, okay?
21:12:20Paolo Guiotto: So let's see, in this case, this can be done, but in general, it can be very difficult to prove formally things. So we will be just happy with an intuition, okay?
21:23:810Paolo Guiotto: So we don't need to prove every time that this is the interior, that is the boundary, but we accept that we are able to understand who is the interior and who is the boundary with some practical work, let's say.
21:39:120Paolo Guiotto: Okay.
21:40:300Paolo Guiotto: Because if, well, if you look at this stupid example, what kind of work we should do to verify, this, this property, that this has not interm points.
21:55:180Paolo Guiotto: Because I should say, if Y is a point of S, then there is a unique possibility. That Y must be X, then Y must be X.
22:06:940Paolo Guiotto: And… if,
22:12:130Paolo Guiotto: Y belongs to the interior of S, so it means it is in S, but with a ball, not just,
22:21:290Paolo Guiotto: Alone, so there should be a ball centered at Y, with the radius R positive, so the existence radius positive, such that this is contended into S. But let's translate this.
22:35:310Paolo Guiotto: Now, the ball centered at Y is an interval, so it is Y minus R, Y plus R.
22:43:680Paolo Guiotto: And this should be contained in S. Moreover, we know that the unit point that is in S is X, so we should say that it is the interval x minus R to X plus R that should be contained into S, which is, however, the X alone. And that's impossible for R positive.
23:03:340Paolo Guiotto: This is impossible.
23:07:10Paolo Guiotto: with the… are positive.
23:11:400Paolo Guiotto: You understand this?
23:13:350Paolo Guiotto: Okay? So, no interior points, as the intuition suggests, there is nothing which is well contented into that set, no?
23:22:70Paolo Guiotto: Let's see about the boundary points. What are the boundary points of this set?
23:29:590Paolo Guiotto: So what that points for which…
23:31:910Paolo Guiotto: whenever I take an eyebrowid of that pointer.
23:38:30Paolo Guiotto: for every… sorry, for every ball centered at that point, that ball contains both points of set S and of its complementary. Here, you see the set S is made of the red point, and the complementary is the black line, let's say, no?
23:55:380Paolo Guiotto: So, what should be the answer about the boundary of S? What are the boundary points of this S?
24:07:130Paolo Guiotto: Gary.
24:09:270Paolo Guiotto: Okay, so we have one answer, which is X. Then, another one says… Would you say?
24:20:500Paolo Guiotto: There you go.
24:24:330Paolo Guiotto: Okay, so you said that this is the interior of S?
24:29:360Paolo Guiotto: Okay, any other proposal?
24:34:10Paolo Guiotto: We're just, doing experience, so don't be scared. We are learning these concepts anew.
24:41:660Paolo Guiotto: Now, think about… because what about… what is the conclusion of this argument about the interior? What is the interior of S?
24:53:270Paolo Guiotto: From what we said here, So, interior is…
25:00:160Paolo Guiotto: empty, okay? There are no interior points missed. So, if you are right, you are saying that there are no boundary points.
25:08:280Paolo Guiotto: So which one is the truth? Because he says the boundary is made of X alone, so it's non-empty in particular, and you say that it is the interior, and it is empty.
25:24:170Paolo Guiotto: And no other proposal, so no other set can be boundary points of this S.
25:31:950Paolo Guiotto: Well, okay, so this is correct.
25:35:680Paolo Guiotto: And this is wrong, of course. Let's see why.
25:40:730Paolo Guiotto: Now, to show that the boundary points of S
25:45:460Paolo Guiotto: is the set made of X only? We have to prove two facts. Number one, X is a boundary point.
25:53:300Paolo Guiotto: Number two, no other point is a boundary point, so it's not sufficient to prove that X is a boundary point. Because if you prove that X is a boundary point, you know that among all the boundary points, there is X.
26:07:860Paolo Guiotto: That's what you say, but you don't say anything else about the boundary points.
26:12:880Paolo Guiotto: Okay? So, to prove this, we prove… we prove. Number one, that, our ex…
26:24:510Paolo Guiotto: is a boundary point of S. Number two, there are… No other.
26:35:340Paolo Guiotto: boundary.
26:38:670Paolo Guiotto: points.
26:41:90Paolo Guiotto: or S. So, these two facts together explains why, this is the unique boundary points, so the set of boundary points is made only of that X.
26:54:60Paolo Guiotto: Well, let's try to see with the graphical argument, okay? It's not a proof from the formal point of view, but we can see why this is the case. So, first of all, this is S. Let's see why X is a boundary point.
27:09:470Paolo Guiotto: What does it mean to be boundary point? Boundary point means that no matter we take a ball centered at point X, this ball contains both
27:19:390Paolo Guiotto: point of S and points of S complemented. So let's take our point X, and let's take a ball. Now, the ball is made by an interval of this type.
27:29:980Paolo Guiotto: So I'm coloring in blue.
27:32:70Paolo Guiotto: This is the interval x minus R, X plus R, and this blue thing is the ball centered at X radius r.
27:41:350Paolo Guiotto: Now, as you can see in this ball, of course, there is this point, which is a point of S, is the unit point of S, and there are all these points, which are the point of S complementary.
27:54:240Paolo Guiotto: So you see that in this blue ball, there are, at the same time, both points of S and of S complementary. And no matter how distinct these are, so if R is very small, you would still have black points and the red points. Red point is for sure there.
28:12:190Paolo Guiotto: You don't have to be confused by the figures.
28:24:880Paolo Guiotto: actually, these are points, so they are not… they are not picked, okay? So, of course, this is…
28:32:140Paolo Guiotto: the pencil makes, like, a circle for a point that is the fancy upon, okay? So, it is clear that,
28:40:320Paolo Guiotto: So here you have this that belongs to S, and also you have these points that belong to S complementary.
28:51:340Paolo Guiotto: And since this is valid for every R, this means that this is a boundary point, okay? So this is the proof of property one.
29:00:670Paolo Guiotto: Let's now prove that the property 2.
29:03:430Paolo Guiotto: There are no other boundary points for S. So what we do is, again, a figure. Imagine that this is our point X, and this coincides with the set S. And now, let's see if another point, different from X, can be a boundary point. So imagine that you pick this white.
29:22:820Paolo Guiotto: Is this Y a boundary pointer?
29:26:310Paolo Guiotto: We have to say if it is true or not that whatever, no matter how we take a ball centered at Y, this ball contains both points of S and points of S complementary, which is the black set, yeah.
29:41:870Paolo Guiotto: Now, you understand that if the ball is bigger, like this one.
29:46:980Paolo Guiotto: So let's imagine that we take a very big radius.
29:56:480Paolo Guiotto: So let's imagine that this is the ball. For this ball.
30:01:250Paolo Guiotto: So this is the ball of Y minus R, Y plus R. So, this ball…
30:12:700Paolo Guiotto: contains…
30:17:390Paolo Guiotto: our X, that is the unique point of S, and also other points, let's say Z, and Z, which are in S complementary.
30:28:620Paolo Guiotto: So for this ball, it is true. But the property says that if you want to be a boundary point, you must verify this for every ball, so no matter how is the radius R.
30:40:240Paolo Guiotto: And now you see that if I change the radius, this property, might not be true, because if I have… this is my Y, let's say this is the point X, represent the set S, if now you take a small enough
30:56:530Paolo Guiotto: ball centered at point Y, like this one. This is, again, a Y minus R, Y plus R. In this ball, it is plenty of points in S complementary, but there is no point of S.
31:10:260Paolo Guiotto: So this means that that point does not verify this property, yeah, because it does not do that for every other piece.
31:19:470Paolo Guiotto: There is some odd, at least one, sufficient, for which one of these two, or maybe both, is placed.
31:27:480Paolo Guiotto: Okay.
31:28:440Paolo Guiotto: So, in this case, we can exclude that boundary point. These two together explains why the boundary of S, the set of all the boundary points, is made of the point X itself.
31:41:550Paolo Guiotto: Okay, this was a first example.
31:47:380Paolo Guiotto: Now, if you look at this example, and this is a wrong idea, okay?
31:52:980Paolo Guiotto: So, let's say… Wrong.
32:00:770Paolo Guiotto: Well, let's say, a common… a common wrong.
32:08:880Paolo Guiotto: idea.
32:11:510Paolo Guiotto: If you are a boundary point, you must be in the set, so… And, let's say, ideas.
32:20:60Paolo Guiotto: some.
32:21:40Paolo Guiotto: Because there are several.
32:23:90Paolo Guiotto: So I'm…
32:24:330Paolo Guiotto: Common wrong ideas. So the first common wrong idea is that if X is a boundary point of S, then X must be in S.
32:37:680Paolo Guiotto: This is not necessarily the case. Look, let's be on the real line still, where…
32:45:300Paolo Guiotto: If possible, it is easier to verify this concept. Imagine that our S is an interval from a certain A to a certain B without the end points. So that's our set. So this notation means that the A is not
33:03:260Paolo Guiotto: There, no? So if you want, formally, this is the set of real numbers such that X is greater than A and less than B, strictly greater and strictly less.
33:16:660Paolo Guiotto: So, A is not in that set, and as well as B.
33:20:740Paolo Guiotto: Now, you see here that point A and point B
33:25:820Paolo Guiotto: And not only, but in these two points, A and B,
33:31:500Paolo Guiotto: are boundary points for this S. If you think intuitively, it seems clear, because if my set is the red side in the continent and the black side, it is clear that at these points, I pass from one side to the other, not to the boundary points.
33:50:480Paolo Guiotto: And in fact, you can intuitively see this equation, because if you take a small ball, doesn't matter how much ball it is, centered at A, you see that in this ball there are black points, so this is a point that does not belong to S.
34:07:300Paolo Guiotto: So, and there are also red points. These are points that belong to us.
34:13:290Paolo Guiotto: Okay? So whatever is the radius of this, so no matter how small is this in the bathroom, always will find points that are in S, and points that are not in S. Okay? And the same also for B.
34:28:370Paolo Guiotto: So this explains why, these are boundary points.
34:34:30Paolo Guiotto: And as you can see, none of the two belongs to the set, okay? And…
34:43:530Paolo Guiotto: A and B.
34:45:580Paolo Guiotto: are not in S. So do not, do not think that if you are a boundary point, you must necessarily be in S, okay?
34:55:949Paolo Guiotto: If you look at this set, are you able to say what are the… what is the interior of this set?
35:05:530Paolo Guiotto: interior of the interval AB is what, and the boundary of the interval AB is what?
35:16:00Paolo Guiotto: What about the boundary? Let's start from the second, since we are talking about,
35:20:810Paolo Guiotto: But we know that, that out of recent yeast importance, AMBR, common importance.
35:25:530Paolo Guiotto: Got it.
35:27:850Paolo Guiotto: Is that Angela?
35:36:640Paolo Guiotto: Is there any other pointer, which is the boundary pointer?
35:45:140Paolo Guiotto: Respondo one.
35:58:440Paolo Guiotto: Okay, so let me redo the figure down here, so… No.
36:03:700Paolo Guiotto: We have this.
36:05:150Paolo Guiotto: Okay?
36:07:280Paolo Guiotto: A, B, And that's our S.
36:11:660Paolo Guiotto: So, do you think that the pointer here
36:15:910Paolo Guiotto: Is… is a bundle point for us.
36:20:230Paolo Guiotto: No, because you take a small enough polar here, no, like this one, you see that there are only black points and no red points. So this means that one of the two conditions is not required.
36:34:490Paolo Guiotto: So it's not true that for every dollar, there are all these points of S and point of S complementary, excitement. This point, on this point, there are only points of S complementary, no point of S. And if I take a point here.
36:50:820Paolo Guiotto: Like this one.
36:53:510Paolo Guiotto: What happens here?
36:55:480Paolo Guiotto: You see that if you take a small enough.
36:58:540Paolo Guiotto: The board, which is more than half like this one.
37:01:780Paolo Guiotto: In this bowl, there are only red balls.
37:04:860Paolo Guiotto: And I'm black points.
37:06:770Paolo Guiotto: So this means that in this box, in this mode, there are only points of S, but not points of S complementary. So this means that even this point is not a boundary point. And actually, none of these points,
37:21:300Paolo Guiotto: For the same boundary, is the boundary point, and all of these points, so the boundary is made just by these two points here.
37:29:580Paolo Guiotto: So you see, the boundary is made of two points, A and B, in this case.
37:37:770Paolo Guiotto: What about the interior?
37:40:20Paolo Guiotto: Well, the interior, we know that
37:44:150Paolo Guiotto: points that are in the interior must be contained in the set, even if the points contained in the point always contained in the center, okay? So the point X must be in X.
37:59:60Paolo Guiotto: So, they stay, in fact, in essence, until they continue together with the bull.
38:05:140Paolo Guiotto: Now, if you look at this PU, Yeah.
38:10:710Paolo Guiotto: What could be the interior point? You should start from point of X,
38:15:300Paolo Guiotto: not work at this point, because they cannot be the employer.
38:19:90Paolo Guiotto: And what about these points in the right interval?
38:23:190Paolo Guiotto: Are they interior points?
38:26:680Paolo Guiotto: Now, this time, you need just to prove that there is one ball. One ball is sufficient.
38:32:170Paolo Guiotto: You don't have to prove that every ball is full-time, okay? So you can take it to a small ball, like this one. This is photang grass, so it means that that point is the heater. And this element works for every point of interaction.
38:49:290Paolo Guiotto: I do not consider, so in this case, I get that the interior is the set AB.
38:56:890Paolo Guiotto: itself.
38:58:60Paolo Guiotto: Okay.
39:03:130Paolo Guiotto: Okay, so,
39:06:340Paolo Guiotto: And also, the other statement, if X is in the boundary, okay, if it is not in S, it must be in S complementary. Of course, also, this is false.
39:21:120Paolo Guiotto: As the previous one,
39:23:290Paolo Guiotto: So I'm just writing not true facts. Do not think that I'm saying you facts which are true, because these are facts which are false.
39:33:480Paolo Guiotto: Now, we just modify our example. Let's take a closed interval instead then an open interval.
39:41:370Paolo Guiotto: like this one.
39:44:270Paolo Guiotto: Do you see what are here the interior points of this interval, ABE?
39:50:800Paolo Guiotto: And what are the boundary points of this AB?
39:59:290Paolo Guiotto: So, the interior is made by the same interval, except A and B. Remind that for the interior, I must be this.
40:09:370Paolo Guiotto: We care about that.
40:10:600Paolo Guiotto: And the new point is to look at these points here. We already discussed the points which are in the hero, we know that they are contained also.
40:20:50Paolo Guiotto: It's the same discussion. The only point is, what about A and B? A and B are in the set in this case. But, if you take a ball here, no matter how small is the ball, you see that there is always something which is outside of the set. You don't find any ball which is anterior contained.
40:38:350Paolo Guiotto: And this means that these two points are not even the. So, this means that the integer is only AB without the two endpoints.
40:48:600Paolo Guiotto: About the boundary, The boundary is the same as before.
40:53:420Paolo Guiotto: Okay, same argument. You take a point here, and you take a small library, you see that there are only red points and non-led points, so this means one point is best, and no points best for the pen.
41:04:580Paolo Guiotto: And for the boundary, you must have, no matter how you take the number, so for every ball, big or small, there must always be points of S and points of the fundamental.
41:16:630Paolo Guiotto: If you find one for which there are only points of X, or only points of X complementary, this means that that point is not a boundary point.
41:25:580Paolo Guiotto: Okay? So these are not boundary points, and for A and B, it follows the same argument. I take a ball here, and no matter how this ball is pulled, I always find points of S and point of S on the menu. So the boundary in this case is still the set made of the two points A and B.
41:45:640Paolo Guiotto: Okay?
41:50:510Paolo Guiotto: Okay.
41:51:470Paolo Guiotto: Now, Let's move a bit, forward.
41:56:920Paolo Guiotto: And let's introduce, This, new definition.
42:03:580Paolo Guiotto: This is the definition of open set. So we say.
42:12:140Paolo Guiotto: a sect.
42:14:720Paolo Guiotto: I am going to do a beat.
42:18:870Paolo Guiotto: We say that… a set S contained in RV.
42:24:940Paolo Guiotto: is open.
42:30:880Paolo Guiotto: If a… Well, a set is safe to be open if every point is an interior point.
42:38:820Paolo Guiotto: Now, we can say this in equivalent ways. If… Every… point.
42:49:770Paolo Guiotto: of S, E's, N.
42:53:750Paolo Guiotto: interior.
42:55:690Paolo Guiotto: point.
42:57:560Paolo Guiotto: So this means that, for every X in S,
43:03:990Paolo Guiotto: there exists a ball, there exists a radius, positive, such that the ball centered attacks with the radius R,
43:13:980Paolo Guiotto: is contented into S.
43:16:840Paolo Guiotto: Or equivalently.
43:18:810Paolo Guiotto: We can state this in this way. Now, we take the interior of S, the set made of the interior points.
43:27:170Paolo Guiotto: We know that the interior points are a subset, in general, of the set, okay? So this is always contended into S.
43:37:330Paolo Guiotto: Now, what we say here…
43:40:210Paolo Guiotto: is exactly saying that the two are the same, because I'm saying if every point of S is an anterior point. So, interior points of S are always contained into S, this is always true, but what they say here is a bit more. I'm saying that they are equal, because this says that
43:58:400Paolo Guiotto: Every point of S belongs to the interior, so it is the individual property, and this property a folder
44:05:790Paolo Guiotto: So, a way to write OpenSet is this one.
44:11:40Paolo Guiotto: Now, since, bullet.
44:14:10Paolo Guiotto: Technical reasons,
44:17:140Paolo Guiotto: We sometimes, we will have to consider also a special set, like the empty set, the set made of no points.
44:26:180Paolo Guiotto: It is convenient to assume that that set is, by definition, open, okay? So, the… Empty sector.
44:41:200Paolo Guiotto: is open.
44:44:220Paolo Guiotto: by definition.
44:46:610Paolo Guiotto: Otherwise, we should do exceptions in certain situations.
44:54:70Paolo Guiotto: Okay, so, let's return to some examples.
45:05:780Paolo Guiotto: So, we introduced the… now I will do the examples in the plane, so we change the environment.
45:15:200Paolo Guiotto: We introduced these two definitions of open and closed ball.
45:21:280Paolo Guiotto: The fact that we call the open and close the ball is not… is not incidental, it's just due to this definition. If we take the open ball, what we call the open ball, this one.
45:34:750Paolo Guiotto: This is an open sector.
45:40:490Paolo Guiotto: according to this definition we just, we just introduced. Now, let's do a figure of this situation in dimension 2, okay? So I…
45:52:920Paolo Guiotto: I am in a Cartesian plane, huh?
45:55:560Paolo Guiotto: And we know that an open ball is a disc center that's at some point, X.
46:03:610Paolo Guiotto: with the radius R…
46:06:460Paolo Guiotto: So it is everything contained into the circle without the edge of the circle, okay? So this is,
46:17:880Paolo Guiotto: the open bowl. Now, intuitively, if you think about the…
46:22:760Paolo Guiotto: To be open means that every point belongs to the interior, so every point is contained into the set with a ball.
46:31:520Paolo Guiotto: with a certain ball. If you look at the figure, this seems to be clear, because I take this point. Is there a ball and chili containing in that set? Yes, take a small ball, like this one, and you are done. Take a point very close to that dashed line.
46:47:910Paolo Guiotto: You can find a very small, teeny ball.
46:51:890Paolo Guiotto: You can do a Zoom, you can check that I'm still inside, and so on, okay? So the idea is that, intuitively, do you think I will respond to a call from Greece?
47:04:800Paolo Guiotto: Well, I don't know anyone. Look.
47:09:890Paolo Guiotto: That's this.
47:11:450Paolo Guiotto: Okay.
47:13:950Paolo Guiotto: Now, let's see formally why this is true. You will see it's a little bit more complicated, but it's a nice exercise for me, okay? So, let's show…
47:25:940Paolo Guiotto: Well, since we are, 117… well, do you want to take the break?
47:32:190Paolo Guiotto: Before we see this, huh?
47:34:460Paolo Guiotto: It will take 5 minutes, if you want, we do this, then we do the break, okay?
47:39:390Paolo Guiotto: So let's show this in general. I will do figures in the plane, just to add the intuition, but what we see works whatever is the dimension. So, that's why I will write this RD, as if it is RD, okay?
47:54:700Paolo Guiotto: So let's imagine… let's redo the figure. This is our X.
48:00:690Paolo Guiotto: This is the bowler. Let's dash the age.
48:05:910Paolo Guiotto: I'm not using the word boundary, because it is the boundary factor, but this will be checked later. So this is the radius R.
48:16:950Paolo Guiotto: Now, what I have to prove, to prove that this is an open set, I have to pick a generic point in the ball.
48:24:940Paolo Guiotto: and show that there is a little ball centered at that point, anteriorly contained into the big ball, okay? So this one is the big ball.
48:38:230Paolo Guiotto: So this is the ball centered at X radius R.
48:44:300Paolo Guiotto: And this red one is a ball centered at point Y, with some radius, let's say, raw, that I have to determine… to be determined in such a way that it is contained, okay? So,
49:00:980Paolo Guiotto: Let's see why. Solution.
49:03:660Paolo Guiotto: So, here, the set S is the bowler, close the baller, BXR.
49:12:240Paolo Guiotto: we… have… 2.
49:16:980Paolo Guiotto: Verifying…
49:19:540Paolo Guiotto: that I'm just rewriting the definition we have given here, specializing on this case, that for every point Y that belongs to the set S, which is in this case the ball centered attacks with the radius R,
49:40:190Paolo Guiotto: Okay, there exists a ball, I do not use the same letter for the radius, first of all, because it is not true that it is the same radius, but in general, I have to show that there exists a ball with a certain radius, so there exists a radius raw.
49:53:690Paolo Guiotto: positive, such that the ball centered at Y with the radius raw is contained into the ball centered at X with the radius R.
50:05:870Paolo Guiotto: This is what I have to prove for this case, because this is set. No matter how you pick a Y here, you can find a bowler, probably the sweet of our greatest raw, which is contained in the set. This guy, down here, is our S for this example.
50:24:670Paolo Guiotto: Now, what is the radius of the ball I'm looking for? So, the radius of the red ball. I need to write a number, okay? I cannot just be convinced by the figure.
50:36:420Paolo Guiotto: I want to do a formal check of this. So, my radius I'm looking for is this quantity, this is raw.
50:46:210Paolo Guiotto: I want that Rob is small enough in such a way that that ball is contended in the big ball. How can I do that?
50:55:510Paolo Guiotto: Well, here, there is a little geometrical argument that helps to find out the solution, because Well, it's…
51:08:770Paolo Guiotto: Because if you look at this figure.
51:11:200Paolo Guiotto: Imagine I start from the center… sorry.
51:16:570Paolo Guiotto: I pass through point Y, Yeah, I'm not passing to the point.
51:23:950Paolo Guiotto: Okay, with a straight line, okay? Now, this segment, this green segment here, is a radius, so it will have length R.
51:35:180Paolo Guiotto: Now, as you can see, my point why, the center of the bowl.
51:39:800Paolo Guiotto: I want to show… it is contained into the big bowler. It's at some distance to X. Now, I know what is this distance, physically this distance. What is this?
51:54:50Paolo Guiotto: This is the distance between the two points, Y and X. So, what is this?
52:02:450Paolo Guiotto: the norm of X minus Y, exactly.
52:05:890Paolo Guiotto: So that quantity is a norm of X minus 1.
52:09:730Paolo Guiotto: Now, what is the radio, the blue radius? The blue radius is something here that must be smaller than this remaining path. You see that there is a remaining length?
52:21:800Paolo Guiotto: Now, what is this remaining gland?
52:24:710Paolo Guiotto: So, if everything is long, R, this is longer. This, this piece will be long.
52:33:570Paolo Guiotto: R minus this quantity. So I know that this green piece down here.
52:40:510Paolo Guiotto: is longer R minus norm of X minus y.
52:46:630Paolo Guiotto: Okay, so my blue number must be smaller than this.
52:51:430Paolo Guiotto: So, for example, I can take this number and divide by 2.
52:55:370Paolo Guiotto: Okay? So this is what I will do. Take as raw this number, R minus distance between X and Y, divided by 2, so the half of that distance. So I'm sure that that number is smaller than that distance.
53:13:660Paolo Guiotto: And now let's check, really, that with that choice of raw.
53:18:130Paolo Guiotto: the ball centered at that Y, with that radius raw, is contended into the big ball.
53:24:130Paolo Guiotto: The ball centered attacks with radius R.
53:27:10Paolo Guiotto: Okay, so, letter… As we say, raw, by definition, I take R minus the distance between X and Y,
53:38:00Paolo Guiotto: Divided by 2.
53:40:690Paolo Guiotto: The motivation of this choice comes from the previous argument.
53:45:330Paolo Guiotto: Now, the goal is to prove Prove that.
53:52:190Paolo Guiotto: We want to prove that the ball, centered at Y with this, exactly this radius, is contented into the ball, centered at the X with radius R.
54:04:380Paolo Guiotto: Now, to show this, we pick a point into this ball, and we show that it belongs to the other ball.
54:12:60Paolo Guiotto: So, let appoint, Z.
54:15:760Paolo Guiotto: In the bowler, Y, Raw.
54:21:610Paolo Guiotto: So, we want to prove that it belongs in the ball center that text with the radius up.
54:28:640Paolo Guiotto: We… want… to Ruba.
54:34:630Paolo Guiotto: that this Z belongs to the ball centered at X radius R. Now, this happens If and only if…
54:44:690Paolo Guiotto: the distance between Z and X…
54:49:160Paolo Guiotto: is less or equal than R.
54:53:340Paolo Guiotto: no, sorry, the ball here is open.
54:59:730Paolo Guiotto: So this is open.
55:03:660Paolo Guiotto: So this is a strict sign. We want to prove this.
55:08:10Paolo Guiotto: Now, how can I prove this? So, in the figure, I will repeat the figure down here.
55:15:170Paolo Guiotto: The figure is, I have the big ball here.
55:18:660Paolo Guiotto: That's centered at point X.
55:21:480Paolo Guiotto: I have a Y somewhere, and I have my supposed small ball. I don't know yet if it is contained. I'm drawing as if it is contained, but I don't know yet. Now, I am picking a point Z somewhere here, so this is our Z.
55:40:280Paolo Guiotto: And I want to show that the distance between Z and X, this one, is less than R.
55:47:800Paolo Guiotto: How can I do that? Well, I do by using the triangle inequality, so I will assess the distance between this and this. I know that's less than raw, because I am in the small ball, and between this and this, this is known. So, in fact, I can say that the distance now
56:06:770Paolo Guiotto: The distance between Z and X,
56:11:160Paolo Guiotto: If you write this as Z minus Y, plus Y.
56:18:420Paolo Guiotto: minus X, so I'm just adding and subtracting Y.
56:24:280Paolo Guiotto: Nothing is changing. Now, you group this in this way.
56:28:430Paolo Guiotto: And you apply the triangular inequality.
56:33:600Paolo Guiotto: norm of the sum is less or equal than sum of the norms. So this is less or equal than norm of Z minus Y,
56:43:330Paolo Guiotto: plus norm of Y minus X.
56:46:470Paolo Guiotto: But now… Norm of Z minus Y is the distance between Z and Y.
56:52:280Paolo Guiotto: I know that my Z has been taken here, in the ball centered at Y radius rho, so the distance between Z and the center won't be greater than raw.
57:04:360Paolo Guiotto: So this number is less or equal raw. Why? This one, I leave there, fixed, so I know that this is less or equal than raw plus norm of Y minus X.
57:19:00Paolo Guiotto: And I'll remind what was the definition of raw, because this raw is not any number, it's this number.
57:25:390Paolo Guiotto: So we just copy that number inside. We get this is R minus distance between Y and X.
57:34:850Paolo Guiotto: Divided by 2.
57:37:270Paolo Guiotto: plus the distance between Y Index.
57:41:550Paolo Guiotto: I'll do the algebra, and what you get is…
57:54:470Paolo Guiotto: news.
57:58:550Paolo Guiotto: Now, there is something which is wrong here.
58:05:10Paolo Guiotto: Because it shouldn't… it should come less than R.
58:10:700Paolo Guiotto: Oh, yes, yes, because I get…
58:14:360Paolo Guiotto: R, if I do the algebra, do the common denominator, it comes plus distance between Y and X.
58:23:70Paolo Guiotto: Divided by 2, right?
58:25:810Paolo Guiotto: If you do the algebra. But this number is the distance between Y and X. What is Y?
58:31:600Paolo Guiotto: Why is, since the beginning, Here, sorry, that's the open bowl, also here.
58:40:160Paolo Guiotto: Because the set was the open ball, right?
58:45:00Paolo Guiotto: So our set was this one, the open board.
58:48:140Paolo Guiotto: So we pick a point at Y in the open bowl, and therefore the distance between Y and X is strictly less than R. So this number here is less than R, so this number is strictly less than R plus R divided 2, and that's finally R.
59:06:350Paolo Guiotto: So at the end, we obtained that the distance between Z
59:10:950Paolo Guiotto: And X is strictly less than R, and this exactly means that Z belongs to the ball.
59:19:300Paolo Guiotto: open, bold, centridadex with the radius R.
59:23:230Paolo Guiotto: So this shows what we needed to show, that the ball, centered at Y with radius raw is contained into the big ball. This works whatever is Y in the big ball, and therefore, this means that every point of the big ball is contained with a small ball anteriorly.
59:43:470Paolo Guiotto: Inside. And that means that this set is open.
59:47:200Paolo Guiotto: Okay.
59:49:110Paolo Guiotto: Want to take the bracket?
59:51:90Paolo Guiotto: Okay.
59:52:630Paolo Guiotto: Let's do the background.
00:01:610Paolo Guiotto: I shouldn't memorize.
00:03:280Paolo Guiotto: No.
00:04:880Paolo Guiotto: So, there is nothing to be based on… No, there will be theoretical questions, but you will ever… to be able to do some proof.
00:13:440Paolo Guiotto: Maybe of something which is not what we do. I don't want to memorize. That's completely used to the standard, wrong way to study mathematics. So that's why I would not ask you to do that, to repeat the book we do in class.
00:26:110Paolo Guiotto: But, try to understand the arguments, yeah, yeah.
00:31:330Paolo Guiotto: I… we will talk about later about this, okay?
00:37:310Paolo Guiotto: It's too early now.
00:43:330Paolo Guiotto: Okay, take your seat, please.
00:51:480Paolo Guiotto: Okay, so we introduced this definition of open Saturn. Now, there is a twin concept, which is the concept of closed Saturn.
01:07:980Paolo Guiotto: and sat… S… Continent in RD.
01:14:120Paolo Guiotto: Jeez.
01:15:140Paolo Guiotto: Closed.
01:18:510Paolo Guiotto: Well, the formal definition is relatively easy, but it's not immediately intuitive. If it's complementary, as complementary is open.
01:37:910Paolo Guiotto: Now, here, a common, Very frequent error.
01:45:10Paolo Guiotto: very frequent, very frequent mistake.
02:01:610Paolo Guiotto: Is that, you see, look at this definition.
02:06:980Paolo Guiotto: The set is closed if its complementary is open. Now, you may think that a set can be only open or closed.
02:16:130Paolo Guiotto: Because there is this alternative, the complementary, you can do this kind of mess. A set… S is. Either.
02:29:940Paolo Guiotto: open.
02:32:470Paolo Guiotto: or closed.
02:34:660Paolo Guiotto: This is a false, completely false belief.
02:41:600Paolo Guiotto: Now, let me explain you with examples that there are sets which are known of the two properties, so that are not open and neither closed.
02:50:910Paolo Guiotto: For example, we go in the real line.
02:54:770Paolo Guiotto: Take a set, which is an interval, but closed in one side, open in the other.
03:03:600Paolo Guiotto: Like this one.
03:05:860Paolo Guiotto: numbers which are greater or equal than A and strictly less than B. This set as… S… ease…
03:15:30Paolo Guiotto: Neither.
03:19:680Paolo Guiotto: open.
03:23:190Paolo Guiotto: No.
03:24:550Paolo Guiotto: Closed.
03:28:470Paolo Guiotto: It is not open because, it is… not… Open.
03:37:960Paolo Guiotto: Because open means every point belongs to the set with a little ball that, in this case, is an interval. And we know that this point, A,
03:48:730Paolo Guiotto: Is not contained with the ball, no?
03:52:420Paolo Guiotto: A is a point of the set which is not in the interior of the set.
03:56:930Paolo Guiotto: Because…
04:01:480Paolo Guiotto: A belongs to S, but…
04:05:20Paolo Guiotto: A is not an interior point of S. We remind that open means that all the interior points of S are the points of S themselves, okay?
04:17:390Paolo Guiotto: So, this set is not open.
04:21:640Paolo Guiotto: It is not… Close them.
04:26:690Paolo Guiotto: Because… So we said that, by definition, closed means the complementary is open.
04:34:970Paolo Guiotto: What is the complementary of this set?
04:39:120Paolo Guiotto: Because S complementary for this case is this. We have the half line up to A excluded, then we take the half line from B to plus to B.
04:52:370Paolo Guiotto: That's the complementary of S.
04:54:970Paolo Guiotto: Now.
04:56:70Paolo Guiotto: Is this set open? Because to be closed, S, we must have that the complementary is open. That's the definition.
05:05:530Paolo Guiotto: Now, is this set open? No, because if you look at point B, now, this point is never anteriorly contended into the set as complementary with a ball. So, the set is not closed, because S complementary
05:25:170Paolo Guiotto: S complementary is not open.
05:30:780Paolo Guiotto: So B belongs to S complementary, but B is not in the interior of S complementary.
05:39:880Paolo Guiotto: So S complementary cannot be open. Otherwise, the interior of S complementary would coincide with S complementary.
05:47:700Paolo Guiotto: So, you cannot say that a set is open or closed because of this definition that can lead you to some error. A set is closed if and only if the complementary is open.
06:01:820Paolo Guiotto: That's, that's, that's it about the definition. Well, another, very frequent error…
06:13:900Paolo Guiotto: Very frequent mistake. Very common mistake.
06:18:820Paolo Guiotto: Is the following.
06:22:620Paolo Guiotto: You may think that… so, this, the first one was a set is either open or closed, so it's sort of alternative.
06:32:720Paolo Guiotto: And that's false. There are at which are none of the two.
06:36:720Paolo Guiotto: And, you can also think that there are no sets that can be both open and closed, because there is this… this history of the complementary that… that…
06:47:330Paolo Guiotto: So, a set… a set S cannot be… Both.
06:57:760Paolo Guiotto: open… and closed.
07:04:340Paolo Guiotto: Also, this is false.
07:08:330Paolo Guiotto: There are sets which are both open and closed at the same time.
07:14:40Paolo Guiotto: Of course, these are exceptional sets, which are the following. Actually, it's only in the following two sets. So, the full space, RD and the empty are both
07:29:230Paolo Guiotto: Open.
07:31:80Paolo Guiotto: And…
07:32:130Paolo Guiotto: closed, and actually it can be proved, but that's difficult, that they are the unique one set that can be both open and closed. Let's see why, for example, empty is at the same time open and closed, and the same is forever, as you will understand. For example…
07:55:800Paolo Guiotto: S equals empty is… Open.
08:01:490Paolo Guiotto: by definition.
08:03:640Paolo Guiotto: We said we can… well, actually, we could say that the characteristic property of the open sets is actually verified, because we said that S is open if every point of S is an interior point. If there are no points, every point is in the interior, because there is nothing to check.
08:23:430Paolo Guiotto: You cannot say that this is false, but there is no point that does not verify this condition. So all points
08:29:880Paolo Guiotto: which are no points, verify this. But, however, we defined that the empty set is open by definition. So, empty set is open, and the complementary of empty set, in this case, is water. The complementary of nothing is everything. So, it's the full space RD.
08:50:29Paolo Guiotto: Now,
08:52:660Paolo Guiotto: why I'm saying that I'm interested in S complementary? Because to say if S is closed, I need to verify if S complementary is open. So, S complementary is RD. RD is open.
09:10:460Paolo Guiotto: Which is… Which is, let's say, trivially.
09:19:310Paolo Guiotto: Open.
09:20:609Paolo Guiotto: Because it is clear that if the set is the full space, RD, imagine the plane, you pick any point, you pick any ball, it is always contained into the set. So, every point is in the interior of RD, for every point of RD.
09:37:740Paolo Guiotto: interior points of RB, R, RB,
09:44:680Paolo Guiotto: is the set of the itself.
09:47:750Paolo Guiotto: So, this means that, since S complementary is open, we deduce that S is closed.
09:59:180Paolo Guiotto: from this. Okay?
10:02:470Paolo Guiotto: So you see that S equals empty is open and closed.
10:07:570Paolo Guiotto: And the same holds for RV, because RV is open, PR, and the complementary of RD is what?
10:15:920Paolo Guiotto: Complementaries of everything is nothing. It's the empty set. So the complementary set of RB is empty, which is open.
10:24:750Paolo Guiotto: So it is true that if your set is arguing.
10:27:790Paolo Guiotto: Both the set and each complementary are open, so the complementary open means the set is closed. So the set is both open and closed.
10:38:300Paolo Guiotto: Okay.
10:40:790Paolo Guiotto: Now, this, definition
10:44:420Paolo Guiotto: Of closed set is the mathematical definition. It's not immediately… well, we could say, how do we verify if a set is open? We take the complementary, we check if it is open… if it is open.
10:58:210Paolo Guiotto: Okay? There is another equivalent property, which is the following, which is called the Kantor theorem.
11:11:330Paolo Guiotto: Which is important to deduce a practical test for concrete cases.
11:18:650Paolo Guiotto: So, the following two properties are equivalent. The following… properties.
11:32:530Paolo Guiotto: are equivalent.
11:36:290Paolo Guiotto: So, number one is the set S is… Closed.
11:43:620Paolo Guiotto: The number 2 is the following test. For every sequence of vectors, Xn, contained in S, such that
11:54:800Paolo Guiotto: there exists the limit for the sequence XN,
12:00:830Paolo Guiotto: when n goes to infinity equal to some vector L of RTE,
12:08:970Paolo Guiotto: then, necessarily, that limit belongs to S.
12:14:710Paolo Guiotto: So the property 2 says, in other words, that the ZS contains
12:21:600Paolo Guiotto: all possible limits of sequences made of each point. So if you take a sequence of points of S that has a limit L, necessarily that limit must be into the set S.
12:35:690Paolo Guiotto: Okay? So, S contains… all… possible.
12:47:50Paolo Guiotto: Limit… points.
12:51:960Paolo Guiotto: All.
12:52:900Paolo Guiotto: sequences.
12:57:840Paolo Guiotto: made… Paul.
13:02:570Paolo Guiotto: vectors.
13:06:520Paolo Guiotto: office.
13:08:930Paolo Guiotto: Now, this is, is not for the practical user. In practice, you can… if S is particularly simple.
13:19:60Paolo Guiotto: it could be done this, but this simple means that if S is finite, this can be done, this check. But if S is infinite, as normally it happens, this is not important for the practical application, but it will become important
13:36:00Paolo Guiotto: to get a test for common types of sets that we will seek.
13:42:300Paolo Guiotto: after this. Well, let's see this proof, because it's basically a combination of the concepts we have seen, so it is a way to practice the definitions. One of you asked me, have we to study, memorize proofs? No.
13:57:440Paolo Guiotto: I repeat, I'm not asking you to memorize proofs, but you should study to have an understanding of what we are doing, to have an understanding, a better understanding, a deeper understanding of definitions, ideas behind all this. So this course is not offering… is not just offering you a number of recipes to solve a certain number of problems.
14:21:800Paolo Guiotto: If you think that this is the coast, you are in the wrong place, okay?
14:26:350Paolo Guiotto: Okay, so this is not a cooking show. I give you recipes, put this 300 grams of flowers, 2 eggs, and 100 grams of butter, and you do this pie. That's not the course that we are doing, okay?
14:40:930Paolo Guiotto: So, we do proofs because we…
14:45:80Paolo Guiotto: We… we… we prove studying these properties helps us to understand the definition, should help us to understand definition, help us to do abstract reasoning, which is a very important, important, skill that you must learn.
15:02:880Paolo Guiotto: Before the end of your academic life.
15:06:250Paolo Guiotto: Because when you want to present a project, you must follow a logical scheme, and you must be able to show that what you do follows, because you have done this and that, because there are dead principles, and it falls from a logical argument that's not just because
15:23:600Paolo Guiotto: I go from the sorcerer, and it tells me that it works, or I go to the priest, and it tells me that I have to
15:30:720Paolo Guiotto: to be… to have faith, or things like that. It's a rational process, that's what… what mathematics should teach us.
15:38:450Paolo Guiotto: button…
15:40:980Paolo Guiotto: I may ask you, I explained to try to do abstract reasoning, not showing me that you memorized the proof, but that you are able to do a little proof by your own. It is clear that if you never study a proof, you will never be able to do a new proof.
15:59:230Paolo Guiotto: Okay? Because, as I told you, most of the work you are doing in your life is adapting known ideas to new problems. So, you must master the old ideas to solve the new problems. Otherwise, you will never be able to solve a new problem.
16:17:70Paolo Guiotto: Okay? If you have never seen how an engine works, you cannot do the project of a new car.
16:22:960Paolo Guiotto: It's simply impossible, and nonsense, even.
16:25:810Paolo Guiotto: Okay, so let's see the little proof of this factor.
16:31:410Paolo Guiotto: So it's an equivalence, so we have to prove that these two properties follows each from the other. So, let's do the first implication. The first implication is this one, so the hypothesis is number one, set S is closed, so this is an assumption.
16:51:130Paolo Guiotto: And the thesis is what we have to prove, so the point we have to reach. We have to prove that that property, which is listed here, that's a star, give a name, we do not copy, star holds
17:07:430Paolo Guiotto: True.
17:09:280Paolo Guiotto: So, what should I prove? I should prove that whenever I take a generic sequence of points of S, such that this sequence has the limit L, then necessarily that limit is in the set S.
17:25:360Paolo Guiotto: We can help the intuition with figures, if possible. So, let's do this. So, we have our set S. Let's say that it is a set like that, okay?
17:36:730Paolo Guiotto: Now, we have to show that whenever we take a sequence of points of S, so we imagine a sequence of points, vectors, moving around in S. So this is our XN.
17:49:140Paolo Guiotto: such that this sequence goes to a limit L,
17:53:130Paolo Guiotto: Now, where do I put this limit? I don't know. The sequence could go here.
17:57:860Paolo Guiotto: Or could go here.
17:59:900Paolo Guiotto: Now, I have to prove that, necessarily, Maya must be inside, so cannot be here.
18:08:170Paolo Guiotto: That's what he's saying. So this sequence converges to something. That something must be inside the S, cannot be outside. Okay?
18:17:410Paolo Guiotto: Why this? This should follow by the fact that, as is called.
18:21:580Paolo Guiotto: Now, let's see why.
18:24:180Paolo Guiotto: Now, as closed means that, by definition, the definition says a set is closed if the complementary is open. That's what we know.
18:35:520Paolo Guiotto: So, now we want to exclude that this happens.
18:40:100Paolo Guiotto: And therefore, typically, when we want to expose something, a way to do an argument is, let's assume that that happens, so the impossible happens. Let's see that we get a logical contradiction. Once we get the contradiction, we can say the impossible cannot happen, okay? So…
18:58:850Paolo Guiotto: Suppose, by contradiction, suppose by… contradiction.
19:10:970Paolo Guiotto: bet.
19:13:80Paolo Guiotto: Our L, in fact, belongs to the complementary of S, so it's not in S.
19:19:620Paolo Guiotto: Okay? And something should go wrong. So imagine that our sequence go here. Let's try to understand why this is not possible.
19:29:130Paolo Guiotto: Because, imagine, if the sequence gets there, what does it mean that the limit is that point? It means that this sequence goes there, so gets close to that end, no? Because the middle limit means that the distance goes to zero. So you have to imagine that the sequence
19:46:500Paolo Guiotto: Should go something like that.
19:49:40Paolo Guiotto: What do you see?
19:53:260Paolo Guiotto: To reach this point, I should…
19:57:400Paolo Guiotto: I should leave S. And this is not allowed, because my secret is… I'm staying.
20:03:330Paolo Guiotto: Now… Why this really happens.
20:08:00Paolo Guiotto: And this is just because of this.
20:11:350Paolo Guiotto: Now, since, Boom.
20:17:80Paolo Guiotto: We know… that the sequence Xana
20:23:680Paolo Guiotto: is, on one side, contained into the set S, okay? And the sequence goes to L.
20:34:570Paolo Guiotto: So, in particular, this means that the distance between XM and the limit Goes to zero, right?
20:44:100Paolo Guiotto: So this means that XN gets closer and closer to the limit L.
20:49:350Paolo Guiotto: But L is, we are supposing, in S complementary, which is open. What does it mean that you are in an open set?
20:57:550Paolo Guiotto: It means just one thing, we have seen the definition. It means that you are in the interior of that set, so it means that there exists a ball centered at point L, which is anteriorly contained not in S, because it is not S to be supposedly unopened, but in S complementary, okay?
21:16:850Paolo Guiotto: Sings that… S complement, S closed.
21:24:280Paolo Guiotto: implies that S complementary is open, And… L belongs to S complementary.
21:34:790Paolo Guiotto: by the characteristics of the open sets, there exists a ball centered at L, with some radius R, entirely contended not in S, but in S complementary, because it is S complementary to be the open set, yeah.
21:50:930Paolo Guiotto: Okay?
21:52:730Paolo Guiotto: And this is the red ball, that's the ball BL.
21:57:590Paolo Guiotto: R. Now you see that we are getting close to the contradiction.
22:01:510Paolo Guiotto: Because the sequence, the black points here, the accent points, to get close to L, sooner or later, we'll have to enter that ball. And if they enter that red ball, it means that they are outside of S, and that's the contradiction. Now, let's make this precise.
22:21:170Paolo Guiotto: And where this comes from. It comes from this.
22:24:110Paolo Guiotto: Since this distance is going to zero, it means that sooner or later, it will be smaller than this number here.
22:34:230Paolo Guiotto: Because that's the definition of limit. In your mind, for every epsilon, Pora, since…
22:42:360Paolo Guiotto: For every epsilon positive, there exists an initial index n, such that the distance between XN and limit, L, is less or equal than epsilon for every n larger than capital N. This is the division of limit we have seen
22:57:280Paolo Guiotto: some classes ago, now take epsilon.
23:03:500Paolo Guiotto: equal R… And this says that…
23:09:50Paolo Guiotto: The distance between XN and the limit will become less or equal than R for every n larger than some initial capital N, but this means that your point XN has gone inside the ball centered at limit L with the radius R.
23:28:750Paolo Guiotto: And that ball is where? It is contained into as complementary. So, your point, as intuitively, we understood, belongs to as complementary, and that's impossible.
23:44:80Paolo Guiotto: Impossible? Why is it impossible? Because we took this sequence into S. And why did we took this into S? Because the property says you have to verify that whenever you take a sequence of point of S that has a limit, that limit is in S. So we… we did this because the sequence was supposed to be in S.
24:03:960Paolo Guiotto: So, since we got a contradiction, it means that our initial assumption, suppose that by contradiction, that L is in S complementary, is wrong.
24:15:250Paolo Guiotto: So it means that L can be only in S. No, see if it is not. So we concluded that L must be in S. And that's the end of the first implication.
24:27:450Paolo Guiotto: Now, the reverse implication is a hypothesis.
24:33:100Paolo Guiotto: We assume that star property, true, and thesis… Is our status is closed.
24:46:300Paolo Guiotto: Which means, by definition, it's complementary is open.
24:53:390Paolo Guiotto: So we have to prove that S complementary is open. So let's do again a figure.
25:00:110Paolo Guiotto: So that's our S.
25:02:960Paolo Guiotto: We are… that's S complementary. We have to verify that S complementary is open if that property star holds. So let's pick a point here.
25:13:590Paolo Guiotto: Okay? Let's call it L. You will understand why we call it L. Sophistic and L in S complementary.
25:22:870Paolo Guiotto: And we have to show that… what?
25:26:320Paolo Guiotto: S complementary is open if every point belongs to the set with an anterior ball. So we have to show that there is a ball anteriorly contained in S complementar.
25:38:840Paolo Guiotto: Okay? So let's see what happens if this is false, if there is no ball.
25:45:190Paolo Guiotto: Okay? So, the thesis is equivalent to prove that there exists a ball, B, let's say, L, R, anteriorly contended, not in S, because the set we have to prove it is open is complementary, S complementary.
26:03:180Paolo Guiotto: If false, huh?
26:08:250Paolo Guiotto: What does it mean?
26:10:470Paolo Guiotto: It means that there is no voluntarily containing.
26:14:600Paolo Guiotto: So it means that whatever is the ball, for every ball B, L, Honor?
26:21:890Paolo Guiotto: what happens? It is not contained anteriorly into S complementary. It doesn't mean that it belongs to S, okay? And it cannot be in S, because the center is in S complementary.
26:36:380Paolo Guiotto: It doesn't mean that this is not the case, the volume is down here.
26:40:960Paolo Guiotto: Nothing.
26:42:80Paolo Guiotto: But it means that this ball, if it is not into S complementary, it must contain points of X. That's the…
26:53:890Paolo Guiotto: the meaning of this. So this means that for every ball, BLR,
27:00:890Paolo Guiotto: This point's intersection with S is non-empty.
27:05:560Paolo Guiotto: Now, you may think it's an accumulation point, because maybe you have memorized that definition. That's not the case, no?
27:14:420Paolo Guiotto: But let's see what this property says. Now, for every ball, why there is for every?
27:20:670Paolo Guiotto: Because there are many balls centered at that point L, what changed the radius?
27:26:180Paolo Guiotto: So it says that if you take this ball, there is a point, but if you take a smaller ball, like this one, there is another point of S still inside here.
27:38:160Paolo Guiotto: You see what is the game, huh?
27:40:940Paolo Guiotto: I take a ball, there is a black point of S to there, but it's here. I take a ball a hole that is a point which is closer.
27:50:230Paolo Guiotto: I think a third ball is still more than this green one. I have another point much more closer than the previous one.
27:58:610Paolo Guiotto: It seems like if I'm giving the people for points that go there.
28:02:800Paolo Guiotto: And that's what we do.
28:04:310Paolo Guiotto: take as radius, R…
28:06:710Paolo Guiotto: Radius 1 divided by N, where N can be 1, 2, 3, 4, etc, the naturals.
28:15:40Paolo Guiotto: So these radios are going down to zero.
28:18:120Paolo Guiotto: So this says that the ball centered at point L with radius 1 over N
28:25:720Paolo Guiotto: has points of S, at least one.
28:29:740Paolo Guiotto: So, this means that there exists a point that we will baptize Xen.
28:36:700Paolo Guiotto: That belongs to the ball, centered at L, radius 1 over N, and, at the same time, is in S.
28:45:930Paolo Guiotto: Okay?
28:47:220Paolo Guiotto: So you have these balls centered, radius 1, radius 1 half, radius 1 3rd, radius 1 fourth, and so on, and we find always a point of S into each of these balls.
29:00:580Paolo Guiotto: So now we have what? This guy here is a sequence.
29:05:420Paolo Guiotto: of points. Where are these points?
29:08:440Paolo Guiotto: By construction, they are in S, so this is a sequence of points of S.
29:13:790Paolo Guiotto: What is this sequence doing? If you compute the distance between XN and that point L,
29:21:900Paolo Guiotto: Well, since you are in this bowl.
29:24:880Paolo Guiotto: This is bounded by the radius of the ball, which is 1 over N.
29:30:410Paolo Guiotto: So the distance between Xn and that point L is controlled above by 1 over n. So what happens when n goes to infinity?
29:41:270Paolo Guiotto: This goes to… Zero.
29:44:10Paolo Guiotto: Because it is positive, is bounded above by this.
29:48:180Paolo Guiotto: squeeze theorem, this quantity goes to zero. But this means that these X times, Doing well.
29:56:20Paolo Guiotto: If this data goes to 0xN, XN goes to L, so…
30:03:700Paolo Guiotto: XN goes to L, and look what we obtained.
30:08:280Paolo Guiotto: We obtained S sequence of points of S.
30:12:290Paolo Guiotto: which is going to hell.
30:16:250Paolo Guiotto: And what is the hypothesis? Here, the hypothesis is the star property. What is the star property?
30:22:910Paolo Guiotto: It says that whenever we have a sequence of points of F that goes to some limit L, that point must be in S, okay? So this must happen also for L.
30:35:450Paolo Guiotto: So, hypothesis… L is in S.
30:41:500Paolo Guiotto: And here we have the contradiction, because this guy is saying that Al is an ass.
30:47:630Paolo Guiotto: But where do… where we started was…
30:51:860Paolo Guiotto: we took a point S, L, which is not in S. It was in S complementary, and that's a contradiction. So it means that if false cannot be
31:06:40Paolo Guiotto: Okay, so that's the contradiction.
31:09:510Paolo Guiotto: impossible.
31:12:710Paolo Guiotto: And so, we have that thesis must be true.
31:18:20Paolo Guiotto: Okay, let's stop here, and you think about this. On Friday, we will return, and we will apply this to get a concrete test to, to distinguish open from closed sets. Okay, so please review this before next class.