October 5, 2013 at 23:30Topology for the smallest. Part 2
In this article, I continue my gentle introduction to topology. The first part is here .
Again, I warn you that everything you read is written twice by the humanities (bachelor and master), so you should not believe blindly. In general, you are warned.
Comments about errors (mathematical) are welcome.
Another warning is a lot of pictures.
Image to attract attention (not related to our text).
How do you think, without breaking these figures, but deforming in any way, is it possible to disconnect them?
Initially, I planned in the second part to talk about metric spaces, but then I decided to postpone it for the future, and now talk in more detail about the surroundings and related concepts, which I only briefly mentioned in the last part. Thus, we are somewhere in the first chapter of some book on General Topology.
The black solid contour in the figures will indicate closed sets, and sets without a contour will be open. Letters TTT I will abbreviate if and only if.
Go.
So, a subset U of a topological space (X, T) is called a neighborhood of a point x mtt, when in U lies an open set containing x
In the figure above, this situation is illustrated. I showed the neighborhood of the point x in the form of a circle bounded by a dotted line. Of course, the neighborhood does not have to be around. It only has to contain the open set to which the point x belongs. In this case, I depicted the sets in red. Since every set belongs to itself, it is obvious that every open set is a neighborhood of each of its points. The converse is not true. The neighborhood does not have to be an open array at all. In the figure below, I illustrated these situations.
U, U 'and X and X' are neighborhoods of x. Moreover, U, U 'and X are open, but X' is not.
All this implies a very important theorem (and maybe not very) - the set is open, um, when it contains neighborhoods of all its points.
I thought for a long time how to draw it. The result is such a picture as below.
Look at this set X. If this set is open, no matter what point you find in it, it will always have a neighborhood belonging to this set. I showed how it will look when you enlarge the picture. For example, the point d lies almost on the very edge, but if you look more closely at this section, you can see that around it you can always describe an open circle. You can imagine such a game. Let the demon and genius choose a random set. The demon puts on the fact that the multitude is not open, but the genius that is open. After that, the demon indicates some point of the set, and the genius tries to draw a neighborhood around it. If everything succeeds, he receives a prize, and the demon makes a new point. If a genius is lucky with a lot, he can win a whole infinity of prizes. Of course, mortal creatures in this game is not too urgent to play.
There is an important type of points called accumulation points or limit points.
A limit point is a point, a subset A of a topological space (X, T), in any neighborhood of which there is a point of the subset A.
Uf. Speaking on the basis of visual intuition, the limit points are those points that lie “inside” the set or on the “edge”. Not every point that belongs to the set is the limit. Take a look at the picture below.
Point a belongs to the reddish set, but it has a neighborhood in which there are no other points of this set. So it’s not the limit. And, on the contrary, there are such points that may not belong to this set, but still be limit for it.
The picture below shows different situations.
The point a contains in any neighborhood of it a point of the subset X. The point b also. Therefore, they are accumulation points. But the point c, although it has neighborhoods which contain points X, but also has other neighborhoods that do not contain points from X.
The neighborhood in this case is very similar to the concept of a neighborhood in the everyday sense - to lie in a neighborhood means to be in proximity. In a sense, the neighborhood expresses proximity. But it is important to understand that while all the spaces we are talking about are not equipped with a metric, i.e. have no distance and we cannot express the proximity numerically. Later, when we talk about metric spaces, we will return to the topic of neighborhood, but from a slightly different side.
All of the above allows us to formulate the definition of closed sets in a new language. In the last chapter, I said that closed sets are those that complement open ones. This is a rather offrmal definition, but the surroundings make it possible to see it in a new light and it suddenly acquires depth and, as it were, new facets.
So, a subset of a topological space is closed when it contains all its limit points.
Look at the picture.
We have a red array, which is a subset of the orange array. It is open, since all its points enter it with its surroundings. And, obviously, there is an infinite number of limit points that do not belong to it. I drew several of them in the figure in the middle. If we add to the set the set of all its limit points, then it becomes closed.
How is this proved? Well, everything is simple here. Suppose A is a subset of a topological space X. There is a point x that does not belong to A and has an open neighborhood U that does not intersect A. since this neighborhood is open, it is a neighborhood of any of its points. Since it does not intersect A, then no point in this neighborhood is the limit point of A. This means that (A is the set of its limit points) is open U = X. Or X \ (A is the set of its limit points) = open many
i.e. And together with the set of its limit points, it is a complement to the open set, which means it is closed. It is amazing how the concept of neighborhood turned out to be deeply connected with the concept of openness and isolation.
Now we define the concept of closure closely related to closeness
The closure of a subset X is the intersection of all closed sets that contain X. The closure of X is denoted, for example, by [X]. Closing a subset is a closed set because the intersection of closed sets is closed. And it is obvious that closure is the smallest closed set that contains X.
Look at the picture.
We have a closed yellow array and an orange open. If we successively intersect all closed sets that contain orange (I showed only some steps), then we get the closure of the orange set.
I think you guessed it, a set is a closed ttt when it coincides with its closure ([X] = X).
A closed set is a set that contains all its limit points; it is also true that a closure is the union of a set and its limit points. This seems trivial, but this fact needs to be proved. It is proved simply that the limit points of the set X are, obviously, the limit points of each set containing X. Therefore, they belong to every closed set containing X. Therefore, the limit points of X are also contained in any intersection of these sets, that is, in [X]. On the other hand, the set together with its limit points is closed, which means that the closure of the set X is the set X together with its limit points.
An operator that associates a closure with a set is called a closure operator. I will denote it like that ⊛. Usually in mathematics they don’t designate it like that, but we have the right to our designations.
It has several properties:
1) ⊛∅ = ∅
This means that the closure of an empty set is empty.
2) X⊆⊛X The
set is contained in its closure.
3) ⊛X = ⊛⊛X The
closure of the closure of a set is equal to the closure of the set
4) ⊛X∪⊛X '= ⊛ (X∪X') The
union of closures is equal to the closure of the union.
These rules are called the axioms of Kuratovsky’s closure.
It seems to me that one of the best ways to understand topology is to look at some very small topological space. For example, one that consists of only a few elements.
Suppose we have a set {a, b, c, d, e} equipped with the topology described in the figure. Consider a subset of this set {a, b, c}. What points are the limit for it in this topology? By definition, these will be the points that have in eachpoints from {a, b, c} in their neighborhood besides themselves. I showed open neighborhoods schematically using circles. Points inside one circle — are in the same neighborhood (or in one open subset of the set). Therefore, we check for each point whether it has a neighborhood in which there are no points from {a, b, c}. Point a has such a neighborhood. This is a neighborhood where only point a is contained. So it is not the limit point. In all neighborhoods of b there are points from {a, b, c} (besides b itself). Similarly for d and e. Etc. thus, from this diagram, we see that b, d, e are limit points for {a, b, c}, but a and c are not.
We equip the same spaces with the discrete topology and see what happens.
In an indiscrete topology, each point will be the limit for the set {a, b, c} (and for any other, except the set itself and the empty set). This topology is not in vain called the topology of stuck together points, in it all points of space are as if stuck together with each other, and you won’t understand where exactly what. I would not want to live in a universe with an indiscrete topology.
In discrete topology, we have the opposite situation. I tried to draw, but honestly, I abandoned this idea. You remember how many open sets there will turn out even in a relatively small set.
Looking at all these pictures, I want to define the concept of the interior of the set and the border. For example, when I was drawing sets outlined in black, I immediately wanted to call this line the boundary of the set. However, so far we have not had the concept of border and interior. And now it will be.
A point is called an interior point of a set when the set is its neighborhood. The interior of a set is the totality of all its internal points. In other words, the interior of a set is the largest open set contained in it. Those. the multitude is open only when it coincides with its interior. Agree, this is very similar to defining a closure.
The interior of any subset of the discrete space is empty. In discrete space, any subset coincides with its interior and with its closure at the same time.
The boundary of the set X (X is the subset of A) is the set of those points of the set A whose neighborhood contains both points X and A / X.
I drew such a picture, a little in the style of abstract art came out.
Here you go. on this note I end the article. For the exercise, you can think about which points are the boundary of the set {a, b, c} from the picture, which are its interior (if any), what is its closure. Well, of course, is it possible to unfasten the figures from the initial image. And how to do it (develops spatial thinking very much). I promise to attach the answer to the last question to the following articles.
That's all for now.
Again, I warn you that everything you read is written twice by the humanities (bachelor and master), so you should not believe blindly. In general, you are warned.
Comments about errors (mathematical) are welcome.
Another warning is a lot of pictures.
Image to attract attention (not related to our text).
How do you think, without breaking these figures, but deforming in any way, is it possible to disconnect them?
Initially, I planned in the second part to talk about metric spaces, but then I decided to postpone it for the future, and now talk in more detail about the surroundings and related concepts, which I only briefly mentioned in the last part. Thus, we are somewhere in the first chapter of some book on General Topology.
The black solid contour in the figures will indicate closed sets, and sets without a contour will be open. Letters TTT I will abbreviate if and only if.
Go.
So, a subset U of a topological space (X, T) is called a neighborhood of a point x mtt, when in U lies an open set containing x
In the figure above, this situation is illustrated. I showed the neighborhood of the point x in the form of a circle bounded by a dotted line. Of course, the neighborhood does not have to be around. It only has to contain the open set to which the point x belongs. In this case, I depicted the sets in red. Since every set belongs to itself, it is obvious that every open set is a neighborhood of each of its points. The converse is not true. The neighborhood does not have to be an open array at all. In the figure below, I illustrated these situations.
U, U 'and X and X' are neighborhoods of x. Moreover, U, U 'and X are open, but X' is not.
All this implies a very important theorem (and maybe not very) - the set is open, um, when it contains neighborhoods of all its points.
I thought for a long time how to draw it. The result is such a picture as below.
Look at this set X. If this set is open, no matter what point you find in it, it will always have a neighborhood belonging to this set. I showed how it will look when you enlarge the picture. For example, the point d lies almost on the very edge, but if you look more closely at this section, you can see that around it you can always describe an open circle. You can imagine such a game. Let the demon and genius choose a random set. The demon puts on the fact that the multitude is not open, but the genius that is open. After that, the demon indicates some point of the set, and the genius tries to draw a neighborhood around it. If everything succeeds, he receives a prize, and the demon makes a new point. If a genius is lucky with a lot, he can win a whole infinity of prizes. Of course, mortal creatures in this game is not too urgent to play.
There is an important type of points called accumulation points or limit points.
A limit point is a point, a subset A of a topological space (X, T), in any neighborhood of which there is a point of the subset A.
Uf. Speaking on the basis of visual intuition, the limit points are those points that lie “inside” the set or on the “edge”. Not every point that belongs to the set is the limit. Take a look at the picture below.
Point a belongs to the reddish set, but it has a neighborhood in which there are no other points of this set. So it’s not the limit. And, on the contrary, there are such points that may not belong to this set, but still be limit for it.
The picture below shows different situations.
The point a contains in any neighborhood of it a point of the subset X. The point b also. Therefore, they are accumulation points. But the point c, although it has neighborhoods which contain points X, but also has other neighborhoods that do not contain points from X.
The neighborhood in this case is very similar to the concept of a neighborhood in the everyday sense - to lie in a neighborhood means to be in proximity. In a sense, the neighborhood expresses proximity. But it is important to understand that while all the spaces we are talking about are not equipped with a metric, i.e. have no distance and we cannot express the proximity numerically. Later, when we talk about metric spaces, we will return to the topic of neighborhood, but from a slightly different side.
All of the above allows us to formulate the definition of closed sets in a new language. In the last chapter, I said that closed sets are those that complement open ones. This is a rather offrmal definition, but the surroundings make it possible to see it in a new light and it suddenly acquires depth and, as it were, new facets.
So, a subset of a topological space is closed when it contains all its limit points.
Look at the picture.
We have a red array, which is a subset of the orange array. It is open, since all its points enter it with its surroundings. And, obviously, there is an infinite number of limit points that do not belong to it. I drew several of them in the figure in the middle. If we add to the set the set of all its limit points, then it becomes closed.
How is this proved? Well, everything is simple here. Suppose A is a subset of a topological space X. There is a point x that does not belong to A and has an open neighborhood U that does not intersect A. since this neighborhood is open, it is a neighborhood of any of its points. Since it does not intersect A, then no point in this neighborhood is the limit point of A. This means that (A is the set of its limit points) is open U = X. Or X \ (A is the set of its limit points) = open many
i.e. And together with the set of its limit points, it is a complement to the open set, which means it is closed. It is amazing how the concept of neighborhood turned out to be deeply connected with the concept of openness and isolation.
Now we define the concept of closure closely related to closeness
The closure of a subset X is the intersection of all closed sets that contain X. The closure of X is denoted, for example, by [X]. Closing a subset is a closed set because the intersection of closed sets is closed. And it is obvious that closure is the smallest closed set that contains X.
Look at the picture.
We have a closed yellow array and an orange open. If we successively intersect all closed sets that contain orange (I showed only some steps), then we get the closure of the orange set.
I think you guessed it, a set is a closed ttt when it coincides with its closure ([X] = X).
A closed set is a set that contains all its limit points; it is also true that a closure is the union of a set and its limit points. This seems trivial, but this fact needs to be proved. It is proved simply that the limit points of the set X are, obviously, the limit points of each set containing X. Therefore, they belong to every closed set containing X. Therefore, the limit points of X are also contained in any intersection of these sets, that is, in [X]. On the other hand, the set together with its limit points is closed, which means that the closure of the set X is the set X together with its limit points.
An operator that associates a closure with a set is called a closure operator. I will denote it like that ⊛. Usually in mathematics they don’t designate it like that, but we have the right to our designations.
It has several properties:
1) ⊛∅ = ∅
This means that the closure of an empty set is empty.
2) X⊆⊛X The
set is contained in its closure.
3) ⊛X = ⊛⊛X The
closure of the closure of a set is equal to the closure of the set
4) ⊛X∪⊛X '= ⊛ (X∪X') The
union of closures is equal to the closure of the union.
These rules are called the axioms of Kuratovsky’s closure.
It seems to me that one of the best ways to understand topology is to look at some very small topological space. For example, one that consists of only a few elements.
Suppose we have a set {a, b, c, d, e} equipped with the topology described in the figure. Consider a subset of this set {a, b, c}. What points are the limit for it in this topology? By definition, these will be the points that have in eachpoints from {a, b, c} in their neighborhood besides themselves. I showed open neighborhoods schematically using circles. Points inside one circle — are in the same neighborhood (or in one open subset of the set). Therefore, we check for each point whether it has a neighborhood in which there are no points from {a, b, c}. Point a has such a neighborhood. This is a neighborhood where only point a is contained. So it is not the limit point. In all neighborhoods of b there are points from {a, b, c} (besides b itself). Similarly for d and e. Etc. thus, from this diagram, we see that b, d, e are limit points for {a, b, c}, but a and c are not.
We equip the same spaces with the discrete topology and see what happens.
In an indiscrete topology, each point will be the limit for the set {a, b, c} (and for any other, except the set itself and the empty set). This topology is not in vain called the topology of stuck together points, in it all points of space are as if stuck together with each other, and you won’t understand where exactly what. I would not want to live in a universe with an indiscrete topology.
In discrete topology, we have the opposite situation. I tried to draw, but honestly, I abandoned this idea. You remember how many open sets there will turn out even in a relatively small set.
Looking at all these pictures, I want to define the concept of the interior of the set and the border. For example, when I was drawing sets outlined in black, I immediately wanted to call this line the boundary of the set. However, so far we have not had the concept of border and interior. And now it will be.
A point is called an interior point of a set when the set is its neighborhood. The interior of a set is the totality of all its internal points. In other words, the interior of a set is the largest open set contained in it. Those. the multitude is open only when it coincides with its interior. Agree, this is very similar to defining a closure.
The interior of any subset of the discrete space is empty. In discrete space, any subset coincides with its interior and with its closure at the same time.
The boundary of the set X (X is the subset of A) is the set of those points of the set A whose neighborhood contains both points X and A / X.
I drew such a picture, a little in the style of abstract art came out.
Here you go. on this note I end the article. For the exercise, you can think about which points are the boundary of the set {a, b, c} from the picture, which are its interior (if any), what is its closure. Well, of course, is it possible to unfasten the figures from the initial image. And how to do it (develops spatial thinking very much). I promise to attach the answer to the last question to the following articles.
That's all for now.