Introduction to topology (for teapots and humanities scholars)
I do not remember when I first learned about topology, but I was immediately interested in this science. The kettle turns into a bagel, the sphere turns inside out. Many have heard about it. But those who want to delve into this topic on a more serious level, often have difficulties. This particularly applies to the development of the most elementary concepts, which are essentially very abstract. Moreover, many sources, as if specifically seeking to confuse the reader. Let's say the Russian wiki gives a rather vague formulation of what topology does. It says that this is a science that studies topological spaces . In the article on topological spaces, the reader can learn that topological spaces are spaces provided with topology . Such explanations in the style of Lemov's sepules do not clarify the essence of the subject. I will try to further outline the basic concepts in a clearer form. In my article there will be no transforming teapots and bagels, but the first steps will be taken that will eventually allow us to learn this magic.
However, since I am not a mathematician, but an absolute humanist, it is quite possible that what is written below is a lie! Well, or at least part.
I first wrote this note as the beginning of a series of articles on topology for my humanitarian friends, but none of them read it. I decided to lay out the corrected and expanded version on habr. It seemed to me that there was a certain interest in this topic here and there were no such articles yet. Thanks in advance for all comments about errors and inaccuracies. I warn you that I use a lot of pictures.
Let's start with a brief repetition of set theory. I think most readers are well acquainted with her, but nevertheless I will remind the basics.
')
So, it is considered that the set has no definition and that we intuitively understand what it is. Cantor said this: “By“ many ”we mean connecting into a certain whole M certain well-defined objects m of our contemplation or our thinking (which will be called the“ elements ”of the set M)”. Of course, this is just an allegorical description, not a mathematical definition.
Set theory is known (I beg your pardon for a pun) by a number of amazing paradoxes. For example . It is also associated with the crisis of mathematics at the beginning of the 20th century.
Set theory exists in several variants, such as ZFC or NBG and others. A variant of the theory is the theory of types , which is very important for programmers. Finally, some mathematicians suggest instead of the theory of sets as the foundation of mathematics use the theory of categories, about which much is written on Habré. The theory of types and the theory of sets describe mathematical objects as if “from the inside”, and the theory of categories is not interested in their internal structure, but only how they interact, that is, gives them an "external" characteristic.
For us, only the most basic principles of set theory are important.
Sets are finite.
There are endless. For example, the set of integers, which is denoted by the letter ℤ (or simply Z, if you do not have curly letters on the keyboard).
Finally, there is an empty set. It is exactly one in the whole universe. There is a simple proof of this fact, but I will not give it here.
If the set is infinite, it is countable . Countable are those sets whose elements can be renumbered with natural numbers. The set of natural numbers itself, as you guessed, is also countable. But how can you number the integers.
With rational numbers more difficult, but they are amenable to numbering. This method is called the diagonal process and looks like the image below.
We zigzag move on rational numbers, starting from 1. In addition, we assign an even number to each number that we have. Negative rational numbers are considered the same way, only odd numbers, starting from 3. Zero traditionally gets the first number. Thus it is clear that all rational numbers can be numbered. All numbers like 4.87592692976340586068 or 1.00000000000001, or -9092, or even 42 get their number in this table. However, not all numbers fall here. For example, √2 will not get a number. Once this is very upsetting the Greeks. They say the guy who discovered the irrational numbers drowned.
A generalization of the concept of size for sets is power . The power of finite sets is equal to the number of their elements. The power of infinite sets is denoted by the Hebrew letter alef with an index. The smallest infinite power is power ℵ 0 . It is equal to the cardinality of countable sets. As we see, thus, there are as many natural numbers as there are whole or rational numbers. Strange, but true. Next is the power of the continuum . It is denoted by a small gothic letter with. This is the power of the set of real numbers ℝ, for example. There is a hypothesis that the power of the continuum is equal to the power 1 . That is, it is the next power after the power of the counting sets, and there is no intermediate power between the counting sets and the continuum.
You can perform various operations on sets and get new sets.
1. Sets can be combined.
2. Sets can be “subtracted”. This operation is called a supplement .
3. You can search for the intersection of sets.
Actually, it's all about the sets that you need to know for the purposes of this note. Now we can proceed to the topology itself.
Topology is a science that studies sets with a specific structure. This structure is also called topology.
Suppose we have some nonempty set S.
Let this set have a certain structure, which is described with the help of the set, which we call T. T is a set of subsets of the set S such that:
1. S and ∅ belong to T.
2. Any union of arbitrary families of elements of T belongs to T.
3. The intersection of an arbitrary finite family of elements T belongs to T.
If these three points hold, then our structure is a topology T on the set S. The elements of the set T are called open sets on S in the topology T. The addition to open sets is closed sets. It is important to note that if the set is open, this does not mean that it is not closed and vice versa. In addition, in a given set with respect to some topology there may be subsets that are neither open nor closed.
Let's give an example. Suppose we have a set consisting of three colored triangles.
The simplest topology on it is called the antidiscrete topology . Here she is.
This topology is also called the topology of stuck points . It consists of the set itself and of the empty set. This really satisfies the axioms of topology.
Several topologies can be defined on one set. Here is another very primitive topology that happens. It is called discrete. This is a topology that consists of all the subsets of a given set.
And here's another topology. It is given on a set of 7 multi-colored stars S, which I have labeled. Verify that this is a topology. I am not sure about this, suddenly I missed, some kind of union or intersection. In this picture there should be the set S itself, the empty set, the intersections and unions of all other elements of the topology should also be in the picture.
A pair of topology and the set on which it is given is called a topological space .
If there are many points in a set (not to mention the fact that there can be infinitely many), then it can be problematic to list all open sets. For example, for a discrete topology on a set of three elements, it is necessary to make a list of 8 sets. And for a 4-element set, the discrete topology will already be 16, for 5 - 32, for 6 —64, and so on. In order not to enumerate all open sets, the abbreviated notation is used, as it were - those elements are written out, the unions of which can give, all open sets. This is called a topology base . For example, for a discrete topology of a space of three triangles - these will be three triangles taken separately, because by combining them, you can get all the other open sets in this topology. The base is said to generate a topology. Sets whose elements generate a base are called a prebase.
Below is an example of a base for a discrete topology on a set of five stars. As you can see, in this case the base consists of only five elements, while in the topology there are as many as 32 subsets. Agree, to use the database to describe the topology is much more convenient.
What are open sets for? In a sense, they give an idea of the "proximity" between the points and the difference between them. If points belong to two different open sets or if one point is in an open set that does not contain the second, then they are topologically different. In the anti-discrete topology, all points in this sense are indistinguishable, they seem to be stuck together. On the contrary, in a discrete topology all points have a difference.
The concept of a neighborhood is inextricably linked with the concept of an open set. Some authors define topology not through open sets, but through neighborhoods. The neighborhood of the point p is a set that contains an open ball with the center at this point. For example, the figure below shows neighborhoods and not neighborhoods of points. The set S 1 is a neighborhood of the point p, and the set S 2 is not.
The relationship between open set and octet can be formulated as follows. An open set is such a set, each element of which has a certain neighborhood lying in a given set. Or, on the contrary, one can say that a set is open if it is a neighborhood of any of its points.
All these are the most basic concepts of topology. From here it is still not clear how to turn the spheres inside out. Perhaps in the future, I can get to this kind of topics (if I figure it out myself).
UPD. Due to the inaccuracy of my speech, there was some confusion regarding the power of the sets. I have corrected my text somewhat and here I want to give an explanation. Cantor, creating his theory of sets, introduced the concept of power, which made it possible to compare infinite sets. Cantor found that the powers of countable sets (for example, rational numbers) and the continuum (for example, real numbers) are different. He suggested that the power of the continuum is the next after the power of countable sets, i.e. equals alef one. Cantor tried to prove this hypothesis, but to no avail. Later it became clear that this hypothesis can neither be refuted nor proved.
However, since I am not a mathematician, but an absolute humanist, it is quite possible that what is written below is a lie! Well, or at least part.
I first wrote this note as the beginning of a series of articles on topology for my humanitarian friends, but none of them read it. I decided to lay out the corrected and expanded version on habr. It seemed to me that there was a certain interest in this topic here and there were no such articles yet. Thanks in advance for all comments about errors and inaccuracies. I warn you that I use a lot of pictures.
Let's start with a brief repetition of set theory. I think most readers are well acquainted with her, but nevertheless I will remind the basics.
')
So, it is considered that the set has no definition and that we intuitively understand what it is. Cantor said this: “By“ many ”we mean connecting into a certain whole M certain well-defined objects m of our contemplation or our thinking (which will be called the“ elements ”of the set M)”. Of course, this is just an allegorical description, not a mathematical definition.
Set theory is known (I beg your pardon for a pun) by a number of amazing paradoxes. For example . It is also associated with the crisis of mathematics at the beginning of the 20th century.
Set theory exists in several variants, such as ZFC or NBG and others. A variant of the theory is the theory of types , which is very important for programmers. Finally, some mathematicians suggest instead of the theory of sets as the foundation of mathematics use the theory of categories, about which much is written on Habré. The theory of types and the theory of sets describe mathematical objects as if “from the inside”, and the theory of categories is not interested in their internal structure, but only how they interact, that is, gives them an "external" characteristic.
For us, only the most basic principles of set theory are important.
Sets are finite.
There are endless. For example, the set of integers, which is denoted by the letter ℤ (or simply Z, if you do not have curly letters on the keyboard).
Finally, there is an empty set. It is exactly one in the whole universe. There is a simple proof of this fact, but I will not give it here.
If the set is infinite, it is countable . Countable are those sets whose elements can be renumbered with natural numbers. The set of natural numbers itself, as you guessed, is also countable. But how can you number the integers.
With rational numbers more difficult, but they are amenable to numbering. This method is called the diagonal process and looks like the image below.
We zigzag move on rational numbers, starting from 1. In addition, we assign an even number to each number that we have. Negative rational numbers are considered the same way, only odd numbers, starting from 3. Zero traditionally gets the first number. Thus it is clear that all rational numbers can be numbered. All numbers like 4.87592692976340586068 or 1.00000000000001, or -9092, or even 42 get their number in this table. However, not all numbers fall here. For example, √2 will not get a number. Once this is very upsetting the Greeks. They say the guy who discovered the irrational numbers drowned.
A generalization of the concept of size for sets is power . The power of finite sets is equal to the number of their elements. The power of infinite sets is denoted by the Hebrew letter alef with an index. The smallest infinite power is power ℵ 0 . It is equal to the cardinality of countable sets. As we see, thus, there are as many natural numbers as there are whole or rational numbers. Strange, but true. Next is the power of the continuum . It is denoted by a small gothic letter with. This is the power of the set of real numbers ℝ, for example. There is a hypothesis that the power of the continuum is equal to the power 1 . That is, it is the next power after the power of the counting sets, and there is no intermediate power between the counting sets and the continuum.
You can perform various operations on sets and get new sets.
1. Sets can be combined.
2. Sets can be “subtracted”. This operation is called a supplement .
3. You can search for the intersection of sets.
Actually, it's all about the sets that you need to know for the purposes of this note. Now we can proceed to the topology itself.
Topology is a science that studies sets with a specific structure. This structure is also called topology.
Suppose we have some nonempty set S.
Let this set have a certain structure, which is described with the help of the set, which we call T. T is a set of subsets of the set S such that:
1. S and ∅ belong to T.
2. Any union of arbitrary families of elements of T belongs to T.
3. The intersection of an arbitrary finite family of elements T belongs to T.
If these three points hold, then our structure is a topology T on the set S. The elements of the set T are called open sets on S in the topology T. The addition to open sets is closed sets. It is important to note that if the set is open, this does not mean that it is not closed and vice versa. In addition, in a given set with respect to some topology there may be subsets that are neither open nor closed.
Let's give an example. Suppose we have a set consisting of three colored triangles.
The simplest topology on it is called the antidiscrete topology . Here she is.
This topology is also called the topology of stuck points . It consists of the set itself and of the empty set. This really satisfies the axioms of topology.
Several topologies can be defined on one set. Here is another very primitive topology that happens. It is called discrete. This is a topology that consists of all the subsets of a given set.
And here's another topology. It is given on a set of 7 multi-colored stars S, which I have labeled. Verify that this is a topology. I am not sure about this, suddenly I missed, some kind of union or intersection. In this picture there should be the set S itself, the empty set, the intersections and unions of all other elements of the topology should also be in the picture.
A pair of topology and the set on which it is given is called a topological space .
If there are many points in a set (not to mention the fact that there can be infinitely many), then it can be problematic to list all open sets. For example, for a discrete topology on a set of three elements, it is necessary to make a list of 8 sets. And for a 4-element set, the discrete topology will already be 16, for 5 - 32, for 6 —64, and so on. In order not to enumerate all open sets, the abbreviated notation is used, as it were - those elements are written out, the unions of which can give, all open sets. This is called a topology base . For example, for a discrete topology of a space of three triangles - these will be three triangles taken separately, because by combining them, you can get all the other open sets in this topology. The base is said to generate a topology. Sets whose elements generate a base are called a prebase.
Below is an example of a base for a discrete topology on a set of five stars. As you can see, in this case the base consists of only five elements, while in the topology there are as many as 32 subsets. Agree, to use the database to describe the topology is much more convenient.
What are open sets for? In a sense, they give an idea of the "proximity" between the points and the difference between them. If points belong to two different open sets or if one point is in an open set that does not contain the second, then they are topologically different. In the anti-discrete topology, all points in this sense are indistinguishable, they seem to be stuck together. On the contrary, in a discrete topology all points have a difference.
The concept of a neighborhood is inextricably linked with the concept of an open set. Some authors define topology not through open sets, but through neighborhoods. The neighborhood of the point p is a set that contains an open ball with the center at this point. For example, the figure below shows neighborhoods and not neighborhoods of points. The set S 1 is a neighborhood of the point p, and the set S 2 is not.
The relationship between open set and octet can be formulated as follows. An open set is such a set, each element of which has a certain neighborhood lying in a given set. Or, on the contrary, one can say that a set is open if it is a neighborhood of any of its points.
All these are the most basic concepts of topology. From here it is still not clear how to turn the spheres inside out. Perhaps in the future, I can get to this kind of topics (if I figure it out myself).
UPD. Due to the inaccuracy of my speech, there was some confusion regarding the power of the sets. I have corrected my text somewhat and here I want to give an explanation. Cantor, creating his theory of sets, introduced the concept of power, which made it possible to compare infinite sets. Cantor found that the powers of countable sets (for example, rational numbers) and the continuum (for example, real numbers) are different. He suggested that the power of the continuum is the next after the power of countable sets, i.e. equals alef one. Cantor tried to prove this hypothesis, but to no avail. Later it became clear that this hypothesis can neither be refuted nor proved.
Source: https://habr.com/ru/post/190860/
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