August 21, 2013 at 15:41Introduction to Topology (for Dummies and Humanities)
I don’t remember when I first learned about topology, but this science immediately interested me. The teapot turns into a bagel, the sphere turns inside out. Many have heard about this. But those who want to delve into this topic at a more serious level often have difficulties. This is especially true for the development of the most basic concepts, which are inherently very abstract. Moreover, many sources seem to intentionally confuse the reader. Say the Russian wiki gives a very vague wording of what topology does. It says that this is a science that studies topological spaces . In an article about topological spaces, the reader can learn that topological spaces are spaces equipped with a topology. Such explanations in the style of Lem's sepules do not really clarify the essence of the subject. I will try to further outline the basic basic concepts in a clearer form. In my article there will be no turning dummies 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 a one hundred percent humanist, it is quite possible that what is written below is a lie! Well, or at least part.
For the first time I wrote this article as the beginning of a series of articles on topology for my humanitarian friends, but none of them began to read it. I decided to lay out the corrected and extended version on the Habr. It seemed to me that there was a certain interest in this topic and there were no articles of this kind just 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 familiar with it, but nevertheless I will remind the basics.
So, it is believed that the set does not have a definition and that we intuitively understand what it is. Kantor said this: "By" multitude "we mean the union into a whole M of certain well-distinguishable 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.
The theory of sets is known (please forgive the pun) with many amazing paradoxes. For example . The crisis of mathematics at the beginning of the 20th century is also associated with it.
Set theory exists in several variants, such as ZFC or NBG and others. A theory variant is type theory, which is very important for programmers. Finally, some mathematicians propose instead of set theory to use category theory, which is much written on Habré, as the foundation of mathematics. Type theory and set theory describe mathematical objects as if “from within”, and category theory is not interested in their internal structure, but only how they interact, i.e. gives their “external” characterization.
For us, only the most basic foundations of set theory are important.
Many are finite.
They are endless. For example, a set of integers, which is indicated 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 entire universe. There is simple evidence of this fact, but I will not cite it here.
If the set is infinite, it is countable . Countable ones are those sets whose elements can be renumbered by natural numbers. The sheer number of natural numbers, you guessed it, is also countable. But here's how to number integers.
Rational numbers are more complicated, but they can also be numbered. This method is called the diagonal process and looks like the image below.
We zigzag along rational numbers, starting with 1. At the same time, we assign an even number to each number that we get. Negative rational numbers are considered in the same way, only the numbers are odd, starting with 3. Zero traditionally gets the first number. Thus, it can be seen 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 receive a number. This once made the Greeks very upset. They say that the guy who discovered the irrational numbers was drowned.
A generalization of the concept of size for sets is the power. The cardinality of finite sets is equal to the number of their elements. The power of infinite sets is denoted by the Hebrew letter aleph with an index. The smallest infinite power is power ℵ 0 . It is equal to the power of countable sets. As we see, in this way, there are as many natural numbers as there are integers or rationals. Strange, but true. The next is the power of the continuum . It is indicated by the small Gothic letter c. This is the cardinality of the set of real numbers ℝ, for example. There is a hypothesis that the power of the continuum is equal to the power ℵ 1 . That is, that this is the next power after the cardinality of countable sets, and there is no intermediate power between the countable sets and the continuum.
You can perform various operations on sets and get new sets.
1. Many can be combined.
2. Many can be "subtracted". This operation is called supplementation .
3. You can search for the intersection of sets.
Actually, this is all about sets that you need to know for the purposes of this article. 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.
Let us have some nonempty set S.
Let this set have some structure, which is described using the set, which we will call T. T is a set of subsets of the set S such that:
1. S and ∅ themselves 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 of T belongs to T.
If these three points hold, then our structure is a topology T on the set S. Elements of the set T are called open sets on S in the topology of T. Addition to open sets are 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 a certain topology, there may be subsets that are neither open nor closed.
We give an example. Let us have a set consisting of three colored triangles.
The simplest topology on it is called the discrete topology . Here she is.
This topology is also called the topology of sticking together points . It consists of the multitude itself and of the empty multitude. This really satisfies the axioms of topology.
On one set, you can specify several topologies. Here is another very primitive topology that happens. It is called discrete. This is a topology that consists of all subsets of a given set.
And here is the topology. It is set on a set of 7 multi-colored stars S, which I marked with letters. Make sure this is the topology. I'm 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 the other elements of the topology should also be in the picture.
A pair of topology and the set on which it is defined is called a topological space .
If the set has many points (not to mention the fact that there can be infinitely many of them), then listing all open sets can be problematic. For example, for a discrete topology on a set of three elements, you need to make a list of 8 sets. And for a 4-element set, the discrete topology will already number 16, for 5 - 32, for 6 - 64 and so on. In order not to list all open sets, a shorthand notation is used, as it were — those elements whose associations can give all open sets are written out. This is called the base.topologies. For example, for a discrete topology of a space of three triangles - these will be three triangles taken separately, because combining them, you can get all the other open sets in this topology. They say that the base generates a topology. The sets whose elements generate the base are called the 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, using the base to describe the topology is much more convenient.
What are open sets for? In a sense, they give an idea of the "proximity" between points and the difference between them. If the points belong to two different open sets, or if one point is in an open set in which the second is not, then they are topologically different. In an indiscrete topology, all points in this sense are indistinguishable, they are as if stuck together. Conversely, in a discrete topology, all points have a difference.
The notion of 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 p is a set that contains an open ball centered at that point. For example, the figure below shows neighborhoods and non-neighborhoods of points. The set S 1 is a neighborhood of p, but the set S 2 is not.
The connection between open set and octest can be formulated as follows. An open set is such a set, each element of which has a certain neighborhood lying in this set. Or vice versa, 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 will be able to get to these kinds of topics (if I figure it out myself).
UPD Due to the inaccuracy of my speech, there was some bewilderment regarding the powers 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 cardinalities of countable sets (e.g., rational numbers) and continuum (e.g., real numbers) are different. He suggested that the power of the continuum is next to the power of countable sets i.e. equal to 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 a one hundred percent humanist, it is quite possible that what is written below is a lie! Well, or at least part.
For the first time I wrote this article as the beginning of a series of articles on topology for my humanitarian friends, but none of them began to read it. I decided to lay out the corrected and extended version on the Habr. It seemed to me that there was a certain interest in this topic and there were no articles of this kind just 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 familiar with it, but nevertheless I will remind the basics.
So, it is believed that the set does not have a definition and that we intuitively understand what it is. Kantor said this: "By" multitude "we mean the union into a whole M of certain well-distinguishable 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.
The theory of sets is known (please forgive the pun) with many amazing paradoxes. For example . The crisis of mathematics at the beginning of the 20th century is also associated with it.
Set theory exists in several variants, such as ZFC or NBG and others. A theory variant is type theory, which is very important for programmers. Finally, some mathematicians propose instead of set theory to use category theory, which is much written on Habré, as the foundation of mathematics. Type theory and set theory describe mathematical objects as if “from within”, and category theory is not interested in their internal structure, but only how they interact, i.e. gives their “external” characterization.
For us, only the most basic foundations of set theory are important.
Many are finite.
They are endless. For example, a set of integers, which is indicated 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 entire universe. There is simple evidence of this fact, but I will not cite it here.
If the set is infinite, it is countable . Countable ones are those sets whose elements can be renumbered by natural numbers. The sheer number of natural numbers, you guessed it, is also countable. But here's how to number integers.
Rational numbers are more complicated, but they can also be numbered. This method is called the diagonal process and looks like the image below.
We zigzag along rational numbers, starting with 1. At the same time, we assign an even number to each number that we get. Negative rational numbers are considered in the same way, only the numbers are odd, starting with 3. Zero traditionally gets the first number. Thus, it can be seen 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 receive a number. This once made the Greeks very upset. They say that the guy who discovered the irrational numbers was drowned.
A generalization of the concept of size for sets is the power. The cardinality of finite sets is equal to the number of their elements. The power of infinite sets is denoted by the Hebrew letter aleph with an index. The smallest infinite power is power ℵ 0 . It is equal to the power of countable sets. As we see, in this way, there are as many natural numbers as there are integers or rationals. Strange, but true. The next is the power of the continuum . It is indicated by the small Gothic letter c. This is the cardinality of the set of real numbers ℝ, for example. There is a hypothesis that the power of the continuum is equal to the power ℵ 1 . That is, that this is the next power after the cardinality of countable sets, and there is no intermediate power between the countable sets and the continuum.
You can perform various operations on sets and get new sets.
1. Many can be combined.
2. Many can be "subtracted". This operation is called supplementation .
3. You can search for the intersection of sets.
Actually, this is all about sets that you need to know for the purposes of this article. 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.
Let us have some nonempty set S.
Let this set have some structure, which is described using the set, which we will call T. T is a set of subsets of the set S such that:
1. S and ∅ themselves 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 of T belongs to T.
If these three points hold, then our structure is a topology T on the set S. Elements of the set T are called open sets on S in the topology of T. Addition to open sets are 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 a certain topology, there may be subsets that are neither open nor closed.
We give an example. Let us have a set consisting of three colored triangles.
The simplest topology on it is called the discrete topology . Here she is.
This topology is also called the topology of sticking together points . It consists of the multitude itself and of the empty multitude. This really satisfies the axioms of topology.
On one set, you can specify several topologies. Here is another very primitive topology that happens. It is called discrete. This is a topology that consists of all subsets of a given set.
And here is the topology. It is set on a set of 7 multi-colored stars S, which I marked with letters. Make sure this is the topology. I'm 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 the other elements of the topology should also be in the picture.
A pair of topology and the set on which it is defined is called a topological space .
If the set has many points (not to mention the fact that there can be infinitely many of them), then listing all open sets can be problematic. For example, for a discrete topology on a set of three elements, you need to make a list of 8 sets. And for a 4-element set, the discrete topology will already number 16, for 5 - 32, for 6 - 64 and so on. In order not to list all open sets, a shorthand notation is used, as it were — those elements whose associations can give all open sets are written out. This is called the base.topologies. For example, for a discrete topology of a space of three triangles - these will be three triangles taken separately, because combining them, you can get all the other open sets in this topology. They say that the base generates a topology. The sets whose elements generate the base are called the 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, using the base to describe the topology is much more convenient.
What are open sets for? In a sense, they give an idea of the "proximity" between points and the difference between them. If the points belong to two different open sets, or if one point is in an open set in which the second is not, then they are topologically different. In an indiscrete topology, all points in this sense are indistinguishable, they are as if stuck together. Conversely, in a discrete topology, all points have a difference.
The notion of 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 p is a set that contains an open ball centered at that point. For example, the figure below shows neighborhoods and non-neighborhoods of points. The set S 1 is a neighborhood of p, but the set S 2 is not.
The connection between open set and octest can be formulated as follows. An open set is such a set, each element of which has a certain neighborhood lying in this set. Or vice versa, 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 will be able to get to these kinds of topics (if I figure it out myself).
UPD Due to the inaccuracy of my speech, there was some bewilderment regarding the powers 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 cardinalities of countable sets (e.g., rational numbers) and continuum (e.g., real numbers) are different. He suggested that the power of the continuum is next to the power of countable sets i.e. equal to 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.