topological space

Topological concepts

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Topological space, a kind of Mathematical structure , which can be formalization To define convergence 、 connected 、 continuity And so on.

Topological space in modern times mathematics It has applications in all branches of. It is a central and unified concept.

Topological space is independent Research The branch of mathematics that studies topological spaces is called topology 。^[1]

Chinese name: topological space
Foreign name: topological space
Discipline: General topology

The earliest researcher: French mathematician Frey
The earliest research time: 1906
Status: Central and unified concept

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brief introduction

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topological space It is a binary group composed of a set X and the topology defined on it

。 The element x of X is usually called topological space

Point of. The term topology covers Open set ， Closed set ， Neighborhood ， Nuclear opening ， closure ， Derived set ， Filter And other concepts. From these concepts, we can give topological space

Make several equivalent definitions.^[2]

Topological space as object, continuous mapping as Morphisms , constitutes Category of topological space It is a basic category in mathematics. The idea of trying to classify this category through invariants has inspired and generated research work in the whole field, including Homotopy theory 、 Homology theory and K theory 。

definition

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Open Set Definition

Let X be a set and 𝓞 be a subset family of X (whose elements are called Open set ）, then (X, 𝓞) is called a topological space , if the following properties are true:

one empty set And X are open sets,

2. The union of any open set is an open set,

3. The intersection of a finite open set is an open set.

At this time, the elements in X are called spot 。 We also call 𝓞 a topology on X.

Neighborhood definition

Let X be a set, 𝔘={𝔘 x } x∈X , then (X, 𝔘) is called a topological space , of which, 𝔘 x Is a subset family, if any U x ∈𝔘 x , called x Neighborhood , the following properties hold:

1. x∈U x , and X is the neighborhood of all points;

2. Including U x The set of is also a neighborhood;

3. The intersection of any two neighborhoods of a point is still the neighborhood of the point;

4.U x It contains a neighborhood of x, which is the neighborhood of all points in it.^[6]

example

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real number Set ℝ constitutes a topological space: all open section It forms a group of topological bases on which the topology is generated. This means that the open set on the real number set ℝ is the union of a group of open intervals (the number of open intervals can be infinite, but it can be further proved that all open sets can be expressed as the union of countable disjoint open intervals). In many ways, the set of real numbers is the most basic topological space, and it also guides us to obtain many intuitive understanding of topological space; But there are also many "strange" topological spaces, which are contrary to our intuitive understanding of the set of real numbers.
More general, n-dimension Euclidean space ℝ constitutes a topological space, on which the open set is generated by the tee.
whatever metric space Can form a topological space if the open set on it is generated by the tee. This situation includes many very useful infinite dimensional spaces, such as functional analysis In the domain Banach space and Hilbert space 。
whatever Local field Have a topology naturally, and this topology can be expanded to vector space 。
In addition to the topology generated by all open intervals, the set of real numbers can also be given another topology - lower limit topology. The open set of this topology consists of the following point sets empty set, total set and all semi open intervals[ a , b ）Generated collection. This topology is strictly finer than the Euclidean topology defined above; In this topological space, a sequence of points converges to a point, if and only if the sequence of points also converges to this point in Euclidean topology. So we give an example of a collection with different topologies.
All manifolds are topological spaces.
Every simplex is a topological space. Simplex is very useful in computational geometry convex set 。 In 0, 1, 2 and 3 dimensional space, the corresponding simplex is point, line segment, triangle and tetrahedron respectively.
Every simple complex is a topological space. A simple complex consists of many simple forms. Many geometries can be modeled by simple complex, see Multicellular form （Polytope）。
Zarisky topology It is a topology defined purely by algebra, which is based on the commutative ring spectrum of a ring or an algebraic variety. yes R perhaps C For example, the closed set defined by the corresponding Zarisky topology is composed of the solution set of all polynomial equations.
A linear graph is a topological space that can generalize many geometric properties of graphs.
Many sets of operators in functional analysis can obtain a special topology in which a class of function sequences converge.
Finite complement topology 。 Let X be a set. The complement of all finite subsets of X plus empty set , forming a topology on X. The corresponding topological space is called Finite complement space 。 A finite complement space is the smallest T on this set one Topology.
Countable complement topology 。 Let X be a set. The complement of all countable subsets of X plus the superset form a topology on X. The corresponding topological space is called Countable complement space 。
If Γ is an ordinal number, then the set [0, Γ] is a topological space, which can be defined by the interval（ a , b ]Build, here a and b Is the element of Γ.

structure

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Any subset of the topological space can be given a subspace topology, and the open set in the subspace topology is the intersection of the open set on the whole space and the subspace.^[3]
For any non empty topological space family, we can construct the topology on the product of these topological spaces, which is called product topology. For a finite product, the open set on the product space can be generated by the product of the open sets of each space in the space family.
Quotient topology can be defined as follows: if X Is a topological space, Y Is a collection if f : X → Y Is a Surjection , then Y Obtain a topology; The open set of the topology can be defined as follows: a set is open if and only if its inverse image is also open. Available f Natural projection determination X On Equivalence class , thus giving the topological space X On equivalence relation 。
Vietoris topology

classification

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According to the degree, size, connectivity, compactness, etc. of point and set separation. Topological spaces can be classified into various categories. And because of these classifications, there are many different terms.

Let's assume that X is a topological space.

Separation axiom

For details, please refer to Separation axiom And relevant bar codes. Some terms have been defined differently in the old literature. Please refer to the history of the separation axiom.

Topological indiscernibility^[4]

The two points x and y in X are called Topologically indistinguishable , if and only if one of the following conclusions is true:

For each open set U in X, either U contains both x and y, or U does not contain both.

The neighborhood system of x is the same as that of y.

, and

。

Countable axiom

The separable X is called Separable space , if and only if it has a countable dense subset.
The first countable X is called First countable space , if and only if any of its points has a countable Neighborhood basis 。
The second countable X is called Second countable space , if and only if it has a countable base 。

Connectivity

Connected X is called Connected space , if and only if it is not the union of two disjoint non empty open sets. (Or equivalently, the closed open sets (both open and closed sets) of the space have only empty sets and full spaces).
Locally connected X is called Locally connected space If and only if every point of it has a special neighborhood base, this neighborhood base is composed of connected sets.
Completely disconnected X is called Completely disconnected , if and only if there is no connected subset of more than one point.
Road connection X is called Road connected space , if and only if any two points x and y , exists from x reach y Road of p , that is, there is a continuous mapping p :[0,1]→ X , satisfied p( 0)= x And p( 1)= y 。 The space connected by roads is always connected.
Local road connection X is called Local road connected space If and only if each point has a special neighborhood base, this neighborhood base is composed of road connected sets. A local road connected space is connected if and only if it is road connected.
The simply connected X is called Simply connected space , if and only if it is path connected and every continuous mapping
Are homotopy with constant mapping.
The contractible X is called Contractible space , if and only if it is homotopy equivalent to a point.
Hyperconnected X is called Hyperconnected , if and only if the intersection of any two non empty open sets is non empty. Hyperconnectivity implies connectivity.
Extremely connected X is called Extremely connected , if and only if the intersection of any two non empty closed sets is non empty. Extremely connected means road connected.
The mediocre X is called Mediocre space , if and only if its open set has only itself and empty set.

Compactness

Tightness X is called Compact space , if and only if any of its open covers has finite open cover refinement.
Lindlev property X is called Lindlev space , if and only if any of its open covers has thinning of countable open covers.
Paracompact X is called Paracompact space If and only if any of its open covers has refinement of locally finite open covers.
Countable compact X is called Countably compact space , if and only if any countable open cover is finite, the refinement of open cover.
Column tight X is called Isotonic , if and only if any of its point columns contain convergent sub columns.
Pseudo compact X is called Pseudo compact space , if and only if any real valued continuous function on it is bounded.

Metrizable

Measurability means that a space can be given a metric to give the topology of the space. There are many versions of the metrization theorem, the most famous of which is Ulysson's degree quantization theorem A second countable regular Hausdorff space can be metrized. Any second countable manifold All can be quantified.

Has algebraic structure

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For any kind of algebraic structure, we can consider the topological structure on it, and require that the relevant algebraic operations are continuous maps. For example, a Topological group G is a topological space with continuous mapping

(group multiplication) and

(inverse element) to make it have group structure.

Similarly, you can define Topological vector space It is a vector space with topological structure, so that addition and scalar multiplication are continuous mapping, which is the theme of functional analysis; We can similarly define topological rings, topological domains, and so on.^[5]

The combination of topology and algebraic structure can often lead to quite rich and practical theories, such as the main homogeneous space of differential geometry. In algebraic number theory and algebraic geometry, people often define appropriate topological structures to simplify the theory, and get more concise statements; Such as the local field (a topological field) in number theory, Galois theory Krull topology (a special topological group) considered in, and I-progressive topology (a topological ring) that is indispensable for defining formal generalities, etc.

Possessive order structure

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Topological space may also have natural Ordinal structure , examples include:

The order structure on the spectral space.

Specialization preorder: definition

。 Common in computer science 。

Separation axiom description

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There are mainly the following items.

T1 separation axiom

Any two different points in the space have a neighborhood without another point.

Hausdorff separation axiom

(T2 separation axiom) Any two different points in space have their own neighborhood disjoint.

Canonical separation axiom

Each point in the space and any closed set without the point have their own neighborhood disjoint.

Completely regular separation axiom

For every point x in space x and any closed set B without x, there exists a continuous mapping ƒ: x → [0,1] such that ƒ (x)=0 and ƒ (y)=1 for every point y in B.

Normal separation axiom

Any two disjoint closed sets in the space have their own neighborhood disjoint.

Meet T1 Separation axiom The space of is called T1 space 。 The space satisfying T2 separation axiom is called T2 space or Hausdorff space 。 If a T1 space also satisfies the regular separation axiom, the full regular separation axiom or the normal separation axiom, it is called Regular space ， Completely regular space and Normal space 。 The implication relationship between the spaces can be expressed as follows by "崊": normal space 崊 completely regular space 崊 regular space 崊 T2 space 崊 T1 space. metric space And the following compact spaces and paracompact spaces are normal spaces.

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