Topological Space

In Mathematical, the topological spaces make it possible to define in a very general context of the concepts like the convergence, the Continuité and the connexity. These concepts appear in almost all the branches of mathematics, they are thus central in the modern vision of mathematics. The branch of mathematics which studies these spaces calls the Topologie.

The concept of space topological is an axiomatic definition, formalized by a structure ensemblist. The axioms are minimal, and in this direction it is the most general structure to study the quoted concepts.

This article is technical, a vision general and historical is given in Topologie.

Concepts

Definitions

A topological space is a couple

(E, \ Tau)

, where

E

is a Ensemble and

\ Tau

a number of parts of

E

which one defines as the open of

(E, \ Tau)

, checking the following properties:

the Empty set and $E \;$ is defined like open.
All meeting of open is open, i.e. if $(O_i) _ {I \ in I}$ is a family (finished or infinite) of elements of $\ Tau$ , then

\ bigcup_ {I \ in I} O_i \ in \ Tau

All intersection finished the open ones is opened, i.e. if $O_1, \ ldots, O_n$ are elements of $\ Tau$ , then

O_1 \ course \ ldots \ course O_n \ in \ Tau

The unit

\ Tau

is then called topology of

E

It results from the elementary set theory that any intersection finished of open is open.

The closed of a topology are defined like the Complémentaire S of the open ones. Consequently, the family of closed contains $E$ and the empty set. It results from the elementary set theory that any intersection from closed one is closed, and that any meeting finished of closed one is closed.

It is of use to recall the presence of the empty part to axiom 1; it is however in good rigor superfluity, since one can obtain it by applying axiom 2 to the meeting indexed by the empty set.

One of the first roles of topology is to describe the Voisinage S of the points. It is a concept-key to include/understand topology. It is used for example with the definition of Continuité or Limite in a point. This concept is formalized in the article Voisinage. Let us recall here simply that part of $E$ is a point neighborhood if and only if it contains open containing this point.

Examples

a simple example is $(\ mathbb {NR}, \ mathcal {P} (\ mathbb {NR}))$ . All let us singletons them are opened, all the points are thus isolated from/to each other. Topology thus defined is called discrete topology . More generally, discrete topology on a unit $X \,$ is that for which $\ Tau = \ mathcal {P} (X)$ .
N the other hand of simplicity, it does not offer much interest.

Another example without interest: coarse topology on $X \,$ is that for which the only open ones are the empty part and $X \,$ itself.

a more interesting example on the entireties is $(\ NR, \ mathcal F) \;$ where $\ mathcal F \;$ indicates the Filtre of Fréchet, i.e. all the complementary ones to finished units and the empty set. This topology gives a direction in the vicinity of infinite and makes it possible for example to define the concept of limit of a continuation.

the article on the Voisinage S shows that there exists a topology associated with all metric Espace. Open a $O \;$ is then a unit which contains for each point $a \;$ of $O \;$ a swell open of center $a \;$ .

the whole of the real numbers is thus provided naturally with a topology resulting from its distance. Open is then a union of open intervals.

the induced topology of a subset $F \;$ of a unit $E \;$ is the topology obtained by intersection of open of $E \;$ with $F \;$ . This definition makes it possible for example to define the topology induced by that of $\ R$ on an interval, and thus to be able to define the properties of continuity and limit in functions defined on an interval of $\ R$ .

Of other examples of more sophisticated topologies is given in the article vicinity.

the Cube of Hilbert $^ \ N$ is a generalization of the cube in infinite dimension.

the Ensemble of Cantor is source of many examples and counterexamples.

Continuous applications

Definitions

One of the first interests of the concept of space topological is to be able to define a continuous application. There exist two approaches, the local approach given in the article Voisinage and which defines continuity in a point, and the comprehensive approach which defines continuity in any point.

total Definition . An application $f \;$ of $A \; \ rightarrow \; B \;$ between two topological spaces is known as continuous if the reciprocal image $f^ {- 1} (U) \;$ of all open $U \;$ of $B \;$ is open of $A \;$ (the reciprocal image $f^ {- 1} (U) \;$ is the whole of all the points of $A \;$ that $f \;$ sends in $U \;$ ).

local Definition . That is to say $f \;$ a function of a topological space $E \;$ in $F \;$ and is $a \;$ a point element of the field of definition of $f \;$ . The function $f \;$ is continuous at the point $a \;$ if and only if the reciprocal image of open containing $f (a) \;$ contains open containing $a \;$ . This statement is equivalent to that given in the article Voisinage.

Equivalence of local continuity in any point and total continuity . If an application is overall continuous the reciprocal image of open containing $f (a) \;$ contains itself which is open containing $a \;$ . The application is thus continuous in any point. Reciprocally, if the application is continuous in any point then its reciprocal image contains for each point open container and included in the reciprocal image. The union of all its open is by definition open and is equal to the reciprocal image. The reciprocal image is thus open.

A continuous bijective mapping and whose reciprocal one is continuous is called a Homéomorphisme.

the concept of continuous application is developed in detail in the article Continuité.

Examples

the identity application of a topological space in itself is continuous. Indeed the reciprocal image of all open is itself thus is open.

a constant application of a topological space in another is continuous. Indeed the reciprocal image is either the empty set or the entire starting whole.

the application $\ left \ {\ begin {matrix} \ R & \ rightarrow & \ R \ \ X & \ mapsto & x^2 \ end {matrix} \ right.$ is continuous. The proof is given by it in the article Continuité.

Limit

Adherence

This concept is developed in a specific article Adhérence. We will develop this concept only insofar as it is necessary to formalize the concept of Limite.

In topology the adherence of a $X$ part of a topological space is the smallest closed unit which contains this part. It is often noted $\ overline X$ .

Another way of defining topological spaces consists in calling upon the pretopologic concept of adherence : one defines a adherence on a unit E as an application which with very part has E associates a part containing has , the adherence of the part empties remaining empty. If adherence is idempotente and where the adherence of the union of two parts is equal to the union of adherences, it is said that adherence is topological . A topological space can be defined as a unit provided with a topological adherence. The open ones are then complementary to stable parts for adherence.

In terms of adherences, an application of a topological space in another is continuous if and only if the image of an adherent point to a part is necessarily adherent with the image of this part.

Definition

The concept of limit, if it exists, describes the behavior which a function should have if it were defined in this point. The simplest example is the case of a function defined on an open interval of

\ R

; the limit is the concept which makes it possible to determine the behavior of the function at the boundaries of this interval.

That is to say $(E, \; \ Tau) \;$ and $(F, \; \ Upsilon) \;$ two topological spaces. That is to say $(E', \; \ Tau') \;$ a subspace of $E \;$ provided with its induced topology and $f \;$ a function of $E' \;$ in $F \;$ . That is to say finally a point $a \;$ of $\ overline {E'} \;$ and $l \;$ a point of $F \;$ . Then $l \;$ is the limiting of the function $f \;$ at the point $a \;$ if and only if the reciprocal image of open containing $l \;$ contains open ${E'} \;$ container $a \;$ . This statement is equivalent to that which is given in the article Voisinage. The statement being simpler with the formalism of the vicinities, it is in general that one which is used.

; Notice 1: The concept of limit is developed in the article Limite.

; Notice 2: If the point $a \;$ is element of the unit ${E'} \;$ , then the limit, if it exists, is equal to $f (a) \;$ and the function $f \;$ is continuous in $a \;$ .

Properties

One says that a topological space is separate or of Hausdorff or $T_2$ when two unspecified distinct points admit disjoined vicinities.
One says that a space checks the Propriété of Borel-Lebesgue when one can extract a under-covering finished from all open Recouvrement. One also speaks about quasi-compact space .
a quasi-compact and separate space is known as compact .

Examples

the first historical example of topological space is the whole of the real numbers. This example is that which is at the base of the theory of topological spaces. It seems a particular case of the second family of examples given here.
the metric spaces and, in particular, the normalized vector spaces are topological spaces.
There exist many classes of topological spaces (topological vector spaces, Espace of Banach, Fréchet, Hilbert, Hausdorff, Kolmogorov, Montel, Baire, compact, quasi-compact, précompacts, paracompacts, connected well, complete, related, simply related, related by arcs, locally compact, locally related, topological Groupe, topological Anneau etc).

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