Completely regular room

In the mathematical sub-area of topology , a completely regular space is understood to be a topological space with special separation properties . These are topological spaces which, in the sense specified below, have a sufficient number of continuous functions to lead to a rich theory. The meaning of this term becomes clear through a large number of equivalent characterizations.

definition

The function f separates the point x from the set A.

A completely regular space is a topological Hausdorff space in which there is a continuous function with and for all for every closed set and every point . ${\displaystyle X}$ ${\displaystyle A\subseteq X}$ ${\displaystyle x\in X\setminus A}$ ${\displaystyle f\colon X\rightarrow {\mathbb {R} }}$ ${\displaystyle f(x)=1}$ ${\displaystyle f(a)=0}$ ${\displaystyle a\in A}$

In the sense of this definition, a completely regular space has a sufficient number of continuous functions to separate closed sets from external points. In addition, there is no restriction to assume that these are restricted functions . That is, if there is a continuous function that separates and , then it is also with , defined by for as well as for and for . ${\displaystyle f}$ ${\displaystyle x}$ ${\displaystyle A}$ ${\displaystyle g\circ f}$ ${\displaystyle g\colon \mathbb {R} \to \mathbb {R} }$ ${\displaystyle g(z)=z}$ ${\displaystyle z\in [0,1]}$ ${\displaystyle g(z)=1}$ ${\displaystyle z>1}$ ${\displaystyle g(z)=0}$ ${\displaystyle z<0}$

Completely regular spaces are also called Tichonow spaces after the Russian mathematician Andrei Nikolajewitsch Tichonow or also T _3½ spaces or T _3a spaces , since the defining property lies between the separation axioms T ₃ and T ₄ . There are authors who do not require the Hausdorff property in the definition of complete regularity and understand a Tichonow space to be a Hausdorff completely regular space.

Examples

• Normal Hausdorff spaces are completely regular, as follows easily from Urysohn's lemma. In particular, all metric spaces are completely regular.
• The Niemytzki room is an example of a completely regular room that is not normal.
• The Mysior level is an example of a regular Hausdorff room that is not completely regular.
• Local compact Hausdorff rooms are completely regular.
• Hausdorff's topological vector spaces are completely regular, the infinite-dimensional ones among them are not locally compact.
• More generally, Hausdorff's topological groups are completely regular.
• Even more general are all Hausdorff's (depending on the definition), uniform rooms Tichonow rooms. In contrast to the other examples, this even provides a characterization (see below).

Permanence properties

• Sub-spaces of fully regular rooms are fully regular again.
• Any products of completely regular rooms are completely regular again.

Characterizations

For a topological space, consider the set of all continuous functions . By definition, for any topological space , the initial topology with respect. Coarser than the original topology on X. The following applies: ${\displaystyle C(X)}$ ${\displaystyle X\rightarrow {\mathbb {R} }}$ ${\displaystyle X}$ ${\displaystyle C(X)}$

• A Hausdorff space is completely regular if and only if its topology coincides with the initial topology . ${\displaystyle C(X)}$

With Stone-Čech compacting it is easy to show:

• The completely regular rooms are exactly the sub-rooms of compact Hausdorff rooms.

Uniform spaces induce a topology on the underlying set, see article uniform space . The following applies:

• A Hausdorff space X is completely regular if and only if its topology is induced by a uniform structure.

The uniform structure is not clearly defined by the completely regular space. Uniform rooms are completely regular rooms with an additional structure, namely the uniform structure. The terms completeness , uniform continuity and uniform convergence defined in the article uniform space depend on the uniform structure; they cannot be treated purely topologically in the context of completely regular spaces.

A topology on a set is generated by a family of pseudometrics if the open sets are precisely those sets for which there is a finite number of pseudometrics and a with for each . The following applies: ${\displaystyle X}$ ${\displaystyle {\mathcal {D}}}$ ${\displaystyle U\subset X}$ ${\displaystyle x\in U}$ ${\displaystyle d_{1},\ldots ,d_{n}\in {\mathcal {D}}}$ ${\displaystyle \epsilon >0}$ ${\displaystyle \{y\in X;d_{j}(x,y)<\varepsilon ,j=1,\ldots ,n\}\subset U}$

• A topological Hausdorff space is completely regular if and only if its topology is generated by a family of pseudometrics.

properties

Completely regular rooms are regular . Hence every point has a neighborhood base of closed sets.

If X is a topological Hausdorff space with a countable basis , then are equivalent:

• X is completely regular.
• X is normal .
• X is paracompact .

literature

• Johann Cigler , Hans-Christian Reichel : Topology. A basic lecture. Bibliographisches Institut, Mannheim et al. 1978, ISBN 3-411-00121-6 ( BI university pocket books 121).
• Wolfgang Franz : Topology. Volume 1: General Topology. 4th edition. de Gruyter, Berlin et al. 1973, ISBN 3-11-004117-0 ( Göschen collection 6181).