Tychonoff space

In the mathematical branch of topology is meant by a completely regular space a topological space with special separation properties. Here are topological spaces that have a sufficient number of continuous functions in the bottom präzisierten senses to lead to a rich theory. The importance of this concept is illustrated by a variety of equivalent characterizations.

Definition

A completely regular space is a topological Hausdorff space, in which there is any closed set and every point is a continuous function with and for all.

For the purposes of this definition, a completely regular space has sufficiently many continuous functions to separate closed sets of outlying points.

Completely regular spaces after the Russian mathematician Andrei Nikolaevich Tikhonov also called Tikhonov spaces or T3 ½ rooms or T3a areas, as the defining characteristic between the separation axioms T3 and T4 is. There are authors who do not require the Hausdorff property in the definition of complete regularity and a Tikhonov space to understand a Hausdorff completely regular space.

Examples

  • Normal spaces are completely regular, as follows easily from the lemma of Urysohn. In particular, all metric spaces are completely regular.
  • The Niemytzki room is an example of a completely regular space which is not normal.
  • The Mysior level is an example of a regular Hausdorff space which is not completely regular.
  • Local Compact spaces are completely regular.
  • Hausdorff'sche topological vector spaces are completely regular, which among them are not locally compact infinite-dimensional.
  • General level, Hausdorff'sche topological groups are completely regular.
  • More generally, all Hausdorff'schen ( depending on definition ), uniform spaces Tychonoff spaces. This provides, in contrast to the other examples, even a characterization ( see below).

Permanence properties

  • Subspaces completely regular spaces are completely regular again.
  • Any products completely regular spaces 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 to coarser than the original topology on X. The following applies:

  • A Hausdorff space is completely regular if and only if its topology with respect coincides with the initial topology.

Using the Stone - Čech compactification shows you can easily:

  • The completely regular spaces are precisely the subspaces of compact Hausdorff spaces.

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

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

The uniform structure is not uniquely determined by the completely regular space. Uniform spaces are completely regular spaces with an additional structure, namely the uniform structure. The operations defined in Article uniform space notions completeness, uniform continuity and uniform convergence depends on the uniform structure, they can not be purely topological handle in the context of completely regular spaces.

A topology on a set is generated by a family of pseudo- metrics, if the open sets are precisely those quantities for which it to any finite number of pseudo- metrics and are with. The following applies:

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

Properties

Completely regular spaces are regular. Therefore, each point has a neighborhood base of closed sets.

If X is a topological Hausdorff space with a countable base, so are equivalent:

  • X is completely regular.
  • X is normal.
  • X is paracompact.
Urysohn's lemma Product topology
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