Regular space

In topology and related areas of mathematics regular rooms are special topological spaces in which every closed subset A and any point not lying in A are separated by x environments.

A T3 - space is a regular space is a Hausdorff space as well.

Definition

Be a topological space. Two subsets and of the hot environments separately, if disjoint open sets and exist with and.

Called a regular space if every closed set and each point separated by environments as well, so with.

Note: In the literature, the term regular room and T3 - space is not unique. Occasionally, the definitions from the version presented here variant are reversed.

Relationships to other separation axioms

• Every regular space which satisfies T ₀, also satisfies T ₂ T ₁ and thus: Consider two points. Without loss of generality exists a neighborhood that does not contain (otherwise reverse the two points). Its complement is completed and includes but is not and can therefore be separated from by disjoint environments and thus separate well.

• Every regular space is präregulär.

• Every regular space is also semi- regular. The regular open sets form a basis of a regular space. This characteristic, however, is weaker than that of regularity. That is, there are topological spaces whose regular open sets form a base but which are not regular.

• A topological space is then a regular room when the Kolmogorov quotient KQ ( 'X') satisfies the separation axiom T3.

• Every completely regular space is regular, the converse is not true, as the example of Mysior - level shows.

• Meets a regular room the second axiom of countability, it is already normal and after Metrisierbarkeitssatz of Urysohn pseudometrisierbar.

Equivalent characterization

A topological space is regular if each of its points has a neighborhood base of closed sets. To be based around a point, means that for every neighborhood an environment and place.

The facts can be quite easily be expressed simply with the basic topological concepts ( openness and degree), without having to introduce environments and surroundings bases: For each open, you will find an open with.