Regular room

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

A T ₃ room is a regular room that is also a Hausdorff room .

definition

Be a topological space. Two subsets and of are called separated by neighborhoods if there are disjoint open sets and with and . ${\displaystyle X}$ ${\displaystyle Y}$ ${\displaystyle Z}$ ${\displaystyle X}$ ${\displaystyle U}$ ${\displaystyle V}$ ${\displaystyle Y\subset U}$ ${\displaystyle Z\subset V}$

${\displaystyle X}$is called regular space if every closed set and every point are separated by neighborhoods from and from , i.e. with . ${\displaystyle A\subset X}$ ${\displaystyle x\notin A}$ ${\displaystyle U\in {\mathfrak {U}}(A)}$ ${\displaystyle A}$ ${\displaystyle V\in {\mathfrak {U}}(x)}$ ${\displaystyle x}$ ${\displaystyle U\cap V=\emptyset }$

Note: In the literature, the designation regular room and T ₃ room is not clear. Occasionally the definitions are reversed compared to the variant presented here.

Examples

• Any indiscreet room with more than one element is regular.
• Every metric space is regular.
• The Niemytzki room is a regular room that is not normal .

Permanence properties

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

Relationships with other axioms of separation

• Every regular room is symmetrical .
• Every regular space that satisfies T _{0 also} satisfies T ₂ and thus T ₁ : Consider two points and . Without loss of generality, there is an open environment of that does not contain (otherwise swap the two points). Their complement is closed and contains , but not and can therefore be separated from by disjoint neighborhoods, which thus also separate and . ${\displaystyle x}$ ${\displaystyle y}$ ${\displaystyle y}$ ${\displaystyle x}$ ${\displaystyle x}$ ${\displaystyle y}$ ${\displaystyle y}$ ${\displaystyle x}$ ${\displaystyle y}$
• Every regular room is pre- regular .
• Every regular room is also semi- regular . The regularly open sets form the basis of a regular space. However, this property is weaker than that of regularity. That is, there are topological spaces whose regular open sets form a basis, but which are not regular.
• A topological space is a regular space if and only if the Kolmogoroff quotient KQ ('X') satisfies the axiom of separation T ₃ .
• Every completely regular room is also regular, the reverse is not true, as the example of the Mysior plane shows.
• If a regular space fulfills the second axiom of countability , then it is already normal and can be pseudometrized according to Urysohn's metrisability theorem .
• Every symmetrical normal space is regular.

Equivalent characterization

A topological space is regular if and only if each of its points has a neighborhood basis made up of closed sets. To be the environment base of a point means that one finds an environment with and for every environment . ${\displaystyle {\mathfrak {B}}}$ ${\displaystyle x\in X}$ ${\displaystyle U\in {\mathfrak {U}}(x)}$ ${\displaystyle B\in {\mathfrak {U}}(x)}$ ${\displaystyle B\in {\mathfrak {B}}}$ ${\displaystyle B\subseteq U}$

The factual situation can also be expressed quite easily with the topological basic terms ( openness and closure ) without having to introduce environments and environment bases: For each , open, one finds an open with . ${\displaystyle x\in O}$ ${\displaystyle O}$ ${\displaystyle V}$ ${\displaystyle x\in V\subseteq {\overline {V}}\subseteq O}$

literature

• Boto von Querenburg : Set theoretical topology (= Springer textbook ). 3rd, revised and expanded edition. Springer, Berlin et al. 2001, ISBN 3-540-67790-9 .

Individual evidence

1. ^ Boto von Querenburg: Set theoretical topology. 3rd, revised and expanded edition. Springer, Berlin et al. 2001, ISBN 3-540-67790-9 , p. 84 ( limited preview in the Google book search).
2. ↑ Lynn Arthur Steen: Counterexamples in Topology. Courier Corporation, 1995, ISBN 978-0-486-68735-3 , p. 100 ( limited preview in Google Book Search).
3. ↑ René Bartsch: General Topology. Walter de Gruyter GmbH & Co KG, 2015, ISBN 978-3-110-40618-4 , p. 118.
4. ↑ René Bartsch: General Topology. Walter de Gruyter GmbH & Co KG, 2015, ISBN 978-3-110-40618-4 , p. 122.