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 .
is called regular space if every closed set and every point are separated by neighborhoods from and from , i.e. with .
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 .
- 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 .
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 .
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
- ^ 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).
- ↑ Lynn Arthur Steen: Counterexamples in Topology. Courier Corporation, 1995, ISBN 978-0-486-68735-3 , p. 100 ( limited preview in Google Book Search).
- ↑ René Bartsch: General Topology. Walter de Gruyter GmbH & Co KG, 2015, ISBN 978-3-110-40618-4 , p. 118.
- ↑ René Bartsch: General Topology. Walter de Gruyter GmbH & Co KG, 2015, ISBN 978-3-110-40618-4 , p. 122.