A completely regular space or a Tychonoff space is a topological space satisfying the axioms of separability T 1 and T 3½ , that is, a topological space in which all one-point sets are closed and for any closed set and a point outside it there is a continuous numerical function equal to one on the set and zero at a point ( A.N. Tikhonov , 1930).
Properties
- Each Tikhonov space is regular .
- The subspace of Tikhonov space is Tikhonov space.
- The product of any number of Tikhonov spaces is Tikhonov.
- A topological space is Tychonoff if and only if it is homeomorphic to a subspace of a Tychonoff cube of some weight .
- A topological space is Tikhonov if and only if it has Hausdorff compactification .
- Topology in space Tikhonovskaya if and only if it is generated by some separable uniformity .
Examples
Tikhonov spaces are:
- Normal spaces , in particular metric spaces
- Locally compact Hausdorff spaces
- Topological groups satisfying the axiom of separability T 0 , in particular topological vector spaces
- Ordinal spaces with ordinal topology
- The Nemytsky plane is an example of a Tikhonov space that is not normal
Literature
- Engelking, R. General Topology. - M .: Mir , 1986 .-- 752 p.