completely Hausdorff
Definition 1.
[1] Let $\mathrm{(}X\mathrm{,}\tau \mathrm{)}$ be a topological space^{}. Suppose that for any two different points $x\mathrm{,}y\mathrm{\in}X\mathrm{,}x\mathrm{\ne}y$, we can find two disjoint neighborhoods^{}
$${U}_{x},{V}_{y}\in \tau ,x\in {U}_{x},y\in {Y}_{y}$$ |
such that their closures^{} are also disjoint:
$$\overline{{U}_{x}}\cap \overline{{V}_{y}}=\mathrm{\varnothing}.$$ |
Then we say that $\mathrm{(}X\mathrm{,}\tau \mathrm{)}$ is a completely Hausdorff space or a ${T}_{\mathrm{2}\mathrm{\u2064}\frac{\mathrm{1}}{\mathrm{2}}}$ space.
Notes
A synonym for functionally Hausdorff space is Urysohn space [1]. Unfortunately, the definition of completely Hausdorff^{} and ${T}_{2\u2064\frac{1}{2}}$ are not as standard as one would like since. For example, the term completely Hausdorff space is also used to mean a functionally Hausdorff space (e.g. [2]). Nevertheless, in the present convention, we have the implication^{}:
$$\text{functionally Hausdorff}\Rightarrow \text{completely Hausdorff}\Rightarrow {T}_{2}=\text{Hausdorff},$$ |
which suggests why the ${T}_{2\u2064\frac{1}{2}}$ name have been used to denote both completely Hausdorff spaces and functionally Hausdorff spaces.
References
- 1 L.A. Steen, J.A.Seebach, Jr., Counterexamples in topology, Holt, Rinehart and Winston, Inc., 1970.
- 2 S. Willard, General Topology, Addison-Wesley Publishing Company, 1970.
Title | completely Hausdorff |
---|---|
Canonical name | CompletelyHausdorff |
Date of creation | 2013-03-22 14:16:03 |
Last modified on | 2013-03-22 14:16:03 |
Owner | PrimeFan (13766) |
Last modified by | PrimeFan (13766) |
Numerical id | 15 |
Author | PrimeFan (13766) |
Entry type | Definition |
Classification | msc 54D10 |
Synonym | completely Hausdorff space |
Synonym | ${T}_{2\u2064\frac{1}{2}}$ |
Synonym | Urysohn space |
Related topic | HausdorffSpaceNotCompletelyHausdorff |