completely Hausdorff

Definition 1.

[1] Let $(X,\tau)$ be a topological space. Suppose that for any two different points $x,y\in X,x\neq y$ , we can find two disjoint neighborhoods

$U_{x},V_{y}\in\tau,\qquad x\in U_{x},y\in Y_{y}$

such that their closures are also disjoint:

$\overline{U_{x}}\cap\overline{V_{y}}=\emptyset.$

Then we say that $(X,\tau)$ is a completely Hausdorff space or a $T_{2\frac{1}{2}}$ space.

Notes

A synonym for functionally Hausdorff space is Urysohn space [1]. Unfortunately, the definition of completely Hausdorff and $T_{2\frac{1}{2}}$ 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:

$\mbox{functionally Hausdorff}\Rightarrow\mbox{completely Hausdorff}\Rightarrow T% _{2}=\mbox{Hausdorff},$

which suggests why the $T_{2\frac{1}{2}}$ 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	$T_{2\frac{1}{2}}$
Synonym	Urysohn space
Related topic	HausdorffSpaceNotCompletelyHausdorff