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# 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.

## Mathematics Subject Classification

54D10*no label found*

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