PlanetMath: completely Hausdorff

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completely Hausdorff

(Definition)

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

$\displaystyle U_x,V_y\in \tau,\qquad x\in U_x, y\in Y_y$

such that their closures are also disjoint:

$\displaystyle \overline{U_x}\cap \overline{V_y}=\emptyset.$

Then we say that $(X,\tau)$ is a completely Hausdorff space or a $T_{2\frac12}$ space.

Notes

A synonym for functionally Hausdorff space is Urysohn space [1]. Unfortunately, the definition of completely Hausdorff and $T_{2\frac12}$ 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:

functionally Hausdorff $\displaystyle \Rightarrow$ completely Hausdorff $\displaystyle \Rightarrow T_2=$ Hausdorff $\displaystyle ,$

which suggests why the $T_{2\frac12}$ name have been used to denote both completely Hausdorff spaces and functionally Hausdorff spaces.

Bibliography

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.

"completely Hausdorff" is owned by PrimeFan. [ full author list (5) | owner history (4) ]

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Other names:

completely Hausdorff space, $T_{2\frac{1}{2}}$ , Urysohn space

Attachments:: Hausdorff space not completely Hausdorff (Example) by drini

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Cross-references: implication, mean, term, Hausdorff space, closures, neighborhoods, disjoint, points, topological space
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This is version 12 of completely Hausdorff, born on 2004-03-17, modified 2007-09-14.
Object id is 5717, canonical name is CompletelyHausdorff.
Accessed 5969 times total.

Classification:

AMS MSC:

54D10 (General topology :: Fairly general properties :: Lower separation axioms )

Pending Errata and Addenda

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