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completely Hausdorff
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(Definition)
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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$$U_x,V_y\in \tau,\qquad x\in U_x, y\in Y_$$ 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 $T_{2\frac12}$ space.
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  completely Hausdorff  Hausdorff 
- 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.
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"completely Hausdorff" is owned by PrimeFan. [ full author list (5) | owner history (4) ]
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Cross-references: implication, mean, term, Hausdorff space, closures, neighborhoods, disjoint, points, topological space
There are 2 references to this entry.
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 6790 times total.
Classification:
| AMS MSC: | 54D10 (General topology :: Fairly general properties :: Lower separation axioms ) |
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Pending Errata and Addenda
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