Login
This is a place holder for potential sponsor logos.
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$$U_x,V_y\in \tau,\qquad x\in U_x, y\in Y_$$ 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.
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
completely Hausdorff
Hausdorff
which suggests why the $T_{2\frac12}$ name have been used to denote both completely Hausdorff spaces and functionally Hausdorff spaces.
completely HausdorffHausdorff
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 John Smith, Pedro Sanchez, Raymond Puzio, matte, Michael Slone, L. H..
None.
[ View all 4 ]