Definition:Completely Hausdorff Space/Definition 1

Definition

Let $T = \struct {S, \tau}$ be a topological space.

$\struct {S, \tau}$ is a completely Hausdorff space or $T_{2 \frac 1 2}$ space if and only if:

$\forall x, y \in S, x \ne y: \exists U, V \in \tau: x \in U, y \in V: U^- \cap V^- = \O$

That is, for any two distinct elements $x, y \in S$ there exist open sets $U, V \in \tau$ containing $x$ and $y$ respectively whose closures are disjoint.

Source of Name

This entry was named for Felix Hausdorff.

Also see

• Equivalence of Definitions of Completely Hausdorff Space

• Results about completely Hausdorff spaces can be found here.