Definition:Completely Hausdorff Space/Definition 2

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:

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

That is:

$\struct {S, \tau}$ is a $T_{2 \frac 1 2}$ space if and only if every two points in $S$ are separated by closed neighborhoods.

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.