Definition:Hausdorff Space/Definition 3

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

$\struct {S, \tau}$ is a Hausdorff space or $T_2$ 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:

for any two distinct elements $x, y \in S$ there exist disjoint neighborhoods $N_x, N_y \subseteq S$ containing $x$ and $y$ respectively.

That is:

$\struct {S, \tau}$ is a $T_2$ space if and only if every two elements in $S$ are separated by neighborhoods.

Also known as

This condition is known as the Hausdorff condition.

For short, $T$ is Hausdorff is used to mean $T$ is a Hausdorff space.

Conveniently, a topological space is Hausdorff if any two distinct points can be housed off from one another in separate disjoint open sets.

Some sources use the term separated space for Hausdorff space but this is discouraged as there already exists considerable confusion and ambiguity around the definition of the word separated in the context of topology.

Some authors require a space to be Hausdorff before allowing it to be classed as a topological space, but this approach is unnecessarily limiting.

Also see

Source of Name

This entry was named for Felix Hausdorff.