Definition:Hausdorff Space

Definition
Let $T = \left({X, \vartheta}\right)$ be a topological space.

Then $X$ is a Hausdorff space or $T_2$ space iff:


 * $\forall x, y \in X: x \ne y: \exists U, V \in \vartheta: x \in U, y \in V: U \cap V = \varnothing$

That is, when every two points in $X$ are separated by neighborhoods.

For short, we can say $T$ is Hausdorff, using the name as an adjective.

This condition is known as the Hausdorff condition.

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

For more on the notation $T_2$, see the page on separation axioms.