Definition:Hausdorff Space

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 by disjoint open sets.

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