Hausdorff space

Definition

A topological space $X$ is Hausdorff when for all points $x \neq y$ of $X$ there are open sets $U$ and $V$ with $x \in U$, $y \in V$ and $U \cap V = \varnothing$.

Remarks

Each metric space is Hausdorff. Subspaces and products of Hausdorff spaces are Hausdorff. In a Hausdorff space, limits of sequences are unique and compact subsets are closed. Each Hausdorff space is T1. An infinite set with the cofinite topology is T1 but not Hausdorff. The Hausdorff condition is also called T2.

Explanations

L1 human

In a Hausdorff space, you can put any two distinct points into two open sets that do not meet. Most spaces of analysis, such as $\mathbb{R}^n$, are Hausdorff.