Hausdorff spaces

Definition

A topological space $X$ is called Hausdorff if any pair of distinct points have disjoint neighborhoods $i$.$e$. distinct points can be separated by open sets

Example

Every metric space is Hausdorff

Proposition

• Any finite subset of a Hausdorff space is closed.
• Convergent sequences have unique limits (main goal why we define Hausdorff space)
• If $p\in X$ is a limit point of $A$, then any neighborhood of $p$ contains infinitely many points of $A$

NOTE

Every metric space is Hausdorff. (the converse is false)