HausdorffSpace

Hausdorff Space (T2 Space)

A Hausdorff space, also known as a $T_2$ space, is a topological space in which any two distinct points can be separated by non-overlapping open sets. Formally, a topological space $(X, \tau)$ is said to be Hausdorff if, for any two distinct points $x, y \in X$ (where $x \neq y$), there exist open sets $U, V \in \tau$ such that:

1. $x \in U$
2. $y \in V$
3. $U \cap V = \emptyset$

In a Hausdorff space, limits of sequences (if they exist) are unique, ensuring that sequences converge to at most one point. The $T_2$ separation axiom is a part of a hierarchy of separation conditions in topology, including $T_1$, $T_3$, $T_3\frac{1}{2}$, and $T_4$, each with its own distinct properties.