T1 Space

Definition

A Topological Space $(X,\mathcal{T})$ is $T_1$ space if $\forall x\in X,(X\setminus x)\in \mathcal{T}⟺\forall x,y\in X\text{s.t.}x\ne y,\exists {\mathcal{N}}_x\text{s.t.}y\notin {\mathcal{N}}_x$

Examples

Finite complement topology on Real Number ${\mathcal{T}}_f=\{U\subset \mathbb{R}|U=\varnothing \text{or}|\mathbb{R}-U|<\infty \}$ satisfies $T_1$ axiom but not $T_2$.

Facts

Consider a Topological Space $(X,\mathcal{T})$ satisfying $T_1$ axiom, and a subset $A\subset X$. $x\in A^{\prime }⟺\forall {\mathcal{N}}_x,|{\mathcal{N}}_x\cap A|=\infty$