Definition 3.1.3 (Seperation Axioms).label Let $(X,\tau)$ be a topological space, then we have the following separation axioms which $X$ might satisfy

1. (T0)

   For any $x\neq y\in X$ there exists $U\in \tau$ such that either $y\nin U\ni x$ or $x\nin U\ni y$

2. (T1)

   For any $x\neq y\in X$ there exists $U\in\tau$ such that $y\nin U\ni x$

3. (T2)

   For any $x\neq y\in X$ there exists $U,V\in\tau$ such that $x\in U,y\in V$ and $U\cap V=\emp$

4. (T3)

   $X$ is (T1) and for any $x\in X,A\suf X$ closed with $x\nin A$ there exists $U,V\in\tau$ such that $x\in U,A\suf V$ and $U\cap V=\emp$

5. (T4)

   $X$ is (T1) and for any $A,B\suf X$ closed with $A\cap B=\emp$ there exists $U,V\in\tau$ such that $A\suf U,B\suf V$ and $U\cap V=\emp$