Completely Regular Space

Definition

A Topological Space $(X,\mathcal{T})$ is completely regular ($T_{3\frac{1}{2}}$, $T_{3.5}$ or Tychnoff) space if it satisfies T1 Axiom and for all disjoint $x$ and Closed Set $B$ in $X$, there exists a continuous real-valued function $f:X\to \mathbb{R}$ which separates $x$ and $B$. $\forall x\in X,(X\setminus B)\in \mathcal{T}\text{s.t.}x\cap B=\varnothing ,\exists (f:X\to \mathbb{R})\in C^0\text{s.t.}f(\{x\})=\{0\},f(B)=\{1\}$

Facts

Consider a completely regular space $(X,\mathcal{T})$. Then the weak topology for $X$ generated by the set of all bounded continuous functions $C(X,\mathbb{R})=\{(f:X\to \mathbb{R})\in C^0|\forall x\in X,\exists M\in \mathbb{R}\text{s.t.}|f(x)|\le M\}$ is given topology $\mathcal{T}$.