Uryshon Lemma

Definition

Consider a Normal Space $(X,\mathcal{T})$ and disjoint closed subsets $A,B\subset X$ of $X$. Then, there exists a continuous map $f:X\to [0,1]\text{s.t.}f(x)=\{\begin{array}{ll}0& x\in A\\ 1& x\in B\end{array}$. $\forall A,B\subset X\text{s.t.}(X\setminus A),(X\setminus B)\in \mathcal{T}\text{and}A\cap B=\varnothing ⟹\exists f\in C^0(X,[0,1])\text{s.t.}f(x)=\{\begin{array}{ll}0& x\in A\\ 1& x\in B\end{array}$