("A and B can be separated by a continuous real-valued function")

Lemma.

Let $X$ be $T_4$ (Separation axiom), and $A,B$ be disjoint closed subset of $X$. Then there exists a Continuous function $f: X \to [0,1]$ ($\subset \mathbb{R}$) s.t. $f(A) = \{0\}$ and $f(B) = \{1\}$.

Observation: $X$ is $T_4$ if and only if any disjoint closed subsets can be separated by a cont. real-valued function

Proof that f separates

Proof that f is continuous