Is $\mathbb R$ a normal topological space?

Question

As in the title, in euclidean space is it always possible two find for two disjoint closed sets $A,B$ two open sets $U,V$ disjoint such that $A \subseteq U$ and $B \subseteq V$ (T4-property, normal)?

Answer 1

Every metric space is normal, in particular $\mathbb R^n$. The proof goes as follows:

For each $a\in A$, let $r_a=\frac{1}{3}d(a,B)$, and for each $b\in B$, let $s_b=\frac{1}{3}d(b,A)$. Now define $U=\bigcup_{a\in A}B(a,r_a)$ and $V=\bigcup_{b\in B}B(b,s_b)$. It is not hard to show that $A\subseteq U,$ $B\subseteq V,$ and $U\cap V=\varnothing$.