Proposition   A connected normal space with more than one point is uncountable.

Proof. Let $X$ be a connected normal space with at least two distinct points $x_1$ and $x_2$ As the sets $\set{x_1}$ and $\set{x_2}$ are closed and disjoint, Urysohn's lemma furnishes a continuous function $f:X\rightarrow[0,1]$ such that $f(x_1)=0$ and $f(x_2)=1$ Because $X$ is connected, the generalized intermediate value theorem implies that $f$ is surjective. Thus $f$ may be suitably restricted to give a bijection between a subset of $X$ and the uncountable set $[0,1]$ from which it follows that $X$ is uncountable.