a connected normal space with more than one point is uncountable

The proof of the following result is an application of the generalized intermediate value theorem (along with Urysohn’s lemma):

Proposition.

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

Proof.

Let $X$ be a http://planetmath.org/node/941connected http://planetmath.org/node/1532normal space with at least two distinct points $x_1$ and $x_2$. As the sets $\{x_1\}$ and $\{x_2\}$ are http://planetmath.org/node/2739closed and disjoint, Urysohn’s lemma furnishes a continuous function $f:X\to [0,1]$ 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 to give a bijection between a subset of $X$ and the uncountable set $[0,1]$, from which it follows that $X$ is uncountable.∎