Conneted normal space having more that one point is uncountable

Question

A connected normal space having more that one point is uncountable.

As I apply Uryshon Lemma to resolve this fact. Thanks.

Answer 1

Let $X$ be a connected normal space and $x$ and $y$ are distinct points of $X$. Then by Uryshon Lemma, there is a continuous function $f:X → [0, 1]$ with $f(x)=0$ and $f(y)=1$. Since the image $f(X)$ is connected, $f(X)=[0, 1]$; suppose not. Then $f(X)$ is a disjoint union of nonempty open sets $f(X)\cap[0, r)$ and $f(X) \cap (r, 1]$ for some $r \notin f(X)$ where $0 < r < 1$. Consequently, the preimage $f^{-1}(s)\neq-1$ for any irrational $s \in[0, 1]$. Thus $X$ is uncountable.