# 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):

###### 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 $f:X\rightarrow[0,1]$ 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 to give a bijection between a subset of $X$ and the uncountable set $[0,1]$, from which it follows that $X$ is uncountable. ∎

Title a connected normal space with more than one point is uncountable AConnectedNormalSpaceWithMoreThanOnePointIsUncountable 2013-03-22 17:17:46 2013-03-22 17:17:46 azdbacks4234 (14155) azdbacks4234 (14155) 10 azdbacks4234 (14155) Theorem msc 54D05 UrysohnsLemma NormalTopologicalSpace Uncountable Bijection ConnectedSpace