Suppose $X$ is a compact Hausdorff space and let $x,y \in X$, and $x \not = y$

Question

Then there is a continuous real-valued function $f$ on $X$ such that $f(x) \not = f(y)$. I'm really stuck upon this problem. Any help?

score 3 · Accepted Answer · answered Feb 20 '20 at 09:42

3

This is a consequence of the theorem of Tietze Uryshon, if $X$ is Hausdorff, $\{x,y\}$ is closed, defined $f$ on $\{x,y\}$ by $f(x)=0, f(y)=1$, it can be extended to $X$.

https://en.wikipedia.org/wiki/Tietze_extension_theorem

answered Feb 20 '20 at 09:42

Tsemo Aristide

89,587

freakish · Answer 2 · 2020-02-20T09:48:08.723

3

Compact Hausdorff spaces are normal. Then apply Urysohn's lemma.

edited Feb 20 '20 at 09:48

answered Feb 20 '20 at 09:42

freakish

47,446

Suppose $X$ is a compact Hausdorff space and let $x,y \in X$, and $x \not = y$

2 Answers2