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?

Answer 1

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

Answer 2

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