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Let $X$ be a compact Hausdorff space. I would like to find a (minimal) condition on $X$, which is sufficient to guarantee that one can always find a non-constant continuous function

$f : X \to \mathbb{C}$.

Clearly we need to assume that $|X|>1$, but what else can we say?

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Urysohn's lemma says that $\vert X \vert > 1$ suffices. – Andrea Nov 19 '11 at 13:40
up vote 3 down vote accepted

Just to give this an answer:

As has been pointed out in the comments, $|X|>1$ is both necessary and sufficient: if $x,y\in X$ are distinct, Urysohn’s lemma ensures that there is a continuous function $f:X\to[0,1]$ such that $f(x)=0$ and $f(y)=1$.

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