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Let $p:C\to X$ be a covering space having a global section $s:X\to C$. I can show that this implies that $s(X)$ is disconnected from the rest of $C$.

Is there any reference where this is explicitly proven? I need this result for a paper and would like to reference it. Thanks!

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I don't know of a reference in print, but I answered a similar question here: math.stackexchange.com/questions/256951/…. Worst case scenario, you could reference MSE itself. –  Jason DeVito Jan 18 '13 at 23:56

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