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Munkres - topology p.278

I exactly followed the argument in the text, and I cannot find where I used hausdorffness. Where in the argument used Hausdorffness?

The reason why I am asking is that the article in wikipedia requires the Hausdorffness:

Wikipedia - Let $X$ be a compact Hausdorff space. Then a subset $F$ of $C(X)$ is relatively compact in the topology induced by the uniform norm if and only if it is equicontinuous and pointwise bounded.

Mark Fantini
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Jj-
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1 Answers1

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There is no need for the Hausdorffness hypothesis. Actually the theorem is true for quite general domain (like X being a convergence space only) and using the continuous convergence on $C(X)$. If you want to use a more general range than $\mathbb{R}$ you have to keep regularity and Hausdorff properties on it.

brunoh
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