Compact connected Hausdorff


> Lemma: If space X is compact and connected. For any open set
> U of X, the closure of some component of U intersect X - U.

The above lemma was present some time ago. It seemed quickly
posted for it's necessary that U be nonnul and proper subset.

I'm of the mind that "of some component" was to be read as
"of any component" and that Hausdorff is tacid assumption.

Is the surmise correct? Anyway, let C be a component of open U and
assume the negation, that cl C /\ (X - U) is empty, ie
(cl C) - U = nulset

>From that comes
C = cl C proper subset U.

However how to prove the lemma eludes me. Even knowing the equivalence of
components and quasi-components in continuums, I see no way of using that.
Have you suggestion?


.

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