An open cover of E is a collection of open sets whose union contains E.

In fact, if E contain a finite number of element, then for any open cover of E, you can find a finite subcover. For example if E = {1,2,3}, then given any open cover (can be infinite of course), choose three sets from it in such a way that the first set contains 1, the second contains 2 and the third contain 3. You are sure that these three exist since we started from an open cover covering all E. Hence, the collection of the three sets found constitute a finite subcover. In you example, yes, any open cover of a single point must have a finite subcover since the given set contain a finite number of points (in fact only one point).

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