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cover of )
:Let be a topological space, and be a subbase of If is compact, then every cover of by elements from has a finite subcover.

Although this proof makes use of Zorn's Lemma, the proof does not need the full strength of choice. Instead, it relies on the intermediate Ultrafilter principle.

Using this theorem with the subbase for above, one can give a very easy proof that bounded closed intervals in are compact. More generally,

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