Show that every T1 spaces is countably compact if and only if every infinite ope

Question

Show that every T1 spaces is countably compact if and only if every infinite open cover (where infinite refers to the indexing set) has a proper subcover.

This question appears in "Elements of Point Set Toplogy" by John D. Baum and this is excersize 3.33. In this text, countably compact spaces are those such that: every infinite subset has a limit point in the space.

The hint is that it is related to the proof that:

A T1 space is countably compact if and only if every countable open cover has a finite subcover. I have a suspicion that one may need to use Zorn's lemma.

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