From an old post in math stackexchange, I read a comment which goes as follows " I like to think of Baire Category Theorem as spiced up version of Cantor's Intersection Theorem". My question -----is it possible to derive the latter one using the former?
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The association between the two is the Baire Category Theorem for locally compact Hausaduff spaces used finite intersection property, whereas Cantor's intersection theorem give an infinite intersection property assuming the space is compact.
For a proof and discussion you can check wikipedia.
Do you have a copy of Rudin's Principles of Mathematical Analysis? If you do, then problems 3.21 and 3.22 outline how this is done. Quoting here:
Proving density isn't actually much harder than proving nonemptiness.