# Cantor's intersection theorem and Baire Category Theorem

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.

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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:

3.21: Prove: If $(E_n)$ is a sequence of closed and bounded sets in a complete metric space $X$, if $E_n \supset E_{n+1}$, and if $\lim_{n\to\infty}\text{diam}~E_n=0$, then $\cap_1^\infty E_n$ consists of exactly one point.

3.22: Suppose $X$ is a complete metric space, and $(G_n)$ is a sequence of dense open subsets of $X$. Prove Baire's theorem, namely, that $\cap_1^\infty$ is not empty. (In fact, it is dense in $X$. Hint: Find a shrinking sequence of neighborhoods $E_n$ such that $\overline{E_n}\subset G_n$, and apply Exercise 21.

Proving density isn't actually much harder than proving nonemptiness.

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yep i have sorry for late reply – Koushik Jan 4 '13 at 5:39