Theorem: If R^k=countable union of closed sets F_n_ then there is one closed F_n_ with nonempty interior.(adsbygoogle = window.adsbygoogle || []).push({});

Equivalent statement: Let G_n_ be a open dense subset of X then take a countable intersection of these G_n_'s, the intersection is nonempty in fact it is dense in R^k.

So, i'm really not sure where to begin. This is a special case of Baire's theorem and is given as the last problem on chapter 2 of Rudin's PMA. Rudin's hint tells me imitate the proof of theorem 2.43 which proves that every perfect set in R^k is uncountable.

Can anyone give me a hint on where to begin with the hint!

It's easy to show that any finite intersection of dense open subsets is also open and dense in R^k. That's as far as I've gotten.

Please do not give the problem away.

**Physics Forums | Science Articles, Homework Help, Discussion**

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

# The Baire Category Theorem (RUDIN)

**Physics Forums | Science Articles, Homework Help, Discussion**