Why are theorems such as the Baire Category Theorem proved for $C([0,1])$ and not more general spaces?

Question

In analysis I see that the proof of the Baire Category Theorem is proved for the set of all continuous functions on $[0,1]$, $C([0,1])$. However, I was wondering if the BCT would also hold for the set of continuous functions $C: \mathbb{R} \to \mathbb{R}$ as well. Is it just simpler to prove results for $C([0,1])$ or does it only hold on the mapping $C: [0,1] \to \mathbb{R}$? Thanks!

Answer 1

The Baire Category Theorem may be proved for

• Complete metric space

• Locally compact Hausdorff space

• Locally countably compact regular space

...and all of these use pretty much the same proof.

Why would a book prove it only for $C([0,1])$? The only reason I can think of is that this book will only use it in that case. Maybe the author does not assume his readers know what is a "complete metric space". But these are only my guesses for the reason.

Answer 2

The Baire Category Theorem as described in Wolfram Alpha is not limited to $C[0,1]$. A proof of the theorem as Wolfram Alpha describes it is found, e.g., in Rudin Functional Analysis 2.2, and doubtless in many other places. The assumptions are just (a) a complete metric space or (b) a locally compact Hausdorff space. For either of those spaces, the intersection of a countable dense collection of open subsets is dense.

Thus it would seem that your view that the Baire Category Theorem is limited in some way to $C[0,1]$ is incorrect.