The Baire Category Theorem

Topology · Axiom Academy

LESSON The Baire Category Theorem Understanding why complete metric spaces cannot be "meager" Before we can understand the Baire Category Theorem, we need to build up some vocabulary about "small" sets in metric spaces. The first concept is that of a nowhere dense set . A subset of a metric space is called nowhere dense if its closure has empty interior. That is: Intuitively, a nowhere dense set is "thin" - it doesn't contain any open balls, no matter how small. Even after we take its closure (add all limit points), we still can't find any open region completely contained in it. Example: The Rationals in the Reals Consider as a subset of . The rationals are not nowhere dense because: The closure is (rationals are dense in the reals) The interior of is all of , which is not empty However, a single point like is nowhere dense in . Its closure is just itself, and a single point has empty interior. Step 2: First and Second Category Now we can classify sets based on whether they can be built from nowhere dense sets. Let be a metric space and a subset. is of the first category (or meager ) if it can be written as a countable union of nowhere dense sets. is of the second category (or non-meager ) if it is not of the first category. The terminology might seem backwards at first! But think of it this way: first category sets are the "first attempt" at smallness, while second category sets are everything else - the complement of this notion of smallness.

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