Math 320 - November 23, 2020 14 Denseness and Baire’s Theorem

Definition 14.1. A set A⊆R is called an Fσ set, if it can be written as the countable union of closed sets. A set B ⊆R is called a Gδ set, if it can be written as the countable intersection of open sets.

One can show that a set is a Gδ set, if and only if its compliment is an Fσ set.

Definition 14.2. A set G⊆R is dense in R, if for all a,b∈R with aInterval Property, by carefully constructing a family of nested intervals {In} with In ⊆Gn, using the fact that Gn is open and dense. Taking compliments, it’s not hard to see that the last theorem also implies that it is impossible to write S∞ R= n=1Fn, where each Fn is a closed set containing no nonempty intervals.

Definition 14.5. A set E ⊆R is nowhere dense in R if E contains no nonempty open intervals. It is not hard to see that the notion of nowhere dense is precisely the opposite of the notion of being dense.

Theorem 14.6. E ⊆R is nowhere dense in R if and only if the compliment of E is dense in R. With the last theorem, it is clear that nowhere denseness is a measure of thinness of a set. And the previous results allow us to claim that R is larger than any countable union of thin sets. Theorem 14.7 (Baire). The set R cannot be written as the countable union of nowhere dense sets. The last fact holds for general complete metric spaces, i.e. those that are equipped with a proper notion of distance of elements (metric space), and where Cauchy sequences must be convergent (completeness). In that case Baire’s Category Theorem states that complete metric spaces cannot be meager sets, i.e. those that can be written as countable unions of nowhere dense sets.