In Gale–Stewart games: set A is endowed with discrete topology, and Aω w resulting product topology, where Aw: a countably infinite topological product of A*A A : {0,1}, topology on Aω is ordinary topology on #Cantor space
A :N , it is ordinary #topology on #Baire space.

- #Cantor space is #homeomorphic to any finite #Cartesian power of itself, and #Baire space is homeomorphic to any finite Cartesian power of itself, the analytical hierarchy applies equally well to finite Cartesian power of one of these spaces.

- If a subset of a Polish space has the property of #Baire, then its corresponding #Banach–#Mazur game is determined. The converse does not hold

- the Boolean prime ideal theorem implies that there is a nonprincipal ultrafilter on the set of natural numbers; each such #ultrafilter induces, via binary representations of reals, a set of reals without the #Baire property.

As typical for existence arguments invoking #Baire category theorem, this #proof is nonconstructive. It shows that family of continuous functions whose #Fourier series converges at a given x is of first Baire category, in #Banach space of continuous functions on circle.

As typical for existence arguments invoking #Baire category theorem, this #proof is nonconstructive. It shows that family of continuous functions whose #Fourier series converges at a given x is of first Baire category, in #Banach space of continuous functions on circle.

Another application of #Baire category theorem
If n \geq 3 , R^n/Q^n is simply connected
- https://chiasme.wordpress.com/2015/08/30/another-application-of-baire-category-theorem/

Another application of #Baire category theorem
If n \geq 3 , R^n/Q^n is simply connected
- https://chiasme.wordpress.com/2015/08/30/another-application-of-baire-category-theorem/