Math 574: Descriptive Set Theory

Name: Anush Tserunyan
Email: anush at illinois dot edu
Office: 257 Altgeld Hall
Office hours: Mon 4pm-5pm

Class info

Luzin-Novikov: any Borel relation with countable fibers is a countable disjoint union of Borel graphs. In particular, its projection is Borel.
Borel isomorphism theorem: any two uncountable Polish spaces are Borel isomorphic.
Turning Borel into clopen: for every Borel subset B of a Polish space, there is a finer Polish topology that has the same Borel sets and makes B clopen.
Baire alternative: if a set A has the BP, then either A is meager or it is comeager in a nonempty open set.
In Baire spaces (in particular in Polish spaces), a set is comeager if and only if it contains a dense G_delta set.
Every nonempty Polish space is a continuous injective image of a closed subset of the Baire space.
Uncountable Polish spaces contain a copy of the Cantor space.
Closed subsets of the Baire space N^N come from trees.
G_delta subsets of Polish spaces are Polish.

Baire necessities (by Artem Mavrin and Dakota Taylor)

Created one night during the UCLA Logic Summer School 2013

Diagonalization (by xkcd)

Original comic: Fetishes