Descriptive Set Theory
What's descriptive set theory? It is the study of sets easily described.
to be more specific. Imagine all sets. There are a bunch of them.
Sometimes; we encounter sets with weird properties. We study sets that
don't have weird properties.
For example; thanks to the axiom of
choice; we could find a "figure" in the plane that doesn't have an
area; and I don't mean in the sense that its area is 0; but rather;
that we cannot define a quantity for its area.
The problem with
this sets; is that it's basically imposible to exhibit. We know that it
exists; but giving out it's members it's not an easy task.
the other side; must "natural" examples of sets we encounter usually
are easily described; at least in a certain way. For example; an
interval; the set of numbers where a particular function is
continuous; all compact perfect subsets of the reals; the set of
all invariant G-ergodic measures on a Polish space X for G a countable
set of homeomorphisms of X.
We study this type of sets, looking
for nice properties they share; and that's descriptive set theory. It's
a nice combination of general topology with logic and set theory; with
applications in areas as diverse as combinatories, probability
theory, topological groups; etc.