Descriptive Set Theory

What's descriptive set theory? It is the study of sets easily described.

Ok; 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.

On 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.