Descriptive Set Theory is the study of structures and the relations of sets in a Polish space. A Polish space is a metric space which is separable and complete. It is easy to see that the cardinality of a Polish space is at most continuum because of the separability condition. The most typical examples of Polish spaces are Rn, ω (the set of natural numbers with discrete topology), the Cantor space ω2, and the Baire space ωω.
For more examples and discussions on Polish spaces, the reader can consult the following
two excellent texts on this topic.
Y. N. Moschovakis, Descriptive Set theory North-Holland, Amsterdam.
A. S. Kechris, Classical Descriptive Set Theory Springer-Verlag