Second Category

A second category subset of a topological space is one which cannot be written as the countable union of subsets which are nowhere dense in , i.e., if writing as a union

implies that at least one subset fails to be nowhere dense in . Said differently, any set which fails to be of first category is necessarily second category and unlike sets of first category, one thinks of a second category subset as a "non-small" subset of its host topological space. Sets of second category are sometimes referred to as nonmeager.

An important distinction should be made between the above-used notion of "category" and category theory. Indeed, the notions of first category and second category sets are independent of category theory.

The irrational numbers are of second category and the rational numbers are of first category in with the usual topology. In general, the host topological space and its topology play a fundamental role in determining category. For example, the set of integers with the relative topology inherited from is (vacuously) of second category relative to itself because every subset of is open in with respect to that topology; on the other hand, is of first category in with its standard topology and in with the relative topology inherited by from . Likewise, the Cantor set is a Baire space (i.e., each of its nonempty open sets is of second category relative to it) even though it is of first category in the interval with the usual topology.