Second Category

A subset  of a topological space  is said to be of second category in  if  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 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 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 space and its topology play a fundamental role in determining category. For example, the set  of integers with the subset 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 subset topology inherited by  from . Likewise, the Cantor set is a Baire space (i.e., each of its open sets are of second category relative to it) even though it is of first category in the interval  with the usual topology.

REFERENCES

Rudin, W. Functional Analysis. New York: McGraw-Hill, 1991.