A first category subset 
of a topological space 
is one that can be written as the countable union of subsets
which are nowhere dense in 
, i.e., 
is expressible as a union
 |
where each subset 
is nowhere dense
in 
.
Informally, one thinks of a first category subset as a "small" subset of the host topological space and indeed, sets of first category are sometimes referred to as thin sets or meager sets. Sets which are not of first category are of second category.
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 rational numbers are of first category and the irrational numbers are of second
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 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 nonempty open
sets is of second category relative to it)
even though it is of first category in the interval ![[0,1]](/images/equations/FirstCategory/Inline17.svg)
with the usual topology.
