A subset 
of a topological space 
is said to be of first category in 
if 
can be written as the countable union of subsets which are nowhere
dense in 
,
i.e., if 
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 space and indeed, sets of first category are sometimes referred to as thin sets or meager set. 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 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 ![[0,1]](/images/equations/FirstCategory/Inline19.svg)
with the usual topology.
Baire Category Theorem, Meager Set, Nowhere Dense, Nonmeager Set, Residual Set, Second Category
