Second Category
المؤلف:
Rudin, W
المصدر:
Functional Analysis. New York: McGraw-Hill, 1991.
الجزء والصفحة:
...
17-1-2022
2186
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 ![[0,1]](https://mathworld.wolfram.com/images/equations/SecondCategory/Inline19.svg)
with the usual topology.
REFERENCES
Rudin, W. Functional Analysis. New York: McGraw-Hill, 1991.
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