Cocountable
In
mathematics, a
cocountable subset of a set
X is a subset
Y whose
complement in
X is a
countable set. In other words,
Y contains all but countably many elements of
X.
The set of all subsets of
X that are either countable or cocountable forms a
σ-algebra, i.e., it is closed under the operations of countable unions, countable intersections, and complementation. This σ-algebra is the
countable-cocountable algebra on
X. It is the smallest σ-algebra containing every
singleton set.
The
cocountable topology (sometimes called the
countable complement topology) on any set
X consists of the
empty set and all cocountable subsets of
X. In the cocountable topology, the only closed subsets are
countable sets, or the whole of
X. Then
X is automatically
Lindelöf in this topology, since every
open set only omits countably many points of
X.
*
cofinite* Lynn Arthur Steen and J. Arthur Seebach, Jr.,
Counterexamples in Topology. Springer-Verlag, New York, 1978. Reprinted by Dover Publications, New York, 1995. ISBN 0-486-68735-X (Dover edition).
(See example 20)
