Definition:Countable Complement Topology

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Definition

Let $S$ be an infinite set whose cardinality is usually taken to be uncountable.

Let $\tau$ be the set of subsets of $S$ defined as:

$H \in \tau \iff \complement_S \left({H}\right)$ is countable, or $H = \varnothing$

where $\complement_S \left({H}\right)$ denotes the complement of $H$ relative to $S$.

In this definition, countable is used in its meaning that includes finite.


Then $\tau$ is the countable complement topology on $S$, and the topological space $T = \left({S, \tau}\right)$ is a countable complement space.


On a Countable Space

It is possible to define the countable complement topology on a countable set set $S$, but as every subset of a countable set has a countable complement, it is clear that this is trivially equal to the discrete space.

This is why the countable complement topology is usually understood to apply to uncountable sets only.


Also known as

A countable complement topology is also known as a co-countable topology, and similarly we have the term co-countable space.

Sometimes the hyphen is omitted, to give the word cocountable.


Also see

  • Results about countable complement topologies can be found here.


Sources