Category:Definitions/Countable Complement Topology

This category contains definitions related to Countable Complement Topology.
Related results can be found in Category:Countable Complement Topology.

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:

$H \in \tau \iff \relcomp S H$ is countable, or $H = \O$

where $\relcomp S H$ 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 $T = \struct {S, \tau}$ is a countable complement space.