Define cofinite topology
Definition:Finite Complement TopologyFrom ProofWiki Jump to navigation Jump to searchDefinitionLet $S$ be a set whose cardinality is usually specified as being infinite. Show Let $\tau$ be the set of subsets of $S$ defined as: $H \in \tau \iff \relcomp S H \text { is finite, or } H = \O$where $\relcomp S H$ denotes the complement of $H$ relative to $S$.
On a Finite SpaceIt is possible to define the finite complement topology on a finite set $S$, but as every subset of a finite set has a finite complement, it is clear that this is trivially equal to the discrete space. This is why the finite complement topology is usually understood to apply to infinite sets only. On a Countable SpaceLet $S$ be countably infinite. Then $\tau$ is a finite complement topology on a countable space, and $\struct {S, \tau}$ is a countable finite complement space. On an Uncountable SpaceLet $S$ be uncountable. Then $\tau$ is a finite complement topology on an uncountable space, and $\struct {S, \tau}$ is a uncountable finite complement space. Also known asThe term cofinite is sometimes seen in place of finite complement. Some sources are more explicit about the nature of this topology, and call it the topology of finite complements.
This is justified by Finite Complement Topology is Minimal $T_1$ Topology.
However, this is not recommended as there is another so named Zariski topology which is unrelated to this one. Also see
Sources
Retrieved from "https://proofwiki.org/w/index.php?title=Definition:Finite_Complement_Topology&oldid=507075" Categories:
|