Definition:Compact Space/Topology

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Definition

Definition 1

A topological space $T = \left({S, \tau}\right)$ is compact if and only if every open cover for $S$ has a finite subcover.


Definition 2

A topological space $T = \left({S, \tau}\right)$ is compact if and only if it satisfies the Finite Intersection Axiom.


Definition 3

A topological space $T = \left({S, \tau}\right)$ is compact if and only if $\tau$ has a sub-basis $\mathcal B$ such that:

from every cover of $S$ by elements of $\mathcal B$, a finite subcover of $S$ can be selected.


Definition 4

A topological space $T = \left({S, \tau}\right)$ is compact if and only if every filter on $S$ has a limit point in $S$.


Definition 5

A topological space $T = \left({S, \tau}\right)$ is compact if and only if every ultrafilter on $S$ converges.


Also defined as

A compact topological space is often additionally required to be Hausdorff. What is called a compact space here is then called a quasicompact space.


Also see

  • Results about compact spaces can be found here.


Sources