Definition:Compact Space/Topology
Definition
Definition 1
A topological space $T = \struct {S, \tau}$ is compact if and only if every open cover for $S$ has a finite subcover.
Definition 2
A topological space $T = \struct {S, \tau}$ is compact if and only if it satisfies the Finite Intersection Axiom.
Definition 3
A topological space $T = \struct {S, \tau}$ is compact if and only if $\tau$ has a sub-basis $\BB$ such that:
- from every cover of $S$ by elements of $\BB$, a finite subcover of $S$ can be selected.
Definition 4
A topological space $T = \struct {S, \tau}$ is compact if and only if every filter on $S$ has a limit point in $S$.
Definition 5
A topological space $T = \struct {S, \tau}$ is compact if and only if every ultrafilter on $S$ converges.
Also defined as
Some sources, in their definition of a compact space, impose the additional criterion that such a space should also be Hausdorff.
What is called a compact space here is then referred to as a quasicompact (or quasi-compact) space.
Examples
Examples of Non-Compact Spaces
Also see
- Results about compact spaces can be found here.