Open Cover

Relevant Classes: Math 297

Suppose $(X,\tau )$ is a Topological Space and $A\subset X$. An open cover for $A$ is a collection of open sets $\{{\mathcal{O}}_{\lambda }\mid \lambda \in A\}\subset \tau$ whose union contains the set $A$; that is $A\subseteq {\bigcup }_{\lambda \in A}{\mathcal{O}}_{\lambda }.$ (Here, $A$ is an indexing set.)

Given an open cover for $A$, a finite subcover is a finite sub-collection of open sets from the original open cover whose union still manages to completely contain $A$. A subset $A$ is said to be compact provided that every open cover for $A$ has a finite subcover.