A description/proof of that for locally finite cover of topological space, compact subset intersects only finite elements of cover
Topics
About: topological space
The table of contents of this article
Starting Context
- The reader knows a definition of locally finite cover of topological space.
- The reader knows a definition of compact subset of topological space.
Target Context
- The reader will have a description and a proof of the proposition that for any locally finite cover of any topological space, any compact subset intersects only finite elements of the cover.
Orientation
There is a list of definitions discussed so far in this site.
There is a list of propositions discussed so far in this site.
Main Body
1: Description
For any topological space, \(T\), and any locally finite cover, \(S_1\), of \(T\), any compact subset, \(S_2 \subseteq T\), intersects only finite elements of \(S_1\).
2: Proof
\(S_2\) can be covered by a neighborhood of each point on \(S_2\) that (the neighborhood) intersects only finite elements of \(S_1\) by the definition of locally finite cover of topological space. The cover has a finite subcover, \(S_3\), because \(S_2\) is compact. \(\cup S_3\) intersects only finite elements of \(S_1\). So, \(S_2 \subseteq \cup S_3\) intersects only finite elements of \(S_1\).
3: Note
\(S_1\) does not need to be any open cover, although it is typically an open cover.
Even if \(S_1\) is an open cover, \(S_2\)'s being covered by finite elements of \(S_1\) does not immediately mean that \(S_2\) intersects only finite elements of \(S_1\), because the existence of the finite elements that cover \(S_2\) does not immediately mean that there is no other element that intersects \(S_2\).
