A definition of compact subset of topological space
Topics
About: topological space
The table of contents of this article
Starting Context
- The reader knows a definition of topological space.
Target Context
- The reader will have a definition of compact subset of topological space.
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: Definition
For any topological space, \(T\), any subset, \(S \subseteq T\), such that each open cover of \(S\) has a finite subcover
2: Note
\(S\) can be regarded as the topological subspace of \(T\), and we can talk about compactness of \(S\) as the topological space, and compactness of \(S\) as the subset and compactness of \(S\) as the topological space are not the same by the definitions (an open cover of \(S\) as the subset is not necessarily an open cover of \(S\) as the subspace; an open cover of \(S\) as the subspace is not necessarily an open cover of \(S\) as the subset), but in fact, each of the 2 concepts implies the other, by the proposition that the compactness of any topological subset as a subset equals the compactness as a subspace.
