A description/proof of that compact topological space has accumulation point of subset with infinite points
Topics
About: topological space
The table of contents of this article
Starting Context
- The reader knows a definition of compact topological space.
- The reader knows a definition of accumulation point.
Target Context
- The reader will have a description and a proof of the proposition that any compact topological space has an accumulation point of any subset with infinite points.
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 compact topological space, \(T\), and any subset, \(S \subseteq T\), with infinite points, \(T\) has an accumulation point, \(p' \in T\) of \(S\).
2: Proof
Suppose that \(T\) had no accumulation point of \(S\). Then, around each point, \(p \in T\), there would be an open set, \(p \in U_p \subseteq T\), such that \(U_p \cap S = \emptyset \text{ or } {p}\). Such open sets would constitute an open cover of \(T\), which would have a finite subcover. As each open set would have only 1 point of \(S\) at most, \(S\) would have only finite points, which is a contradiction.
