A description/proof of that continuous map from compact topological space into Hausdorff topological space is proper
Topics
About: topological space
The table of contents of this article
Starting Context
- The reader knows a definition of continuous map.
- The reader knows a definition of compact topological space.
- The reader knows a definition of Hausdorff topological space.
- The reader knows a definition of proper map.
- The reader admits the proposition that any compact subset of any Hausdorff topological space is closed.
- The reader admits the proposition that any closed subset of any compact topological space is compact.
Target Context
- The reader will have a description and a proof of the proposition that any continuous map from any compact topological space into any Hausdorff topological space is proper.
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, \(T_1\), and any Hausdorff topological space, \(T_2\), any continuous map, \(f: T_1 \rightarrow T_2\), is proper.
2: Proof
Let \(S \subseteq T_2\) be any compact subset. \(S\) is closed, by the proposition that any compact subset of any Hausdorff topological space is closed. \(f^{-1} (S)\) is closed. \(f^{-1} (S)\) is compact, by the proposition that any closed subset of any compact topological space is compact.
