A description/proof of that for injective closed map between topological spaces, inverse of codomain-restricted-to-range map is continuous
Topics
About: topological space
The table of contents of this article
Starting Context
- The reader knows a definition of topological space.
- The reader knows a definition of continuous map.
- The reader admits the proposition that for any bijection, the preimage of any subset under the inverse of the map is the image of the subset under the map.
- The reader admits the proposition that if the preimage of any closed set under a topological spaces map is closed, the map is continuous.
- The reader admits the proposition that any subset on any topological subspace is closed if and only if there is a closed set on the base space whose intersection with the subspace is the subset.
Target Context
- The reader will have a description and a proof of the proposition that for any injective closed map between any topological spaces, the inverse of the codomain-restricted-to-range map is continuous.
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 spaces, \(T_1\) and \(T_2\), any injective closed map, \(f: T_1 \rightarrow T_2\), and \(f\)'s restriction on the codomain, \(f': T_1 \rightarrow f (T_1)\), \(f'^{-1}: f (T_1) \rightarrow T_1\) is continuous.
2: Proof
For any closed set, \(C \subseteq T_1\), is \({f'^{-1}}^{-1} (C)\) closed on \(f (T_1)\)? As \(f'\) is a bijection, \({f'^{-1}}^{-1} (C) = f' (C)\), by the proposition that for any bijection, the preimage of any subset under the inverse of the map is the image of the subset under the map. As \(f\) is closed, \(f (C) = f' (C)\) is closed on \(T_2\), and \(f' (C) = f' (C) \cap f (T_1)\) is closed on \(f (T_1)\), by the proposition that any subset on any topological subspace is closed if and only if there is a closed set on the base space whose intersection with the subspace is the subset. By the proposition that if the preimage of any closed set under a topological spaces map is closed, the map is continuous, \(f'^{-1}\) is continuous.
