A description/proof of that inverse of closed bijection is continuous
Topics
About: topological space
The table of contents of this article
Starting Context
- The reader knows a definition of bijection.
- The reader knows a definition of closed map.
- The reader knows a definition of continuous 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.
Target Context
- The reader will have a description and a proof of the proposition that the inverse of any closed bijection 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, \(T_1, T_2\), and any closed bijection, \(f: T_1 \rightarrow T_2\), the inverse, \(f^{-1}: T_2 \rightarrow T_1\), is continuous.
2: Proof
For any closed set, \(C \subseteq T_1\), \((f^{-1})^{-1} (C) = f (C)\), which is closed by the definition of closed map. 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.
