A description/proof of that map between topological spaces is continuous if domain restriction of map to each closed set of finite closed cover is continuous
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 closed set.
- The reader admits the proposition that any closed set on any closed topological subspace is closed on the base space.
- 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 any map between topological spaces is continuous if the domain restriction of the map to each closed set of a finite closed cover 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 map, \(f: T_1 \rightarrow T_2\), if there is a finite closed cover of \(T_1\), \(\{C_i \subseteq T_1\}, \cup_{i} C_i = T_1\), such that each \(f|_{C_i}: C_i \rightarrow T_2\) is continuous, \(f\) is continuous.
2: Proof
For any closed set, \(C \subseteq T_2\), \({f|_{C_i}}^{-1} (C)\) is closed on \(C_i\), and on \(T_1\), by the proposition that any closed set on any closed topological subspace is closed on the base space. \(f^{-1} (C) = \cup_i {f|_{C_i}}^{-1} (C)\), because for any \(p \in f^{-1} (C)\), \(f (p) \in C\), but \(p \in \cup_i C_i\), so, \(f|_{C_i} (p) \in C\) for an \(i\), \(p \in {f|_{C_i}}^{-1} (C)\); for any \(p \in \cup_i {f|_{C_i}}^{-1} (C)\), \(p \in {f|_{C_i}}^{-1} (C)\) for an \(i\), so, \(f|_{C_i} (p) \in C\), so, \(f (p) \in C\), so, \(p \in f^{-1} (C)\). So, \(f^{-1} (C)\) is close as the finite union of closed sets. By the proposition that if the preimage of any closed set under a topological spaces map is closed, the map is continuous, \(f\) is continuous.
