A definition of continuous map
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 at point.
- The reader admits the local criterion for openness.
Target Context
- The reader will have a definition of continuous map.
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: Definition
For any topological spaces, \(T_1\) and \(T_2\), any map, \(f: T_1 \to T_2\), that is continuous at each point
2: Note
The definition is in fact equivalent with this definition: for any topological spaces, \(T_1\) and \(T_2\), any map, \(f: T_1 \to T_2\), such that the preimage of any open set on \(T_2\) is an open set. That is because for \(\rightarrow\), for any open set, \(U \subseteq T_2\) and the preimage, \(S\), any point, \(p \in S\), satisfies \(f (p) \in U\), but \(U\) is a neighborhood of \(f (p)\), so, there is a neighborhood, \(U_p\), of \(p\) such that \(f (U_p) \subseteq U\), which means that \(U_p \subseteq S\), so, by the local criterion for openness, \(S\) is open; for \(\leftarrow\), for any neighborhood, \(U\), of \(f (p)\), the preimage, \(S\), is an open set, which is a neighborhood of \(p\) and \(f (S) \subseteq U\).
