definition of continuous, topological spaces 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 topological spaces map continuous at point.
- The reader admits the local criterion for openness.
Target Context
- The reader will have a definition of continuous, topological spaces 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: Structured Description
Here is the rules of Structured Description.
Entities:
\( T_1\): \(\in \{\text{ the topological spaces }\}\)
\( T_2\): \(\in \{\text{ the topological spaces }\}\)
\(*f\): \(: T_1 \to T_2\)
//
Conditions:
\(\forall t \in T_1 (f \in \{\text{ the maps that are continuous at } t\})\)
//
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 subset of \(T_2\) is an open subset. That is because supposing this definition, for any open subset, \(U \subseteq T_2\) 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; supposing the alternative definition, for any neighborhood, \(U\), of \(f (p)\), the preimage, \(S\), is an open subset, which is a neighborhood of \(p\) and \(f (S) \subseteq U\).
