definition of topological spaces map continuous at point
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 map.
Target Context
- The reader will have a definition of topological spaces map continuous at point.
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:
\( T_1\): \(\in \{\text{ the topological spaces }\}\)
\( T_2\): \(\in \{\text{ the topological spaces }\}\)
\( t\): \(\in T_1\)
\(*f\): \(: T_1 \to T_2\)
//
Conditions:
\(\forall N_{f (t)} \subseteq T_2 \in \{\text{ the neighborhoods of } f (t)\} (\exists N_t \in \{\text{ the neighborhoods of } t\} (f (N_t) \subseteq N_{f (t)}))\)
2: Note
"neighborhood" in the definition can be replace with "open neighborhood" without changing the concept, because supposing this definition, for each open neighborhood of \(f (t)\), \(U_{f (t)}\), as \(U_{f (t)}\) is a neighborhood of \(f (t)\), there is a neighborhood of \(t\), \(N_t\), such that \(f (N_t) \subseteq U_{f (t)}\), but there is an open neighborhood of \(t\), \(U_t \subseteq N_t\), and \(f (U_t) \subseteq f (N_t) \subseteq U_{f (t)}\); on the other hand, supposing the "open neighborhood" version, for each neighborhood of \(f (t)\), \(N_{f (t)}\), there is an open neighborhood of \(f (t)\), \(U_{f (t)} \subseteq N_{f (t)}\), and there is an open neighborhood of \(t\), \(U_t\), such that \(f (U_t) \subseteq U_{f (t)} \subseteq N_{f (t)}\), but \(U_t\) is a neighborhood of \(t\).
