definition of proper map
Topics
About: topological space
The table of contents of this article
- Starting Context
- Target Context
- Orientation
- Main Body
- 1: Structured Description
- 2: Natural Language Description
Starting Context
- The reader knows a definition of map.
- The reader knows a definition of compact subset of topological space.
Target Context
- The reader will have a definition of proper 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:
\( T_1\): \(\in \{\text{ the topological spaces }\}\)
\( T_2\): \(\in \{\text{ the topological spaces }\}\)
\(*f\): \(: T_1 \to T_2\)
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Conditions:
\(\forall S \in \{\text{ the compact subsets of } T_2\} (f^{-1} (S) \in \{\text{ the compact subsets of } T_1\})\)
//
2: Natural Language Description
For any topological spaces, \(T_1, T_2\), any map, \(f: T_1 \to T_2\), such that for each compact subset, \(S \subseteq T_2\), \(f^{-1} (S)\) is a compact subset of \(T_1\)
