778: Proper Map

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definition of proper map

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Topics

About: topological space

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The table of contents of this article

• Starting Context
• Target Context
• Orientation
• Main Body
• 1: Structured Description
• 2: Natural Language Description

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Starting Context

• The reader knows a definition of map.
• The reader knows a definition of compact subset of topological space.

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Target Context

• The reader will have a definition of proper map.

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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.

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Main Body

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1: Structured Description

Here is the rules of Structured Description.

Entities:
$T_1$: $\in \{\text{ the topological spaces }\}$
$T_2$: $\in \{\text{ the topological spaces }\}$
$\ast f$: $:T_1\to T_2$
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Conditions:
$\mathrm{\forall }S\in \{\text{ the compact subsets of }{T}_2\}(f^{-1}(S)\in \{\text{ the compact subsets of }{T}_1\})$
//

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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 $T_1$

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References

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