393: Continuous Map from Compact Topological Space into Hausdorff Topological Space Is Proper

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A description/proof of that continuous map from compact topological space into Hausdorff topological space is proper

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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: Description
• 2: Proof

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Starting Context

• The reader knows a definition of continuous map.
• The reader knows a definition of compact topological space.
• The reader knows a definition of Hausdorff topological space.
• The reader knows a definition of proper map.
• The reader admits the proposition that any compact subset of any Hausdorff topological space is closed.
• The reader admits the proposition that any closed subset of any compact topological space is compact.

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Target Context

• The reader will have a description and a proof of the proposition that any continuous map from any compact topological space into any Hausdorff topological space is proper.

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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: Description

For any compact topological space, $T_1$, and any Hausdorff topological space, $T_2$, any continuous map, $f:T_1\to T_2$, is proper.

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2: Proof

Let $S\subseteq T_2$ be any compact subset. $S$ is closed, by the proposition that any compact subset of any Hausdorff topological space is closed. $f^{-1}(S)$ is closed. $f^{-1}(S)$ is compact, by the proposition that any closed subset of any compact topological space is compact.

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References

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