699: Exhaustion Function on Topological Space

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definition of exhaustion function on topological space

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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 compact subset of topological space.

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Target Context

• The reader will have a definition of exhaustion function on topological space.

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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:
$T$: $\in \{\text{ the topological spaces }\}$
$\mathbb{R}$: $=\text{ the Euclidean topological space }$
$\ast f$: $:T\to \mathbb{R}$, $\in \{\text{ the continuous maps }\}$
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Conditions:
$\mathrm{\forall }r\in \mathbb{R}(f^{-1}((-\mathrm{\infty },r])\in \{\text{ the compact subsets of }T\})$
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2: Natural Language Description

For any topological space, $T$, and the Euclidean topological space, $\mathbb{R}$, any continuous map, $f:T\to \mathbb{R}$, such that for each $r\in \mathbb{R}$, $f^{-1}((-\mathrm{\infty },r])$ is a compact subset of $T$

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References

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