698: Exhaustion of Topological Space by Compact Subsets

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definition of exhaustion of topological space by compact subsets

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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.
• The reader knows a definition of interior of subset of topological space.

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Target Context

• The reader will have a definition of exhaustion of topological space by compact subsets.

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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:
$T$: $\in \{\text{ the topological spaces }\}$
$\ast s$: $:\mathbb{N}\setminus \{0\}\to \{\text{ the compact subsets of }T\}$
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Conditions:
$T={\cup }_{j\in \mathbb{N}\setminus \{0\}}s(j)$
$\wedge$
$s(j)\subseteq {s(j+1)}^{\circ }$, where ${s(j+1)}^{\circ }$ denotes the topological interior of $s(j+1)$
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It is possible that $s(j)=s(j+1)$ for each $j$ such that $n\le j$ for an $n\in \mathbb{N}\setminus \{0\}$, then, practically, $s$ is finite. That means that $s(j)=T$, so, $T$ is compact. That is possible because when $s(j+1)=T$, ${s(j+1)}^{\circ }=s(j+1)$ because $T$ is open.

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2: Natural Language Description

For any topological space, $T$, any sequence of subsets, $s:\mathbb{N}\setminus \{0\}\to \{\text{ the compact subsets of }T\}$, such that $T={\cup }_{j\in \mathbb{N}\setminus \{0\}}s(j)$ and $s(j)\subseteq {s(j+1)}^{\circ }$, where ${s(j+1)}^{\circ }$ denotes the topological interior of $s(j+1)$

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References

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