317: For Locally Finite Cover of Topological Space, Compact Subset Intersects Only Finite Elements of Cover

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A description/proof of that for locally finite cover of topological space, compact subset intersects only finite elements of cover

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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
• 3: Note

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Starting Context

• The reader knows a definition of locally finite cover of topological space.
• The reader knows a definition of compact subset of topological space.

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Target Context

• The reader will have a description and a proof of the proposition that for any locally finite cover of any topological space, any compact subset intersects only finite elements of the cover.

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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 topological space, $T$, and any locally finite cover, $S_1$, of $T$, any compact subset, $S_2\subseteq T$, intersects only finite elements of $S_1$.

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2: Proof

$S_2$ can be covered by a neighborhood of each point on $S_2$ that (the neighborhood) intersects only finite elements of $S_1$ by the definition of locally finite cover of topological space. The cover has a finite subcover, $S_3$, because $S_2$ is compact. $\cup S_3$ intersects only finite elements of $S_1$. So, $S_2\subseteq \cup S_3$ intersects only finite elements of $S_1$.

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3: Note

$S_1$ does not need to be any open cover, although it is typically an open cover.

Even if $S_1$ is an open cover, $S_2$'s being covered by finite elements of $S_1$ does not immediately mean that $S_2$ intersects only finite elements of $S_1$, because the existence of the finite elements that cover $S_2$ does not immediately mean that there is no other element that intersects $S_2$.

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References

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