299: For Locally Compact Hausdorff Topological Space, Around Point, There Is Open Neighborhood Whose Closure Is Compact

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A description/proof of that for locally compact Hausdorff topological space, around point, there is open neighborhood whose closure is compact

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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 locally compact topological space.
• The reader knows a definition of Hausdorff topological space.
• The reader knows a definition of neighborhood of point.
• The reader knows a definition of compact subset of topological space.
• The reader knows a definition of subspace topology.
• The reader admits the proposition that any compact subset of any Hausdorff topological space is closed.
• The reader admits the proposition that any subset on any topological subspace is closed if and only if there is a closed set on the base space whose intersection with the subspace is the subset.
• The reader admits the proposition that the compactness of any topological subset as a subset equals the compactness as a subspace.
• The reader admits the proposition that any closed subset of any compact topological space is compact.
• The reader admits the proposition that for any topological space, any compact subset of any subspace is compact on the base space.

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Target Context

• The reader will have a description and a proof of the proposition that for any locally compact Hausdorff topological space, around any point, there is an open neighborhood whose closure is compact.

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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 locally compact Hausdorff topological space, $T$, and any point, $p\in T$, there is an open neighborhood, $U_p\subseteq T$, of $p$ such that the closure, $\overline{U_p}$, is compact on $T$.

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2: Proof

There is a compact neighborhood, $N_p$, of $p$. $N_p$ is closed, by the proposition that any compact subset of any Hausdorff topological space is closed. There is an open neighborhood, $U_p\subseteq N_p$. $\overline{U_p}\subseteq N_p$ because $\overline{U_p}$ is the smallest closed set that contains $U_p$ while $N_p$ is a closed set that contains $U_p$. $\overline{U_p}$ is closed on $N_p$ by the proposition that any subset on any topological subspace is closed if and only if there is a closed set on the base space whose intersection with the subspace is the subset. $N_p$ is a compact topological space, by the proposition that the compactness of any topological subset as a subset equals the compactness as a subspace. $\overline{U_p}$ is compact on $N_p$, by the proposition that any closed subset of any compact topological space is compact. $\overline{U_p}$ is compact on $T$, by the proposition that for any topological space, any compact subset of any subspace is compact on the base space.

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References

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