298: For Topological Space, Intersection of Compact Subset and Subspace Is Not Necessarily Compact on Subspace

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A description/proof of that for topological space, intersection of compact subset and subspace is not necessarily compact on subspace

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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 compact subset of topological space.
• The reader knows a definition of subspace topology.
• The reader knows a definition of Euclidean topology.

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Target Context

• The reader will have a description and a proof of the proposition that for a topological space, the intersection of a compact subset and a subspace is not necessarily compact on the subspace.

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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 a topological space, $T$, a compact subset, $S\subseteq T$, and a subspace, $T_1\subseteq T$, the intersection, $S\cap T_1$, is not necessarily compact on $T_1$.

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2: Proof

A counterexample suffices. Take $T={\mathbb{R}}^2$ with the Euclidean topology, $S=\overline{B_{p-ϵ}}$, and $T_1=B_{p-ϵ}$ where $B_{p-ϵ}$ is the $ϵ$-radius open ball centered at $p$ and the over line denotes the closure. $S\cap T_1=B_{p-ϵ}$ is not compact on $B_{p-ϵ}$.

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3: Note

If $S\subseteq T_1$, $S$ is necessarily compact on $T_1$, by the proposition that for any topological space, any subspace subset that is compact on base space is compact on the subspace. Let us note the difference.

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References

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