296: For Topological Space, Subset of Compact Subset Is Not Necessarily Compact

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A description/proof of that for topological space, subset of compact subset is not necessarily 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
• 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 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, a subset of a compact subset is not necessarily 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 a topological space, $T$, and a compact subset, $S\subseteq T$, a subset, $S_1\subseteq S$, is not necessarily compact on $T$.

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2: Proof

A counterexample suffices. Take $T={\mathbb{R}}^2$ with the Euclidean topology, $S=\overline{B_{p-ϵ}}$, and $S_1=B_{p-ϵ}$ where $B_{p-ϵ}$ is the $ϵ$-radius open ball centered at $p$ and the over line denotes the closure.

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3: Note

The word, "compact", may sound like about being small, but a smaller subset of a compact subset is not necessarily compact.

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References

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