362: Compact Topological Space Has Accumulation Point of Subset with Infinite Points

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A description/proof of that compact topological space has accumulation point of subset with infinite points

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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 compact topological space.
• The reader knows a definition of accumulation point.

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Target Context

• The reader will have a description and a proof of the proposition that any compact topological space has an accumulation point of any subset with infinite points.

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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 compact topological space, $T$, and any subset, $S\subseteq T$, with infinite points, $T$ has an accumulation point, $p^{\prime }\in T$ of $S$.

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2: Proof

Suppose that $T$ had no accumulation point of $S$. Then, around each point, $p\in T$, there would be an open set, $p\in U_p\subseteq T$, such that $U_p\cap S=\mathrm{\varnothing }\text{ or }p$. Such open sets would constitute an open cover of $T$, which would have a finite subcover. As each open set would have only 1 point of $S$ at most, $S$ would have only finite points, which is a contradiction.

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References

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