363: Closed Discrete Subspace of Compact Topological Space Has Only Finite Points

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A description/proof of that closed discrete subspace of compact topological space has only finite 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 discrete topological space.
• The reader admits the proposition that any compact topological space has an accumulation point of any subset with infinite points.
• The reader admits the proposition that any topological subset is closed if and only it equals its closure.
• The reader admits the proposition that the closure of any topological subset equals the union of the subset and the set of the accumulation points of the subset.

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Target Context

• The reader will have a description and a proof of the proposition that any closed discrete subspace of any compact topological space has only finite 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 closed discrete subspace, $S\subseteq T$, $S$ has only finite points.

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2: Proof

Suppose that $S$ had some infinite points. By the proposition that any compact topological space has an accumulation point of any subset with infinite points, $T$ would have an accumulation point, $p\in T$, of $S$. But by the proposition that any topological subset is closed if and only it equals its closure, $S=\bar{S}$, but by the proposition that the closure of any topological subset equals the union of the subset and the set of the accumulation points of the subset, the accumulation point would be contained in $S$, which is impossible, because as $S$ is discrete, each point is open, so also the accumulation point would be an open set by itself, which is against the definition of accumulation point, a contradiction.

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References

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