375: For Metric Space, 1 Point Subset Is Closed

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A description/proof of that for metric space, 1 point subset is closed

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Topics

About: metric 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 metric space.
• The reader knows a definition of closed subset.

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Target Context

• The reader will have a description and a proof of the proposition that for any metric space, any 1 point subset is closed.

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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 metric space, $M$, any 1 point subset, $\{p\}\subseteq M$, is closed.

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2: Proof

Think of $M\setminus \{p\}$ and any point, $p^{\prime }\in M\setminus \{p\}$. As $p\ne p^{\prime }$, by the definition of distance between 2 points, $d(p,p^{\prime })>0$. So, for any $0<ϵ<d(p,p^{\prime })$, the open ball around $p^{\prime }$, $B_{p^{\prime }-ϵ}$, does not contain $p$, so, $B_{p^{\prime }-ϵ}\subseteq M\setminus \{p\}$. So, $M\setminus \{p\}$ is open, so, $\{p\}$ is closed.

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References

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