306: 1 Point Subset of Hausdorff Topological Space Is Closed

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A description/proof of that 1 point subset of Hausdorff topological space is closed

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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 Hausdorff topological space.
• The reader knows a definition of closed set.
• The reader admits the local criterion for openness.

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Target Context

• The reader will have a description and a proof of the proposition that for any Hausdorff topological 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 Hausdorff topological space, $T$, any 1 point set, $\{p\}$, is closed.

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2: Proof

Is $T\setminus \{p\}$ open? For any point, $p^{\prime }\in T\setminus \{p\}$, there are some open neighborhoods, $U_p\subseteq T$ and $U_{p^{\prime }}\subseteq T$, of $p$ and $p^{\prime }$ such that $U_p\cap U_{p^{\prime }}=\mathrm{\varnothing }$. $U_{p^{\prime }}\subseteq T\setminus \{p\}$, because for any $p^{″}\in U_{p^{\prime }}$, $p^{″}\notin U_p$, $p^{″}\notin \{p\}$, so, $p^{″}\in T\setminus \{p\}$. So, by the local criterion for openness, $T\setminus \{p\}$ is open.

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References

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