140: Open Set Minus Closed Set Is Open

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A description/proof of that open set minus closed set is open

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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 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 any open set minus any closed set is open.

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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 topological space, $T$, any open set, $U\subseteq T$, and any closed set, $C\subseteq T$, the open set minus the closed set, $U\setminus C$, is open on $T$.

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2: Proof

For any point, $p\in U\setminus C$, $p\in U$ and $p\notin C$, which means that $p\in T\setminus C$ while $T\setminus C$ is open. There are open sets, $p\in U_1\subseteq U$ and $p\in U_2\subseteq T\setminus C$. $p\in U_1\cap U_2$ while $U_1\cap U_2$ is open and $U_1\cap U_2\subseteq U$ and $U_1\cap U_2\subseteq T\setminus C$, which means that $U_1\cap U_2\cap C=\mathrm{\varnothing }$, so, $U_1\cap U_2\subseteq U\setminus C$. As at any point, $p\in U\setminus C$, there is an open set, $U_1\cap U_2$, such that $p\in U_1\cap U_2\subseteq U\setminus C$, by the local criterion for openness, $U\setminus C$ is open.

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References

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