359: Superset of Residual Subset Is Residual

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A description/proof of that superset of residual subset is residual

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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 residual subset.
• The reader admits the proposition that for any topological space, any subset of any 1st category subset is of the 1st category.

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Target Context

• The reader will have a description and a proof of the proposition that for any topological space, any superset of any residual subset is residual.

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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, $T$, and any residual subset, $S\subseteq T$, any superset, $S^{\prime }$, such that $S\subseteq S^{\prime }$, is residual.

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2: Proof

$S=T\setminus S_0$ where $S_0$ is of the 1st category. $S^{″}:=S^{\prime }\setminus S$ where $S^{″}\subseteq T\setminus S=S_0$. $S^{\prime }=S\cup S^{″}=(T\setminus S_0)\cup S^{″}=T\setminus (S_0\setminus S^{″})$. $S_0\setminus S^{″}$ is of the 1st category, by the proposition that for any topological space, any subset of any 1st category subset is of the 1st category.

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References

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