358: Subset of 1st Category Subset Is of 1st Category

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A description/proof of that subset of 1st category subset is of 1st category

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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 1st category subset.

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Target Context

• The reader will have a description and a proof of the proposition that for any topological space, any subset of any 1st category subset is of the 1st category.

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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 1st category subset, $S\subseteq T$, any subset, $S^{\prime }\subseteq S$, is of the 1st category.

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2: Proof

$S={\cup }_{i\in I}{S}_i$ where $I$ is any countable indices set and $S_i$ is any nowhere dense subset. For $S^{″}=S\setminus S^{\prime }$, $S^{\prime }={\cup }_{i\in I}(S_i\setminus S^{″})$, while $S_i\setminus S^{″}$ is nowhere dense.

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References

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