495: For Intersection of 2 Subsets of Topological Space, Its Regarded as Subspace of a Subset as Subspace, Its Regarded as Subspace of Other Subset as Subspace, and Its Regarded as Subspace of Basespace Are Same

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A description/proof of that for intersection of 2 subsets of topological space, its regarded as subspace of a subset as subspace, its regarded as subspace of other subset as subspace, and its regarded as subspace of basespace are same

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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 subspace topology.
• The reader admits the proposition that in any nest of topological subspaces, the openness of any subset on any subspace does not depend on the superspace of which the subspace is regarded to be a subspace.

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Target Context

• The reader will have a description and a proof of the proposition that for the intersection of any 2 subsets of any topological space, its regarded as the subspace of a subset as the subspace, its regarded as the subspace of the other subset as the subspace, and its regarded as the subspace of the basespace are same.

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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 subsets, $S_1,S_2\subseteq T$, $S_1\cap S_2$ as the subspace of $S_1$ as the subspace of $T$, $S_1\cap S_2$ as the subspace of $S_2$ as the subspace of $T$, and $S_1\cap S_2$ as the subspace of $T$ are the same.

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2: Proof

This proof uses the proposition that in any nest of topological subspaces, the openness of any subset on any subspace does not depend on the superspace of which the subspace is regarded to be a subspace profusely, so, we will not mention it any more.

Let $U\subseteq S_1\cap S_2$ be any open subset with $S_1\cap S_2$ as the subspace of $S_1$ as the subspace of $T$.

$U$ is an open subset of $S_1\cap S_2$ with $S_1\cap S_2$ as the subspace of $T$. $U$ is an open subset with $S_1\cap S_2$ as the subspace of $S_2$ as the subspace of $T$.

Let $U\subseteq S_1\cap S_2$ be any open subset with $S_1\cap S_2$ as the subspace of $S_2$ as the subspace of $T$.

$U$ is an open subset of $S_1\cap S_2$ with $S_1\cap S_2$ as the subspace of $T$. $U$ is an open subset with $S_1\cap S_2$ as the subspace of $S_1$ as the subspace of $T$.

Let $U\subseteq S_1\cap S_2$ be any open subset with $S_1\cap S_2$ as the subspace of $T$.

$U$ is an open subset with $S_1\cap S_2$ as the subspace of $S_1$ as the subspace of $T$. $U$ is an open subset with $S_1\cap S_2$ as the subspace of $S_2$ as the subspace of $T$.

So, the 3 topologies are the same.

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References

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