301: For Topological Space and Point on Subspace, Intersection of Neighborhood of Point on Base Space and Subspace Is Neighborhood on Subspace

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A description/proof of that for topological space and point on subspace, intersection of neighborhood of point on base space and subspace is neighborhood on subspace

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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 neighborhood of point.
• The reader knows a definition of subspace topology.

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Target Context

• The reader will have a description and a proof of the proposition that for any topological space and any point on any subspace, the intersection of any neighborhood of the point on the base space and the subspace is a neighborhood on the subspace.

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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 subspace, $T_1\subseteq T$, any point, $p\in T_1$, and any neighborhood, $N_p\subseteq T$, of $p$ on $T$, $N_p\cap T_1$ is a neighborhood of $p$ on $T_1$.

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2: Proof

$p\in N_p\cap T_1$. There is an open neighborhood, $U_p\subseteq N_p$, of $p$ on $T$. $U_p\cap T_1\subseteq N_p\cap T_1$ is an open neighborhood of $p$ on $T_1$, because $p\in U_p\cap T_1$ and $U_p\cap T_1$ is open on $T_1$.

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References

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