232: Subspace of 2nd Countable Topological Space Is 2nd Countable

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A description/proof of that subspace of 2nd countable topological space is 2nd countable

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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 2nd countable topological space.
• The reader admits the proposition that for any topological space, the intersection of any basis and any subspace is a basis for the subspace.

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Target Context

• The reader will have a description and a proof of the proposition that any subspace of any 2nd countable topological space is 2nd countable.

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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 2nd countable topological space, $T$, any subspace, $T_1\subseteq T$, is 2nd countable.

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2: Proof

$T$ has a countable basis, $B$. By the proposition that for any topological space, the intersection of any basis and any subspace is a basis for the subspace, $T_1$ has the intersection of $B$ and $T_1$ as a basis, $B_1$. As each element of $B_1$ corresponds to an element of $B$, $B_1$ is countable.

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References

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