1102: For Separable Topological Space Induced by Metric, Topological Subspace Is Separable

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description/proof of that for separable topological space induced by metric, topological subspace is separable

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Topics

About: metric space

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The table of contents of this article

• Starting Context
• Target Context
• Orientation
• Main Body
• 1: Structured Description
• 2: Note
• 3: Proof

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Starting Context

• The reader knows a definition of topology induced by metric.
• The reader knows a definition of separable topological space.
• The reader knows a definition of topological subspace.
• The reader admits the proposition that for any topological space induced by any metric, the space has a countable basis if and only if the space is separable.
• The reader admits the proposition that for any topological space, the intersection of any basis and any subspace is a basis for the subspace.
• The reader admits the proposition that for any topological space induced by any metric and any subset, the subset as the topological subspace equals the subset as the topological space induced by the metric subspace.

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Target Context

• The reader will have a description and a proof of the proposition that for any separable topological space induced by any metric, any topological subspace is separable.

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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: Structured Description

Here is the rules of Structured Description.

Entities:
$T^{\prime }$: $\in \{\text{ the separable topological spaces }\}$ induced by any metric, $dist:T\times T\to \mathbb{R}$
$T$: $\in \{\text{ the topological subspaces of }{T}^{\prime }\}$
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Statements:
$T\in \{\text{ the separable topological spaces }\}$
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2: Note

It is crucial for $T$ to be induced by a metric, as is seen in Proof.

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3: Proof

Whole Strategy: Step 1: see that $T^{\prime }$ has a countable basis; Step 2: see that $T$ has a countable basis; Step 3: see that $T$ is the topological space induced by the metric subspace, and is separable.

Step 1:

$T^{\prime }$ has a countable basis, by the proposition that for any topological space induced by any metric, the space has a countable basis if and only if the space is separable.

Step 2:

$T$ has a countable basis, by the proposition that for any topological space, the intersection of any basis and any subspace is a basis for the subspace.

Step 3:

$T$ is the topological space induced by the metric subspace, by the proposition that for any topological space induced by any metric and any subset, the subset as the topological subspace equals the subset as the topological space induced by the metric subspace.

So, $T$ is separable, by the proposition that for any topological space induced by any metric, the space has a countable basis if and only if the space is separable.

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References

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