1057: Topological Space Induced by Metric Is Hausdorff

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description/proof of that topological space induced by metric is Hausdorff

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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: Structured Description
• 2: Proof

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Starting Context

• The reader knows a definition of topology induced by metric.
• The reader knows a definition of Hausdorff topological space.

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Target Context

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

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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:
$T$: $\in \{\text{ the topological spaces }\}$, induced by any metric, $dist:T\times T\to \mathbb{R}$
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Statements:
$T\in \{\text{ the Hausdorff topological spaces }\}$
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2: Proof

Whole Strategy: Step 1: for each $t,t^{\prime }\in T$ such that $t\ne t^{\prime }$, take the open neighborhood of $t$, $B_{t,dist(t,t^{\prime })/2}$, and the open neighborhood of $t^{\prime }$, $B_{t^{\prime },dist(t,t^{\prime })/2}$,, and see that $B_{t,dist(t,t^{\prime })/2}\cap B_{t^{\prime },dist(t,t^{\prime })/2}=\mathrm{\varnothing }$.

Step 1:

Let $t,t^{\prime }\in T$ be any such that $t\ne t^{\prime }$.

$0<dist(t,t^{\prime })$.

Let us take the open neighborhood of $t$, $B_{t,dist(t,t^{\prime })/2}$, and the open neighborhood of $t^{\prime }$, $B_{t^{\prime },dist(t,t^{\prime })/2}$,.

Let us see that $B_{t,dist(t,t^{\prime })/2}\cap B_{t^{\prime },dist(t,t^{\prime })/2}=\mathrm{\varnothing }$.

Let $u\in B_{t,dist(t,t^{\prime })/2}$ be any.

$dist(t,t^{\prime })\le dist(t,u)+dist(u,t^{\prime })$, so, $dist(t,t^{\prime })-dist(t,u)\le dist(u,t^{\prime })$, but $dist(t,t^{\prime })/2=dist(t,t^{\prime })-dist(t,t^{\prime })/2<dist(t,t^{\prime })-dist(t,u)$, so, $dist(t,t^{\prime })/2<dist(u,t^{\prime })$, which means that $u\notin B_{t^{\prime },dist(t,t^{\prime })/2}$.

So, $B_{t,dist(t,t^{\prime })/2}\cap B_{t^{\prime },dist(t,t^{\prime })/2}=\mathrm{\varnothing }$.

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References

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