1059: Metric Is Continuous w.r.t. Topology Induced by Metric

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description/proof of that metric is continuous w.r.t. topology induced by metric

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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: Proof

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Starting Context

• The reader knows a definition of metric.
• The reader knows a definition of topology induced by metric.
• The reader knows a definition of product topology.
• The reader knows a definition of continuous map.

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Target Context

• The reader will have a description and a proof of the proposition that any metric is continuous with respect to the topology induced by the metric.

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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:
$T$: $\in \{\text{ the metric spaces }\}$, with metric, $dist:T\times T\to \mathbb{R}$ with the topology induced by $dist$
$\mathbb{R}$: $=\text{ the Euclidean topological space }$
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Statements:
$dist\in \{\text{ the continuous maps }\}$
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2: Proof

Whole Strategy: Step 1: take any $p=(p^1,p^2)\in T\times T$, any neighborhood of $dist(p^1,p^2)$, $N_{dist(p^1,p^2)}\subseteq \mathbb{R}$, and an open ball around $dist(p^1,p^2)$, $B_{dist(p^1,p^2),ϵ}\subseteq \mathbb{R}$, such that $B_{dist(p^1,p^2),ϵ}\subseteq N_{dist(p^1,p^2)}$; Step 2: take the open neighborhood of $p$, $B_{p^1,ϵ/2}\times B_{p^2,ϵ/2}\subseteq T\times T$, and see that $dist(B_{p^1,ϵ/2}\times B_{p^2,ϵ/2})\subseteq B_{dist(p^1,p^2),ϵ}\subseteq N_{dist(p^1,p^2)}$.

Step 1:

Let $p=(p^1,p^2)\in T\times T$ be any.

Let $N_{dist(p^1,p^2)}\subseteq \mathbb{R}$ be any neighborhood of $dist(p^1,p^2)$.

There is an open ball around $dist(p^1,p^2)$, $B_{dist(p^1,p^2),ϵ}\subseteq \mathbb{R}$, such that $B_{dist(p^1,p^2),ϵ}\subseteq N_{dist(p^1,p^2)}$.

Step 2:

Let us take the open neighborhood of $p$, $B_{p^1,ϵ/2}\times B_{p^2,ϵ/2}\subseteq T\times T$, which is indeed open on $T\times T$ by the definition of topology induced by metric and the definition of product topology.

For any point, $p^{\prime }=(p^{\prime 1},p^{\prime 2})\in B_{p^1,ϵ/2}\times B_{p^2,ϵ/2}$, $dist(p^{\prime 1},p^{\prime 2})\le dist(p^{\prime 1},p^1)+dist(p^1,p^{\prime 2})\le dist(p^{\prime 1},p^1)+dist(p^1,p^2)+dist(p^2,p^{\prime 2})<ϵ/2+dist(p^1,p^2)+ϵ/2$, so, $dist(p^{\prime 1},p^{\prime 2})-dist(p^1,p^2)<ϵ$; $dist(p^1,p^2)\le dist(p^1,p^{\prime 1})+dist(p^{\prime 1},p^2)\le dist(p^1,p^{\prime 1})+dist(p^{\prime 1},p^{\prime 2})+dist(p^{\prime 2},p^2)<ϵ/2+dist(p^{\prime 1},p^{\prime 2})+ϵ/2$, so, $dist(p^1,p^2)-dist(p^{\prime 1},p^{\prime 2})<ϵ$; so, $|dist(p^{\prime 1},p^{\prime 2})-dist(p^1,p^2)|<ϵ$.

That means that $dist(B_{p^1,ϵ/2}\times B_{p^2,ϵ/2})\subseteq B_{dist(p^1,p^2),ϵ}\subseteq N_{dist(p^1,p^2)}$. So, $dist$ is continuous.

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References

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