513: Topology Induced by Metric

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definition of topology induced by metric

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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: Natural Language Description
• 3: Note

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Starting Context

• The reader knows a definition of metric.
• The reader knows a definition of topology.

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Target Context

• The reader will have a definition of topology induced by 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:
$T$: $\in \{\text{ the metric spaces }\}$
$\ast O$: $\in \{\text{ the topologies of }T\}$
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Conditions:
$\mathrm{\forall }S\subseteq T(S\in O⟺(\mathrm{\forall }p\in S(\mathrm{\exists }ϵ\in \mathbb{R}(0<ϵ\wedge (B_{p,ϵ}\subseteq S)))))$, where $B_{p,ϵ}$ is the open ball around $p$ of radius $ϵ$
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2: Natural Language Description

For any metric space, $T$, the topology, $O$, such that for any subset, $S\subseteq T$, $S$ is open if and only if for each $p\in S$, there is a positive, $ϵ\in \mathbb{R}$, such that $B_{p,ϵ}\subseteq S$

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3: Note

It is indeed a topology: $\mathrm{\varnothing }\in O$; $T\in O$; for any possibly uncountable number of open subsets, $\{U_{\alpha }|\alpha \in A\}$, and $U:={\cup }_{\alpha \in A}{U}_{\alpha }$, for any $p\in U$, $B_{p,{ϵ}_{\alpha }}\subseteq U_{\alpha }$ for any fixed $\alpha$, and $B_{p,{ϵ}_{\alpha }}\subseteq U_{\alpha }\subseteq U$; for any finite number of open subsets, $U_1,...,U_k$, and $U:={\cap }_{j=1,..,k}{U}_k$, for any $p\in U$, $B_{p,{ϵ}_j}\subseteq U_j$ for each $j$, and for $ϵ:=min\{{ϵ}_j\}$, $0<ϵ$ and $B_{p,ϵ}\subseteq U$.

For each $p\in T$ and each $ϵ$, $B_{p,ϵ}\in O$, because for any $p^{\prime }\in B_{p,ϵ}$, $dist(p,p^{\prime })\le ϵ$, so, for $B_{p^{\prime },ϵ-dist(p,p^{\prime })}$, for each $p^{″}\in B_{p^{\prime },ϵ-dist(p,p^{\prime })}$, $dist(p^{″},p^{\prime })<ϵ-dist(p,p^{\prime })$, but $dist(p^{″},p)\le dist(p^{″},p^{\prime })+dist(p^{\prime },p)<ϵ-dist(p,p^{\prime })+dist(p^{\prime },p)=ϵ$, which means that $B_{p^{\prime },ϵ-dist(p,p^{\prime })}\subseteq B_{p,ϵ}$.

In the definition, 'open ball' can be replaced by 'open cube' without any change of the concept, because if there is an open ball, there will be an open cube contained in it, and if there is an open cube, there will be an open ball contained in it.

Sometimes, 'open cube' is more convenient, and we can safely use 'open cube' instead of 'open ball'.

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References

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