192: 2 Metrics with Condition with Each Other Define Same Topology

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A description/proof of that 2 metrics with condition with each other define same topology

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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: Description
• 2: Proof

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Starting Context

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

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Target Context

• The reader will have a description and a proof of the proposition that for any set, any 2 metrics on the set with a condition with each other define the same topology.

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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 set, $S$, any metrics, $dis{t}_1$ and $dis{t}_2$, on $S$ that satisfy the condition that for any point, $p\in S$, and any number, $0<ϵ$, there is a number, $0<\delta$, such that for any $p^{\prime }$ that satisfies $dis{t}_1(p,p^{\prime })<\delta$, $dis{t}_2(p,p^{\prime })<ϵ$ and for any $p^{\prime }$ that satisfies $dis{t}_2(p,p^{\prime })<\delta$, $dis{t}_1(p,p^{\prime })<ϵ$ define the same topology.

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2: Proof

Suppose that $U$ is open with respect to $dis{t}_1$. At any point, $p\in S$, there is an open ball with respect to $dis{t}_1$, $B_{1-p-ϵ}\subseteq U$. Let us show that there is a $0<\delta$ such that $B_{2-p-\delta }\subseteq B_{1-p-ϵ}$ where $B_{2-p-\delta }$ is the open ball with respect to $dis{t}_2$. Let us take $\delta$ as satisfies the condition with respect to $p$ and $ϵ$. For any $p^{\prime }\in B_{2-p-\delta }$, $dis{t}_2(p,p^{\prime })<\delta$, so, $dis{t}_1(p,p^{\prime })<ϵ$, so, $p^{\prime }\in B_{1-p-ϵ}$, so, $B_{2-p-\delta }\subseteq B_{1-p-ϵ}\subseteq U$. So, $U$ is open with respect to $dis{t}_2$.

By the symmetry, any $U$ that is open with respect to $dis{t}_2$ is open with respect to $dis{t}_1$.

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References

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