- #1

bedi

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## Homework Statement

Let M be a metric space with metric [itex]d[/itex], and let [itex]d_{1}[/itex] be the metric defined below. Show that the two metric spaces [itex](M,d)[/itex], [itex](M,d_{1})[/itex] have the same open sets.

## Homework Equations

[itex]d_1:\frac{d(x,y)}{1+d(x,y)}[/itex]

## The Attempt at a Solution

I tried to show that the neighborhoods around their elements are the same.

An open ball in [itex](M,d): B_{r}(x)=\{y \in R: |x-y|<r\}[/itex]

An open ball in [itex](M,d_{1}): B_{r}(x)=\{y \in R: \frac{|x-y|}{1+|x-y|}<r\}=\{y \in R: (|x-y|)(1-r)<r\}[/itex], let [itex]1-r=n[/itex], hence [itex]n(|x-y|)<r[/itex].

I'm stuck.