1. ## Topology question

Helo,I need confirmation on this concept:
Let (M,d) be a metric space and $\displaystyle \phi:M\rightarrow N$ be a bijective map ,then we can make N a metric space isometric to M by defining a metric $\displaystyle d_1$ by : distance between
$\displaystyle \phi(x)$ and $\displaystyle \phi(y)$ equals distance between x and y
$\displaystyle d_1(\phi(x),\phi(y))=d(x,y)$

2. ## Re: Topology question

Originally Posted by facenian
Helo,I need confirmation on this concept:
Let $\displaystyle (M,d)$ be a metric space and $\displaystyle \phi:M\rightarrow N$ be a bijective map ,then we can make N a metric space isometric to M by defining a metric $\displaystyle d_1$ by : distance between
$\displaystyle \phi(x)$ and $\displaystyle \phi(y)$ equals distance between x and y
$\displaystyle d_1(\phi(x),\phi(y))=d(x,y)$
Yes, it's true. You have to show that indeed $\displaystyle d_1$ is a metric, and by definition of $\displaystyle d_1$, $\displaystyle \phi$ is an isometric map.