1. ## Topology question

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

2. ## Re: Topology question

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