Topology question

**facenian** · August 3rd 2011, 03:19 AM

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
$d_1(\phi(x),\phi(y))=d(x,y)$

**girdav** · August 3rd 2011, 04:13 AM

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.

Thread: Topology question

LinkBack

Thread Tools

Display