Supoose that f -> X where D is any set and x is any metric space with metric d. We can define a function d*xD->R given by:
d*(x,y) = d(f(x),f(y)) for any x,y in D. If f is injective prove that d* is a metric on D!
Supoose that f -> X where D is any set and x is any metric space with metric d. We can define a function d*xD->R given by:
d*(x,y) = d(f(x),f(y)) for any x,y in D. If f is injective prove that d* is a metric on D!
There is really one of the three properties that is not trival.
If $\displaystyle d^*(x,y)=0$ that mean $\displaystyle d(f(x),f(y))=0$.
Does that mean f(x)=f(y)? And what does that mean?