space of continuous functions on a compact space with uniform metric
i have read the following assertion in a book, but i can't prove it.
Let be a compact metrizable space and a metrizable space.
We denote by the space of continous functions from into with the topology induced by the sup or uniform metric
, where is a compatible metric for .
Now the assertion that i want to prove, but don't know how to:
A simple compactness argument shows that this topology is independent of the choice of
Any ideas how to prove this?
Re: space of continuous functions on a compact space with uniform metric
Yes, I have proved it and posted it on my blog. Here is a link to the post:
Uniform metrics define same topology on C(X,Y) | alanmath
As to Jose27's response, he hasn't understood that is any compatible metric for . So that is any metric that gives the topology on . So, of course, if the metric gives the same topology, then the set of continuous functions will not change.