Hello,

i have read the following assertion in a book, but i can't prove it.

Let $\displaystyle X$ be a compact metrizable space and $\displaystyle Y$ a metrizable space.

We denote by $\displaystyle C(X,Y)$ the space of continous functions from $\displaystyle X$ into $\displaystyle Y$ with the topology induced by thesuporuniform metric$\displaystyle d_{u}(f,g)=\sup_{x\in X} d_{Y}(f(x),g(x))$ , where $\displaystyle d_{Y}$ is a compatible metric for $\displaystyle Y$.

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 $\displaystyle d_{Y}$

Any ideas how to prove this?