I just did an exercise that was the following :
Let $X$ be paracompact and $Y$ metric. For each $\epsilon\in C(X,\mathbb{R}^+)$ define a metric in $C(X,Y)$ as the following $d_{\epsilon}(f,g)=min\{1,\sup\frac{d(f(x),g(x))}{\epsilon(x)}\}$. The strong topology on $C(X,Y)$ is that induced by the family of metrics $\{d_{\epsilon}:\epsilon\in C(X,\mathbb{R}^+)\}$.
Now I thought I was able to do this but I nowhere used the fact that $X$ is paracompact. Does anyone have any insight where this is necessary ? Thanks in advance.