I just wondered whether there is a more general theorem behind claims like 'if a sequence of equicontinuos functions $f_i:[a,b]\rightarrow{\bf R}$ converges pointwise to a continuous function $f$ then it converges uniformly?'
I thought of something like 'if $K$ is compact, $Y$ a uniform space, $g:K\times X\rightarrow Y$ continuous and $(x_\alpha)_\alpha$ a net in $X$ converging to $x$ then $(g(s,x_\alpha))_\alpha$ converges uniformly w.r.t $s$ to $g(s,x)$.'
In the first example $K$ would be $[a,b]$, $X=\{0\}\cup\{1/k\}_{k\ge 1}$, $Y={\bf R}$, $g(1/k,x)=f_k(x)$ and $g(0,x)=f(x)$.
