Let $C(E,F)$ be the set of continouos maps between metric spaces $E$ and $F$. Suppose we are given the $C^0$ fine topology and a metric topology on $C(E,F)$. We know that the fine topology is finer than the compact-open topology, but is it finer than metric topology?
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How is there a metric topology on $C(E,F)$? Is $E$ compact? – Ted Shifrin Apr 27 '15 at 13:58