When is $[0,1]^K$ submetrizable or even metrizable? Let $I=[0,1]$ and $K$ is a compact space. Then could the function space $I^K$ be submetrizable, even metrizable? In other words, in general, if $I^A$ can be submetrizable (metrizable) for some space $A$, what's condition that $A$ should satisfying?

If $A$ is compact, $I^A$ is metrizable with the metric being the uniform norm.  That is, $d(f,g):=\sup_{a\in A} d(f(a),g(a))$.
