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?
When is $[0,1]^K$ submetrizable or even metrizable?
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general-topology
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0Really, I want to know the results on the topology of uniform convergence, the topology of pointwise convergence and the compact-open topology. – 2011-12-31
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If $A$ is compact, $I^A$ is metrizable with the metric being the uniform norm. That is, $d(f,g):=\sup_{a\in A} d(f(a),g(a))$.
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0@Henno Brandsma, could you give me a proof for it? – 2011-12-31