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Thomas Rot
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Function spaces are sometimes not metrizable. Let $X$ and $Y$ be topological spaces, and $C(X,Y)$ be the space of continuous maps from $X$ to $Y$, topologized in the compact open topology. Then $C(X,Y)$ need not be metrizable (it is if $X$ is compact, and $Y$ is a metric space though).

Thomas Rot
  • 7.6k
  • 2
  • 32
  • 54