Definition:Compact Convergence
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Definition
Let $X$ be a topological space.
Let $M$ be a metric space.
Let $\left\langle{f_n}\right\rangle$ be a sequence of mappings $f_n : X \to M$.
Let $f: X \to M$ be a mapping.
Then $f_n$ converges compactly to $f$ if and only if $f_n$ converges uniformly to $f$ on every compact subset of $X$.