Definition:Locally Uniform Convergence/General Definition
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Definition
Let $\struct {X, \tau}$ be a topological space.
Let $\struct {M, d}$ be a metric space.
Let $\sequence {f_n}$ be a sequence of mappings $f_n: X \to M$.
Then $f_n$ converges locally uniformly to $f: X \to M$ if and only if every point of $X$ has a neighborhood on which $f_n$ converges uniformly to $f$.