Definition:Locally Uniform Convergence/General Definition

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 $f: X \to M$ every point of $X$ has a neighborhood on which $f_n$ converges uniformly to $f$.

Also see

 * Definition:Compact Convergence