Let $X$ be a [topological space](/page/Topological%20Space) and let $(Y,e)$ be a [complete metric space](/page/Complete%20Metric%20Space). Let $C_b(X,Y)$ denote the set of all continuous bounded maps $f:X\to Y$, where bounded means that $f(X)$ has finite diameter in $(Y,e)$. Equip $C_b(X,Y)$ with the uniform metric inherited from the space $B(X,Y)$ of bounded maps $X\to Y$: if $X\neq\varnothing$, then $d_\infty(f,g):=\sup_{x\in X} e(f(x),g(x))$, and if $X=\varnothing$, then $d_\infty(f,g):=0$ for the unique maps $f,g:\varnothing\to Y$. Then the [metric space](/page/Metric%20Space) $(C_b(X,Y),d_\infty)$ is complete.