The pointwise and uniform-convergence uniformities on a function set $Y^X$

Let $Y$ be uniform and let $Y^X$ be the set of maps $X\to Y$. For an entourage $V$ of $Y$ and finite $F\subseteq X$ (The cardinality $\mid A\mid$ of a finite set), define

$P(F,V)=\{(f,g):(f(x),g(x))\in V\text{ for every }x\in F\},Q(V)=\{(f,g):(f(x),g(x))\in V\text{ for every }x\in X\}\mathrm{.}$

The relations $P(F,V)$ form an entourage base: finite intersections are refined by replacing $F$ with a finite union and $V$ with a common refinement; inverses replace $V$ by $V^{-1}$; and a square root of $V$ gives a square root of $P(F,V)$. The same verification with the fixed coordinate set $X$ proves the axioms for the $Q(V)$. The uniformities they generate are respectively the pointwise-convergence and uniform-convergence uniformities on $Y^X$.