= Free pro-p group
{title2=$F_p(X)$}
For finite $X$, this is the <pro-p completion> of the <free group> on $X$. Every function from $X$ to a <pro-p group> extends uniquely to a continuous <group homomorphism>. More generally one can require a basis tending to the identity and impose the universal property for maps with that convergence condition; alternatively the pro-$p$ completion of the abstract free group represents unrestricted maps from an arbitrary set. These infinite-rank conventions should not be conflated when interpreting generator cardinalities.
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