= Infinite-rank failure of the first-cohomology generator formula
Let $I$ be the set of infinite binary sequences and $G=\mathbb F_p^I$ with its <product topology>. Index the finite binary strings by the positive integers. For each $n$, take the element whose $r$ coordinate is one exactly when the $n$th string is a prefix of $r$. These countably many elements generate a dense <subgroup>: given finitely many distinct sequences, sufficiently long prefixes distinguish them individually. No finite family generates $G$, so its topological generator number is countable. Every continuous linear functional depends on finitely many coordinates, however, so $H^1(G,\mathbb F_p)$ has a basis of size $|I|=2^{\aleph_0}$. Thus the arbitrary-rank identity is false for unrestricted topological generating sets.
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