= One-cell bound on next-degree cohomology
{title2=$\dim_kH^{n+1}(X;k)\leq1$}
If an $n$-dimensional <CW complex> is extended by exactly one cell in dimension $n+1$, then its cellular <cochain> group in that degree is the coefficient <field> $k$. Regardless of how many higher cells are attached, the <cohomology> is a quotient of a subspace of that one-dimensional group. This gives the displayed bound. Finiteness of the higher cell sets is unnecessary for this particular conclusion.
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