One-cell bound on next-degree cohomology (${\dim }_k{H}^{n+1}(X;k)\le 1$)

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.