Cohomological Künneth theorem over a field ($H^{\ast }(X\times F;k)\cong H^{\ast }(X;k){\otimes }_k{H}^{\ast }(F;k)$)

If $k$ is a field and $H^{\ast }(F;k)$ has finite total dimension, the cohomology cross product gives the displayed graded-ring isomorphism for arbitrary spaces. Over a field, every chain complex splits into its homology with zero differential and contractible summands. The Eilenberg–Zilber theorem thus replaces the fiber chain complex by a bounded finite-dimensional complex; dualizing its tensor product with $C_{\ast }(X;k)$ gives a finite direct sum of shifted cochain complexes of $X$. Its cohomology is the displayed tensor product. Finiteness prevents the infinite-product obstruction that can occur for two infinite discrete spaces. The standard product formula and its finiteness hypothesis are discussed in pi.math.cornell.edu/~hatcher/AT/ATch3.pdf, Theorem 3.15.