  We consider complexes $(\X, d)$ of nuclear Fr\'echet spaces and continuous
boundary maps $d_n$ with closed ranges and prove that, up to topological
isomorphism, $ (H_{n}(\X, d))^*$ $\iso$ $H^{n}(\X^*,d^*),$ where
$(H_{n}(\X,d))^*$ is the strong dual space of the homology group of $(\X,d)$
and $ H^{n}(\X^*,d^*)$ is the cohomology group of the strong dual complex
$(\X^*,d^*)$. We use this result to establish the existence of topological
isomorphisms in the K\"{u}nneth formula for the cohomology of complete nuclear
$DF$-complexes and in the K\"{u}nneth formula for continuous Hochschild
cohomology of nuclear $\hat{\otimes}$-algebras which are Fr\'echet spaces or
$DF$-spaces for which all boundary maps of the standard homology complexes have
closed ranges. We describe explicitly continuous Hochschild and cyclic
cohomology groups of certain tensor products of $\hat{\otimes}$-algebras which
are Fr\'echet spaces or nuclear $DF$-spaces.
