Compact maps surjective on cohomology give finite Hausdorff target cohomology

Write $Z_E^q=\ker (d^q)$, $B_E^q=\mathrm{im}(d^{q-1})$, and likewise $Z_F^q,B_F^q$, using the actual degree $q-1$ term, or the zero space when that term is absent. The cocycle spaces are closed kernels of continuous maps, so they are Fréchet by [F1]. Define continuous linear maps on the Fréchet space $Z_E^q\oplus F^{q-1}$ by $g(x,y)={\rho }^q(x)+{\delta }^{q-1}(y)$ and $h(x,y)=-{\rho }^q(x)$, with codomain $Z_F^q$. The cochain-map identity ensures both images lie in $Z_F^q$.

The map $g$ is surjective: for $z\in Z_F^q$, surjectivity of $H^q({\rho }^{\bullet })$ gives $x\in Z_E^q$ and $y\in F^{q-1}$ with $z-{\rho }^q(x)={\delta }^{q-1}(y)$. It is therefore a quasi-epimorphism by [F3]. The map $h$ is compact because it factors the compact map ${\rho }^q$ restricted to the closed subspace $Z_E^q$ through the finite product projection.

Apply [F2] to $g$ and $h$. Their sum is $(g+h)(x,y)={\delta }^{q-1}(y)$, whose image is exactly $B_F^q$. Hence $B_F^q$ is closed and finite-codimensional in $Z_F^q$. The quotient $Z_F^q\mathrm{/}{B}_F^q=H^q(F^{\bullet })$ is consequently Hausdorff and finite-dimensional.

For the openness assertion, let ${\sigma }^{\bullet }:E^{\bullet }\to F^{\bullet }$ be any continuous cochain map whose induced $q$th cohomology map is surjective, and put $Z_E^q=\ker d^q$, $Z_F^q=\ker {\delta }^q$. The map $G:Z_E^q\oplus F^{q-1}\to Z_F^q$, $G(x,y)={\sigma }^q(x)+{\delta }^{q-1}(y)$, is surjective by the same representative argument as in step 2.1, hence open by [F3]. If $O$ is an open subset of $H^q(E^{\bullet })$, its inverse image $\tilde{O}$ under the quotient map $Z_E^q\to H^q(E^{\bullet })$ is open and saturated by coboundaries. Then $G(\tilde{O}\times F^{q-1})={\pi }_F^{-1}(H^q({\sigma }^{\bullet })(O))$, where ${\pi }_F:Z_F^q\to H^q(F^{\bullet })$ is the quotient map. The left side is open because $G$ is open; by the quotient topology the image $H^q({\sigma }^{\bullet })(O)$ is open.

Steps 3.1 and 3.2 prove the two conclusions in the Statement. The argument uses no bound on the complex length and also applies in degree $q=0$; for a complex concentrated in nonnegative degrees, its absent degree $-1$ term is zero.