  \v{C}ech cohomology $H^n(X)$ of a separable metrizable space $X$ is defined
in terms of cohomology of its nerves (or ANR neighborhoods) $P_\beta$ whereas
Steenrod-Sitnikov homology $H_n(X)$ is defined in terms of homology of compact
subsets $K_\alpha\subset X$.
  We show that one can also go vice versa: in a sense, $H^n(X)$ can be
reconstructed from $H^n(K_\alpha)$, and if $X$ is finite dimensional, $H_n(X)$
can be reconstructed from $H_n(P_\beta)$.
  The reconstruction is via a Bousfield-Kan/Araki-Yoshimura type spectral
sequence, except that the derived limits have to be "corrected" so as to take
into account a natural topology on the indexing set. The corrected derived
limits coincide with the usual ones when the topology is discrete, and in
general are applied not to an inverse system but to a "partially ordered
sheaf".
  The "correction" of the derived limit functors in turn involves constructing
a "correct" (metrizable) topology on the order complex $|P|$ of a partially
ordered metrizable space $P$ (such as the hyperspace $K(X)$ of nonempty compact
subsets of $X$ with the Hausdorff metric). It turns out that three natural
approaches (by using the space of measurable functions, the space of
probability measures, or the usual embedding $K(X)\to C(X;\mathbb R)$) all lead
to the same topology on $|P|$.
