  We show a rigidity theorem for the Seiberg-Witten invariants mod 2 for
families of spin 4-manifolds. A mechanism of this rigidity theorem also gives a
family version of 10/8-type inequality. As an application, we prove the
existence of non-smoothable topological families of 4-manifolds whose fiber,
base space, and total space are smoothable as manifolds. These non-smoothable
topological families provide new examples of 4-manifolds $M$ for which the
inclusion maps $\operatorname{Diff}(M) \hookrightarrow \operatorname{Homeo}(M)$
are not weak homotopy equivalences. We shall also give a new series of
non-smoothable topological actions on some spin 4-manifolds.
