Irreducible restrictions of closed mappings

A regular Lindelöf space X and a closed continuous surjection f: X → Y are constructed such that f is not inductively irreducible, i.e., no restriction of f to a closed subset of X is an irreducible surjection. We also show that if there are no weakly inaccessible cardinals, then every closed continuous surjection f: X → Y for paracompact X is inductively irreducible provided Y does not contain a dense-in-itself clopen P-subspace. We obtain ZFC results under certain further restrictions on X or Y, e.g., if the Lindelöf degree of X is less than, or the tightness of Y is less than or equal to, the first weakly inaccessible cardinal.