Weakly discontinuous and resolvable functions between topological spaces

Abstract

Abstract. We prove that a function f : X → Y from a first-countable (more generally, Preiss-Simon) space X to a regular space Y is weakly discontinuous (which means that every subspace A ⊂ X contains an open dense subset U ⊂ A such that f |U is continuous) if and only if f is open-resolvable (in the sense that for every open subset U ⊂ Y the preimage f −1 (U ) is a resolvable subset of X) if and only if f is resolvable (in the sense that for every resolvable subset R ⊂ Y the preimage f −1 (R) is a resolvable subset of X). For functions on metrizable spaces this characterization was announced (without proof) by Vinokurov in 1985.

Extending Baire-one functions on compact spaces

“…For a deeper discussion of properties and applications of fragmented maps and their analogs we refer the reader to [1,2,7,13,15].…”

Topological properties preserved by weakly discontinuous maps and weak homeomorphisms

“…By definition, a map f : X → Y between topological spaces is weakly discontinuous if each subspace A ⊂ X contains an open dense subspace U ⊂ A such that the restriction f |U is continuous. Such maps were introduced by Vinokurov [30] and studied in details in [4,5,6,7,8,11,12,13,14,22,23]. Also they appear naturally in Analysis, see [9,16,19].…”