A map f : X → Y between topological spaces is defined to be scatteredly continuous if for each subspace A ⊂ X the restriction f |A has a point of continuity. We show that for a function f : X → Y from a perfectly paracompact hereditarily Baire Preiss-Simon space X into a regular space Y the scattered continuity of f is equivalent to (i) the weak discontinuity (for each subset A ⊂ X the set D(f |A) of discontinuity points of f |A is nowhere dense in A), (ii) the piecewise continuity (X can be written as a countable union of closed subsets on which f is continuous), (iii) the G δ -measurability (the preimage of each open set is of type G δ ). Also under Martin Axiom, we construct a G δ -measurable map f : X → Y between metrizable separable spaces, which is not piecewise continuous. This answers an old question of V.Vinokurov.1991 Mathematics Subject Classification. 54C08.
Given a topological space X, we study the structure of ∞-convex subsets in the space SCp(X) of scatteredly continuous functions on X. Our main result says that for a topological space X with countable strong fan tightness, each potentially bounded ∞-convex subset F ⊂ SCp(X) is weakly discontinuous in the sense that each non-empty subset A ⊂ X contains an open dense subset U
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.
A map f : X → Y between topological spaces is called weakly discontinuous if each subspace A ⊂ X contains an open dense subspace U ⊂ A such that the restriction f |U is continuous. A bijective map f : X → Y between topological spaces is called a weak homeomorphism if f and f −1 are weakly discontinuous. We study properties of topological spaces preserved by weakly discontinuous maps and weak homeomorphisms. In particular, we show that weak homeomorphisms preserve network weight, hereditary Lindelöf number, dimension. Also we classify infinite zero-dimensional σ-Polish metrizable spaces up to a weak homeomorphism and prove that any such space X is weakly homeomorphic to one of 9 spaces: ω, 2 ω , N ω , Q, Q ⊕ 2 ω , Q × 2 ω , Q ⊕ N ω , (Q × 2 ω ) ⊕ N ω , Q × N ω .
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