On scatteredly continuous maps between topological spaces

Abstract

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.

On composition of Baire functions

“…The implication (6)⇒(2) follows from [1, Theorem 8.1]. The implication (2)⇒(1) can be proved completely similarly to the proof of Theorem 6.3 from [1], which shows this implication for a normal space X and a path-connected Y ∈ σAE f (X).…”

“…The positive answer to this question was given independently in [2] and [14]. Observe that crucial auxiliary results were a Jayne-Rogers theorem [8,Theorem 1] in [2] and a Banakh-Bokalo theorem [1,Theorem 8.1] in [14] on the equivalence of the G δ -measurability of a function to the piecewise continuity.…”

Weakly discontinuous and resolvable functions between topological spaces

“…By induction on the tree ω <ω we shall construct sequences (x s ) s∈ω <ω of points of the set D \ AC(f ), and sequences (V s ) s∈ω <ω and (U s ) s∈ω <ω , (W s ) s∈ω <ω of sets so that the following conditions hold for every finite number sequence s ∈ ω <ω : This fact and the Preiss-Simon property of X at x s allows us to construct a sequence (V ′ k ) k∈ω of open subsets of V s \ cl X f −1 (W s ) that converges to x s in the sense that each neighborhood of x contains all but finitely many sets V ′ k . Applying Lemma 2.5 (2) to the map f |V s : V s → U s , we can find an infinite subset N ⊂ ω and a sequence (U ′ k ) k∈N of pairwise disjoint open sets of U s such that each set f −1 (U ′ k ) ∩ V ′ k , k ∈ N , has non-empty interior in X. Let N = {k n : n ∈ ω} be the increasing enumeration of the set N .…”

Topological properties preserved by weakly discontinuous maps and weak homeomorphisms

“…The following upper bound for the closed decomposition number of a weakly discontinuous map was given in Proposition 5.3 and Theorem 5.4 of [5].…”