On scatteredly continuous maps between topological spaces
T.Banakh, B.Bokalo

TL;DR
This paper explores the concept of scatteredly continuous maps between topological spaces, establishing their equivalence to several other continuity properties under certain conditions, and constructs a counterexample under Martin Axiom.
Contribution
It characterizes scatteredly continuous maps on specific spaces and provides a counterexample showing such maps need not be piecewise continuous.
Findings
Scattered continuity is equivalent to weak discontinuity, $\sigma$-continuity, and $G_\delta$-measurability under certain conditions.
Constructs a $G_\delta$-measurable map that is not piecewise continuous, answering an open question.
Provides conditions under which scattered continuity aligns with other forms of continuity in topological spaces.
Abstract
A map between topological spaces is defined to be {\em scatteredly continuous} if for each subspace the restriction has a point of continuity. We show that for a function from a perfectly paracompact hereditarily Baire Preiss-Simon space into a regular space the scattered continuity of is equivalent to (i) the weak discontinuity (for each subset the set of discontinuity points of is nowhere dense in ), (ii) the -continuity ( can be written as a countable union of closed subsets on which is continuous), (iii) the -measurability (the preimage of each open set is of type ). Also under Martin Axiom, we construct a -measurable map between metrizable separable spaces, which is not piecewise continuous. This answers an old question of V.Vinokurov.
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Taxonomy
TopicsAdvanced Topology and Set Theory · Mathematical Dynamics and Fractals · Rings, Modules, and Algebras
