On scatteredly continuous maps between topological spaces |
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Authors: | Taras Banakh Bogdan Bokalo |
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Institution: | aDepartment of Mathematics, Ivan Franko National University of Lviv, Universytetska 1, Lviv, Ukraine;bInstytut Matematyki, Akademia Świëtokrzyska, Świëtokrzyska 15, Kielce, Poland |
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Abstract: | A map f:X→Y between topological spaces is defined to be scatteredly continuous if for each subspace AX 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 AX 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 Gδ-measurability (the preimage of each open set is of type Gδ). Also under Martin Axiom, we construct a Gδ-measurable map f:X→Y between metrizable separable spaces, which is not piecewise continuous. This answers an old question of V. Vinokurov. |
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Keywords: | Scatteredly continuous map Weakly discontinuous map Piecewise continuous map color:black" href="/science?_ob=MathURL&_method=retrieve&_udi=B6V1K-4W8TJB0-8&_mathId=mml12&_user=10&_cdi=5677&_rdoc=15&_acct=C000069468&_version=1&_userid=6189383&md5=44400394a5e6a9cc5acbcffaad65fa09" title="Click to view the MathML source" Gδ" target="_blank">alt="Click to view the MathML source">Gδ -measurable map Preiss– Simon space |
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