Weakly discontinuous and resolvable functions between topological spaces

Abstract

We prove that a function $f:X\to Y$ from a first-countable (more generally, Preiss-Simon) space $X$ to a regular space $Y$ is weakly discontinuous (which means that every subspace $A\subset X$ contains an open dense subset $U\subset A$ such that $f|U$ is continuous) if and only if $f$ is open-resolvable (in the sense that for every open subset $U\subset Y$ 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 $R\subset Y$ 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.

Keywords

• Weakly discontinuous function,resolvable function,Preiss-Simon space

References

1. A.V. Arkhangelskii, B.M. Bokalo, Tangency of topologies and tangential properties of topological spaces, Tr.Mosk.Mat. Ob-va 54 (1992), 160185, (in Russian); English transl.: Trans. Mosk. Math. Soc. 54 (1993),139-163.
2. T. Banakh, B. Bokalo, On scatteredly continuous maps between topological spaces, Topology Appl. 157:1 (2010), 108122.
3. T. Banakh, S. Kutsak, V. Maslyuchenko, O. Maslyuchenko, Direct and inverse prob- lems of the Baire classifications of integrals dependent on a parameter, Ukr. Mat. Zhurn. 56:11 (2004), 14431457 (in Ukrainian).
4. B. Bokalo, O. Malanyuk, On almost continuous mappings, Matem. Studii. 9:1 (1995), 9093 (in Ukrainian).
5. Á. Császár, M.Laczkovich, Discrete and equal convergence, Studia Sci. Math. Hungar. 10 (1975), 463472.
6. Á. Császár, M.Laczkovich, Some remarks on discrete Baire classes, Acta Math. Acad. Sci. Hungar. 33 (1979), 5170.
7. J. Jayne, C.A. Rogers, First level Borel functions and isomorphisms, J. Math. Pures Appl.(9) 61:2 (1982), 177205.
8. O. Karlova, V. Mykhaylyuk, On composition of Baire functions, Topology Appl. 216 (2017) 824.

9. B. Kirchheim, Baire one star functions, Real Analysis Exchange, 18:2 (1992/93), 385399.
10. K. Kuratowski, Topology, I, PWN, Warszawa, 1966.
11. R. O'Malley, Baire 1, Darboux functions, Proc. Amer. Math. Soc. 60 (1976), 187 192.
12. D. Preiss, P. Simon, A weakly pseudocompact subspace of Banach space is weakly compact, Comment. Math. Univ. Carol. 15 (1974), 603609.
13. S. Solecki, Decomposing Borel sets and functions and the structure of Baire clas 1 functions, J. Amer. Math. Soc. 11:3 (1998), 521550.
14. V.A. Vinokurov, Strong regularizability of discontinuous functions, Dokl. Akad. Nauk SSSR 281 (1985), no. 2, 265269 (in Russian).