Weakly discontinuous and resolvable functions between topological spaces

Year 2017, Volume: 46 Issue: 1, 103 - 110, 01.02.2017

Taras Banakh , Bogdan Bokalo

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

• 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.
• T. Banakh, B. Bokalo, On scatteredly continuous maps between topological spaces, Topology Appl. 157:1 (2010), 108122.
• 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).
• B. Bokalo, O. Malanyuk, On almost continuous mappings, Matem. Studii. 9:1 (1995), 9093 (in Ukrainian).
• Á. Császár, M.Laczkovich, Discrete and equal convergence, Studia Sci. Math. Hungar. 10 (1975), 463472.
• Á. Császár, M.Laczkovich, Some remarks on discrete Baire classes, Acta Math. Acad. Sci. Hungar. 33 (1979), 5170.
• J. Jayne, C.A. Rogers, First level Borel functions and isomorphisms, J. Math. Pures Appl.(9) 61:2 (1982), 177205.
• O. Karlova, V. Mykhaylyuk, On composition of Baire functions, Topology Appl. 216 (2017) 824.
• B. Kirchheim, Baire one star functions, Real Analysis Exchange, 18:2 (1992/93), 385399.
• K. Kuratowski, Topology, I, PWN, Warszawa, 1966.
• R. O'Malley, Baire 1, Darboux functions, Proc. Amer. Math. Soc. 60 (1976), 187 192.
• D. Preiss, P. Simon, A weakly pseudocompact subspace of Banach space is weakly compact, Comment. Math. Univ. Carol. 15 (1974), 603609.
• S. Solecki, Decomposing Borel sets and functions and the structure of Baire clas 1 functions, J. Amer. Math. Soc. 11:3 (1998), 521550.
• V.A. Vinokurov, Strong regularizability of discontinuous functions, Dokl. Akad. Nauk SSSR 281 (1985), no. 2, 265269 (in Russian).