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There are \(2^{\mathfrak {c}}\) Quasicontinuous Non Borel Functions on Uncountable Polish Space

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Abstract

S. Marcus showed that there is a quasicontinuous function from the interval [0, 1] to \(\mathbb {R}\) which is not Lebesgue measurable. We prove that if X is either an uncountable Polish space or a locally pathwise connected perfectly normal topological space with at least one non isolated point, then there is a quasicontinuous non Borel measurable function from X to [0, 1]. We also found new conditions under which for every quasicontinuous function there is an equivalent Borel measurable quasicontinuous function. If X is a Baire space and Y is a separable metric space, then every Borel measurable function \(f: X \rightarrow Y\) of the first class is cliquish.

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References

  1. Baire, R.: Sur les functions des variables reelles. Ann. Mat. Pura Appl. 3, 1–122 (1899)

    Article  Google Scholar 

  2. Beer, G.: Topologies on Closed and Closed Convex Sets. Kluwer Academic Publishers, Amsterdam (1993)

    Book  Google Scholar 

  3. Bledsoe, W.W.: Neighbourly functions. Proc. Am. Math. Soc. 3, 114–115 (1952)

    Article  MathSciNet  Google Scholar 

  4. Borsík, J.: Points of continuity, quasicontinuity and cliquishness. Rend. Ist. Math. Univ. Trieste 26, 5–20 (1994)

    MathSciNet  MATH  Google Scholar 

  5. Borsík, J., Holá, Ľ, Holý, D.: Baire spaces and quasicontinuous mappings. Filomat 25, 69–83 (2011)

    Article  MathSciNet  Google Scholar 

  6. Bouziad, A.: Every Čech-analytic Baire semitopological group is a topological group. Proc. Am. Math. Soc. 124, 953–959 (1996)

    Article  MathSciNet  Google Scholar 

  7. Crannell, A., Frantz, M., LeMasurier, M.: Closed relations and equivalence classes of quasicontinuous functions. Real Anal. Exchange 31, 409–424 (2006/2007)

  8. Engelking, R.: General Topology. PWN (1977)

  9. Fuller, R.V.: Relations among continuous and various noncontinuous functions. Pac. J. Math. 25, 495–509 (1968)

    Article  Google Scholar 

  10. Giles, J.R., Bartlett, M.O.: Modified continuity and a generalization of Michael’s selection theorem. Set-Valued Anal 1, 247–268 (1993)

    Article  MathSciNet  Google Scholar 

  11. Hansell, R.W.: On Borel mappings and Baire functions. Trans. Am. Math. Soc. 174, 195–211 (1994)

    MathSciNet  MATH  Google Scholar 

  12. Holá, Ľ: Functional characterization of \(p\)-spaces. Central Eur. J. Math. 11, 2197–2202 (2013)

    MathSciNet  MATH  Google Scholar 

  13. Holá, Ľ.: There are \(2^{c}\) quasicontinuous non-Lebesgue measurable functions. Am. Math. Monthly (to appear)

  14. Holá, Ľ, Holý, D.: Minimal USCO maps, densely continuous forms and upper semicontinuous functions. Rocky Mount. Math. J. 39, 545–562 (2009)

    Article  Google Scholar 

  15. Holá, Ľ, Holý, D.: Pointwise convergence of quasicontinuous mappings and Baire spaces. Rocky Mount. Math. J. 41, 1883–1894 (2011)

    Article  MathSciNet  Google Scholar 

  16. Holá, Ľ, Holý, D.: New characterization of minimal CUSCO maps. Rocky Mount. Math. J. 44, 1851–1866 (2014)

    Article  MathSciNet  Google Scholar 

  17. Holá, Ľ, Holý, D.: Quasicontinuous subcontinuous functions and compactness. Mediterranean J. Math. 13, 4509–4518 (2016)

    Article  MathSciNet  Google Scholar 

  18. Holá, Ľ, Holý, D.: Quasicontinuous functions and compactness. Mediterranean J. Math. 14, 219 (2017). https://doi.org/10.1007/s00009-017-1014-7

    Article  MathSciNet  MATH  Google Scholar 

  19. Holá, Ľ, Holý, D.: Minimal USCO and minimal CUSCO maps and compactness. J. Math. Anal. Appl. 439, 737–744 (2016)

    Article  MathSciNet  Google Scholar 

  20. Holý, D.: Ascoli-type theorems for locally bounded quasicontinuous functions, minimal USCO and minimal CUSCO maps. Ann. Funct. Anal. 6(3), 29–41 (2015)

    Article  MathSciNet  Google Scholar 

  21. Jayne, J.E., Orihuela, J., Pallarés, A.J., Vera, G.: \(\sigma \)-fragmentability of multivalued maps and selection theorems. J. Funct. Anal. 117(2), 243–273 (1993)

    Article  MathSciNet  Google Scholar 

  22. Jayne, J.E., Rogers, C.A.: Upper semi-continuous set-valued functions. Acta Math. 149, 87–125 (1982)

    Article  MathSciNet  Google Scholar 

  23. Kelley, J.L.: General Topology. Van Nostrand, New York (1955)

    MATH  Google Scholar 

  24. Kechris, A.S.: Classical Descriptive Set Theory. Springer, New York (1995)

    Book  Google Scholar 

  25. Kempisty, S.: Sur les fonctions quasi-continues. Fundamenta Mathematicae 19, 184–197 (1932)

    Article  Google Scholar 

  26. Kenderov, P.S., Kortezov, I.S., Moors, W.B.: Continuity points of quasi-continuous mappings. Topol. Appl. 109, 321–346 (2001)

    Article  MathSciNet  Google Scholar 

  27. Matejdes, M.: Minimality of multifunctions. Real Anal. Exchange 32, 519–526 (2007)

    Article  MathSciNet  Google Scholar 

  28. Marcus, S.: Sur les fonctions quasi-continues au sense Kempisty. Colloq. Math. 8, 45–53 (1961)

    Article  Google Scholar 

  29. McCoy, R.A., Ntantu, I.: Topological Properties of Space of Continuous Functions. Springer, Berlin (1988)

    Book  Google Scholar 

  30. Moors, W.B.: Any semitopological group that is homeomorphic to a product of Čech-complete spaces is a topological group. Set-Valued Var. Anal. 21, 627–633 (2013)

    Article  MathSciNet  Google Scholar 

  31. Moors, W.B.: Fragmentable mappings and CHART groups. Fundamenta Math. 234, 191–200 (2016)

    MathSciNet  MATH  Google Scholar 

  32. Moors, W.B.: Semitopological groups, Bouziad spaces and topological groups. Topol. Appl. 160, 2038–2048 (2013)

    Article  MathSciNet  Google Scholar 

  33. Moors, W.B., Somasundaram, S.: A weakly Stegall space that is not a Stegall space. Proc. Am. Math. Soc. 131, 647–654 (2003)

    Article  MathSciNet  Google Scholar 

  34. Neubrunn, T.: Quasi-continuity. Real Anal. Exchange 14, 259–306 (1988)

    Article  MathSciNet  Google Scholar 

  35. Thielman, H.P.: Types of functions. Am. Math. Mon. 60, 156–161 (1953)

    Article  MathSciNet  Google Scholar 

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Acknowledgements

The author would like to thank to grant Vega 2/0048/21.

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  1. Institute of Mathematics, Academy of Sciences, Štefánikova 49, 81473, Bratislava, Slovakia

    Ľubica Holá

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Holá, Ľ. There are \(2^{\mathfrak {c}}\) Quasicontinuous Non Borel Functions on Uncountable Polish Space. Results Math 76, 126 (2021). https://doi.org/10.1007/s00025-021-01440-3

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