Borsik’s Properties of Topological Spaces and Their Applications

Abstract

Let X be an uncountable Polish space. L̆ubica Holá showed recently that there are 2c\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$2^{\\mathfrak {c}}$$\\end{document} quasi-continuous real valued functions defined on the uncountable Polish space X that are not Borel measurable. Inspired by Holá’s result, we are extending it in two directions. First, we prove that if X is an uncountable Polish space and Y is any Hausdorff space with |Y|≥2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$|Y|\\ge 2$$\\end{document} then the family of all non-Borel measurable quasi-continuous functions has cardinality ≥2c\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\ge 2^{{\\mathfrak {c}}}$$\\end{document}. Secondly, we show that the family of quasi-continuous non Borel functions from X to Y may contain big algebraic structures.