Borsik’s Properties of Topological Spaces and Their Applications

Let X be an uncountable Polish space. L̆ubica Holá showed recently that there are \(2^{\mathfrak {c}}\) 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 \(|Y|\ge 2\) then the family of all non-Borel measurable quasi-continuous functions has cardinality \(\ge 2^{{\mathfrak {c}}}\). Secondly, we show that the family of quasi-continuous non Borel functions from X to Y may contain big algebraic structures.