Internal descriptions of absolute Borel classes (Q1878515)

For every topological space~\(X\) let \(\mathcal P(X)\) be a~class of subsets of \(X\). A Tikhonov space is an absolute \(\mathcal P\)-space if \(X\) is in \(\mathcal P(Y)\) whenever \(X\) is embedded in a Tikhonov space~\(Y\). In particular, if \(\mathcal P(X)\) is the class of Borel subsets of \(X\) we obtain the definition of an absolute Borel space. The authors consider the classes of certain natural hierarchies of Borel sets and scattered Borel sets and for these classes \(\mathcal P\) prove internal descriptions of absolute \(\mathcal P\)-spaces in terms of complete sequences of covers. They also briefly discuss the preservation of the obtained classes of spaces under open continuous maps and under perfect maps.