Maps of Borel Sets

Questions concerning the structure of Borel sets were raised in special cases by Luzin, Aleksandrov, and Uryson as the problems of distinguishing the sets with certain homogeneous properties in Borel classes and determining the number of such pairwise nonhomeomorphic sets. The universal homogeneity, i.e., the property to contain an everywhere closed copy of any Borel set of the same or smaller class, was considered by L. V. Keldysh. She called the sets of classes $\Pi _{\alpha }^0$, $\alpha > 2$, of first category in themselves that possess this homogeneity property canonical and proved their uniqueness. Thus she revealed the central role of the universality property when describing homeomorphic Borel sets. These investigations led her to the problem of universality of Borel sets and to the problem of finding conditions under which there exists an open map between Borel sets. In this paper, such conditions are presented and similar questions are considered for closed, compact-covering, harmonious, and other stable maps.