Totally analytic spaces under 𝑉=𝐿

Abstract

The following results obtain under the axiom of constructibility ( V -L): Assume that every subset of a topological space -V is analytic. Then A' is o-left-separatcd. Moreover, if the character of X is « u,, then X is o-discrete.Assume that X is a perfectly normal space of character « u, such that every subset of X belongs to the o-algebra generated by the analytic subsets of X. Then X is o-discrete.1. Introduction. By an analytic subset of a topological space A", we mean a set that can be obtained from a family of closed subsets of X by the A -operation, i.e., a set -f the form LI " D c FM", where the sets F,, s G^"(j, are closed in X. We say that X is a totally analytic space if each subset of X is analytic. This notion generalizes that of a space in which each subset is an F"-set.A space X is o-discrete (Fa-discrete) if X is the countable union of discrete (and closed) subspaces. X is left-separated if there exists a well-order < on X such that all the initial segments {y G X: y < x}, x G A", are closed in X. We say that X is o-left-separated if X is the countable union of left-separated subspaces. Note that the concept of an /^-left-separated space is superfluous, since such a space is actually left-separated; in particular, every /"-discrete space is left-separated.The character of a space X is the least cardinal number k such that every point of X has a neighbourhood-base consisting of at most k sets; the character of X is

There is a 𝑄-set space in ZFC

“…First of all, CH obviously implies that there are no Q-sets. Furthermore, extending results of M. Reed [8] and R. Hansell [4], the author and H. Junnila [1] showed that under V = L there are no Q-sets of character < c. They also showed [ 1 ] under V = L that if every subset of a space X is a G^-set, then X has to be fairly close to being cr-discrete: it has to be a -left-separated. These results made use of a method of Fleissner [2].…”