On Analogues of Some Classical Subsets of the Real Line

A theorem on the existence of separable supports of σ-finite Borel measures given on metric spaces with small topological weights is applied to constructions of certain analogues of special subsets of the real line. A well-known result from topological measure theory states that every σ-finite Borel measure on a metric space whose weight is not real-valued measurable possesses a separable support (cf. [7] and [10]). The main goal of the present article is to demonstrate several applications of this result to some generalized analogues of classical subsets of the real line R (e.g. Luzin sets, Sierpiński sets and so on). We assume that all topological spaces E under consideration below satisfy the following condition: all one-element subsets of E are Borel (in E). The symbol ω denotes, as usual, the first infinite cardinal, ω1 — the first uncountable cardinal and c denotes the cardinality continuum. We recall that E is a universal measure zero space (or universally negligible space) if there is no nonzero σ-finite diffused Borel measure on E. The so-called Luzin subsets of the real line R yield nontrivial examples of uncountable universal measure zero spaces. According to the classical 1991 Mathematics Subject Classification. Primary 28A05, 28D05.