On Luzin spaces

V. I. Malykhin

Sbornik. Mathematics, Tome 39 (1981) no. 3, pp. 405-415

Résumé

The main results of the paper are the following theorems: Theorem 1. {\it The following proposition is consistent with the system $ZFC$: $\mathscr {PMS}$. In a product of a family of not more than $2^\mathfrak c$ separable complete metric spaces without isolated points, there exists a dense Luzin subspace of cardinal $\mathfrak c$; if the family is uncountable, then every countable subset of the Luzin subspace is closed.} Theorem 2 [CH]. In a nondiscrete topological group every element of which has order 2, and whose space satisfies the Suslin conditions, has the Baire property and has $\pi$-weight not greater than $\mathfrak c$, there exists a dense Luzin subgroup. Theorem 3. The system $ZFC$ is consistent with the assertion that in any generalized Cantor discontinuum $D^m$ of infinite weight $m$ not greater than $2^\mathfrak c$, considered as a topological group, there exists a dense Luzin subgroup of cardinal $\mathfrak c$. Bibliography: 14 titles.