On Luzin spaces

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 $\mathfrak c$.
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