LAUSR

Abstract By the classical Hewitt –  Marczewski –  Pondiczery theorem (see [2] , [3] ) the Tychonoff product of 2 ω many separable spaces is separable [2] , [3] . We consider the problem of the existence in the Tychonoff product of 2 ω many separable spaces a dense countable subset Q, which contains no nontrivial convergent sequences. We prove ( Theorem 4.1 ) that such dense set exists in the product ∏ α ∈ 2 ω Z α of separable spaces { Z α : α ∈ 2 ω } if in every space Z α ( α ∈ 2 ω ) there are two closed disjoint not empty sets, we call such spaces decomposable. The class of decomposable spaces includes not single point T 1 -spaces, but also some T 0 -spaces and some spaces, which are not even T 0 -spaces.