To the Hewitt–Marczewski–Pondiczery theorem

Abstract

The classical Hewitt–Marczewski–Pondiczery theorem states that if d(Xs)⩽τ(ω⩽τ) for every s∈S and |S|⩽2τ then d(∏s∈SXs)⩽τ.We consider the product of decomposable spaces, which have two disjoint closed not empty sets. Not single point T1-spaces are decomposable.We prove for a regular uncountable cardinal τ that if d(Xα)=τ, Xα is decomposable space for all α∈2τ, then ∏α∈2τXα contains a dense set Q, |Q|=τ such that in every subset P⊆Q, |P|=τ, there is P′⊆P, |P′|=τ, which contains no convergent sequences and therefore is sequentially closed (Theorem 3.1).