On dense subsets of Tychonoff products of T1-spaces

Abstract

We consider dense subsets of Zc=∏α∈2ωZα, where Z=Zα is a separable not single point T1-space.We construct in Zc, |Z|≥ω, (Theorem 4.1) a countable dense set Q⊆Z such that every countable subset of Q contains a countable subset, which can be project on “many” subsets of Z.From this theorem follow some facts.Theorem 4.2 states that in the product Zc of a not single point separable T1-space Z there is a countable dense set which contains no non-trivial convergent in Zc sequences.The existence of such set in the product Ic, where I=[0,1] was proved [13] by W. H. Priestley.In [9] we proved that such set exists in a product of 2ω separable not single point T2-spaces.Theorem 4.4 states that if Z is a separable not countably compact T1-space, then there is a countable dense subset Q⊆Zc, satisfying the following condition: if E⊆Q is a countable set and E converges to a set F⊆Zc, then |E∖F|<ω.