On dense subsets of Tychonoff products of T1-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.