Continuous extensions of functions defined on subsets of productsI,II

A subset Y of a space X is Gδ-dense if it intersects every nonempty Gδ-set. The Gδ-closure of Y in X is the largest subspace of X in which Y is Gδ-dense. The space X has a regular Gδ-diagonal if the diagonal of X is the intersection of countably many regular-closed subsets of X ×X. Consider now these results: (a) [N. Noble, 1972] every Gδ-dense subspace in a product of separable metric spaces is C-embedded; (b) [M. Ulmer, 1970/73] every Σ-product in a product of first-countable spaces is C-embedded; (c) [R. Pol and E. Pol, 1976, also A. V. Arhangel′skĭı, 2000; as corollaries of more general theorems], every dense subset of a product of completely regular, first-countable spaces is C-embedded in its Gδ-closure. The present authors’ Theorem 3.10 concerns the continuous extension of functions defined on subsets of product spaces with the κ-box topology. Here is the case κ = ω of Theorem 3.10, which simultaneously generalizes the abovementioned results. Theorem. Let {Xi : i ∈ I} be a set of T1-spaces, and let Y be dense in an open subspace of XI := Πi∈I Xi. If χ(qi, Xi) ≤ ω for every i ∈ I and every q in the Gδ-closure of Y in XI , then for every regular space Z with a regular Gδ-diagonal, every continuous function f : Y → Z extends continuously over the Gδ-closure of Y in XI . Some examples are cited to show that the hypothesis χ(qi, Xi) ≤ ω cannot be replaced by the weaker hypothesis ψ(qi, Xi) ≤ ω.