Continuous extensions of functions defined on subsets of products

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 [18], ) every Gδ-dense subspace in a product of separable metric spaces is C-embedded; (b) (M. Ulmer, 1970 [22], , 1973 [23], ) every Σ-product in a product of first-countable spaces is C-embedded; (c) (R. Pol and E. Pol, 1976 [20], , also A.V. Arhangelʼskiĭ, 2000 [3]; 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 authorsʼ 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 above-mentioned results. Theorem – Let {Xi:i∈I} be a set of T1-spaces, and let Y be dense in an open subspace of XI:=∏i∈IXi. 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.