Continuous extensions of functions defined on subsets of products

Abstract

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 regularGδ-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. TheoremLet{Xi:i∈I}be a set ofT1-spaces, and let Y be dense in an open subspace ofXI:=∏i∈IXi. Ifχ(qi,Xi)⩽ωfor everyi∈Iand every q in theGδ-closure of Y inXI, then for every regular space Z with a regularGδ-diagonal, every continuous functionf:Y→Zextends continuously over theGδ-closure of Y inXI.Some examples are cited to show that the hypothesis χ(qi,Xi)⩽ω cannot be replaced by the weaker hypothesis ψ(qi,Xi)⩽ω.