Continuous mappings on subspaces of products with the κ-box topology

Much of General Topology addresses this issue: Given a function f ∈ C ( Y , Z ) with Y ⊆ Y ′ and Z ⊆ Z ′ , find f ¯ ∈ C ( Y ′ , Z ′ ) , or at least f ¯ ∈ C ( cl Y ′ Y , Z ′ ) , such that f ⊆ f ¯ ; sometimes Z = Z ′ is demanded. In this spirit the authors prove several quite general theorems in the context Y ′ = ( X I ) κ = ∏ i ∈ I X i in the κ-box topology (that is, with basic open sets of the form ∏ i ∈ I U i with U i open in X i and with U i ≠ X i for <κ-many i ∈ I ). A representative sample result, extending to the κ-box topology some results of Comfort and Negrepontis, of Noble and Ulmer, and of Hušek, is this. Theorem ⩽ κ ≪ α (that means: κ < α , and [ β < α and λ < κ ] ⇒ β λ < α ) with α regular, { X i : i ∈ I } be a set of non-empty spaces with each d ( X i ) < α , π [ Y ] = X J for each non-empty J ⊆ I such that | J | < α , and the diagonal in Z be the intersection of <α-many regular-closed subsets of Z × Z . Then (a) Y is pseudo- ( α , α ) -compact, (b) for every f ∈ C ( Y , Z ) there is J ∈ [ I ] < α such that f ( x ) = f ( y ) whenever x J = y J , and (c) every such f extends to f ¯ ∈ C ( ( X I ) κ , Z ) .