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 , or at least , such that ; sometimes Z=Z′ is demanded. In this spirit the authors prove several quite general theorems in the context Y′=(XI)κ=∏i∈IXi in the κ-box topology (that is, with basic open sets of the form ∏i∈IUi with Ui open in Xi and with Ui≠Xi 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 – Let ω⩽κ≪α (that means: κ<α, and ) with α regular, be a set of non-empty spaces with each d(Xi)<α, π[Y]=XJ 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 xJ=yJ, and (c) every such f extends to .