Functional Dependence on Small Sets of Indices

Let f : Y → Z with Y ⊆ XI := Πi∈I Xi. Then (a) J ⊆ I is essential if there are x, y ∈ Y such that d(x, y) = J and f(x) 6= f(y), where d(x, y) := {i ∈ I : xi 6= yi}; Jf := {i : {i} is essential}; an essential J is optimally essential if no essential J ′ ⊆ J satisfies |J ′| < |J |; J ∈ Jf if J is a maximal family of pairwise disjoint optimally essential sets; λf := sup{|J | : J ∈ Jf}. (b) f depends on J ⊆ I if [x, y ∈ Y, xJ = yJ ] ⇒ f(x) = f(y); Df := {J ⊆ I : f depends on J}; μf := min{|J | : J ∈ Df}. Theorem 1. J ∈ Df ⇒ λf ≤ |J |. Theorem 2. J ∈ Jf ⇒ ⋃ J ∈ Df . That context is strictly set-theoretic. Henceforth let XI and Z be spaces with Z Hausdorff, and let f ∈ C(Y,Z). This is known: (*) if Y contains a σ-product then J ∈ Df iff Jf ⊆ J . The authors give examples to show: Jf ∈ Df in (*) can fail, if any one of the three hypotheses are omitted; J ,J ′ ∈ Jf , with |J | 6= |J ′|, can occur; J ∈ Df ⇒ |J | > λf can occur; J ∈ Jf ⇒ |J | < λf can occur; |J | ≤ λf ⇒ J / ∈ Df (hence, λf < μf ) can occur. The authors’ interest in (*) is motivated by their observation that when XI has the κ-box topology, its obvious analogue, say (*)κ, can fail. They propose and prove (what seem to be) appropriate modifications of (*)κ.