On the continuity of factorizations

Abstract

Let {Xi : i ∈ I} be a set of sets, XJ :=Пi∈J Xi when Ø ≠ J ⊆ I; Y be a subset of XI , Z be a set, and f : Y → Z. Then f is said to depend on J if p, q ∈ Y , pJ = qJ ⇒ f(p) = f(q); in this case, fJ : πJ [Y ] → Z is well-defined by the rule f = fJ ◦ πJ|Y When the Xi and Z are spaces and f : Y → Z is continuous with Y dense in XI , several natural questions arise: (a) does f depend on some small J ⊆ I? (b) if it does, when is fJ continuous? (c) if fJ is continuous, when does it extend to continuous fJ : XJ → Z? (d) if fJ so extends, when does f extend to continuous f : XI → Z? (e) if f depends on some J ⊆ I and f extends to continuous f : XI → Z, when does f also depend on J? The authors offer answers (some complete, some partial) to some of these questions, together with relevant counterexamples. Theorem 1. f has a continuous extension f : XI → Z that depends on J if and only if fJ is continuous and has a continuous extension fJ : XJ → Z. Example 1. For ω ≤ k ≤ c there are a dense subset Y of [0, 1]k and f ∈ C(Y, [0, 1]) such that f depends on every nonempty J ⊆ k, there is no J ∈ [k]<ω such that fJ is continuous, and f extends continuously over [0, 1]k. Example 2. There are a Tychonoff space XI, dense Y ⊆ XI, f ∈ C(Y ), and J ∈ [I]<ω such that f depends on J, πJ [Y ] is C-embedded in XJ , and f does not extend continuously over XI .

Keywords

• Product space
• Dense subspace
• Continuous factorization
• Continuous extensions of maps