On the continuity of factorizations

Authors

• W. W. Comfort Wesleyan University
• Ivan S. Gotchev Central Connecticut State University
• Luis Recoder-Nuñez Central Connecticut State University

Keywords:

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

Abstract

Let {Xi : i âˆˆ I} be a set of sets, XJ :=ПiâˆˆJ Xi when Ø â‰  J âŠ† I; Y be a subset of XI , Z be a set, and f : Y â†’ Z. Then f is said to depend on J if p, q âˆˆ Y , pJ = qJ â‡’ f(p) = f(q); in this case, fJ : πJ [Y ] â†’ Z is well-defined by the rule f = fJ â—¦ πJ|Y

When the Xi and Z are spaces and f : Y â†’ Z is continuous with Y dense in XI , several natural questions arise:

(a) does f depend on some small J âŠ† I?

(b) if it does, when is fJ continuous?

(c) if fJ is continuous, when does it extend to continuous fJ : XJ â†’ Z?

(d) if fJ so extends, when does f extend to continuous f : XI â†’ Z?

(e) if f depends on some J âŠ† I and f extends to continuous f : XI â†’ 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 â†’ Z that depends on J if and only if fJ is continuous and has a continuous extension fJ : XJ â†’ Z.

Example 1. For ω ≤ k ≤ c there are a dense subset Y of [0, 1]k and f âˆˆ C(Y, [0, 1]) such that f depends on every nonempty J âŠ† k, there is no J âˆˆ [k] such that fJ is continuous, and f extends continuously over [0, 1]k.

Example 2. There are a Tychonoff space XI, dense Y âŠ† XI, f âˆˆ C(Y ), and J âˆˆ [I] such that f depends on J, πJ [Y ] is C-embedded in XJ , and f does not extend continuously over XI .

Author Biographies

W. W. Comfort, Wesleyan University

Department of Mathematics and Computer Science

Ivan S. Gotchev, Central Connecticut State University

Department of Mathematical Sciences

Luis Recoder-Nuñez, Central Connecticut State University

Department of Mathematical Sciences

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