On the continuity of factorizations
DOI:
https://doi.org/10.4995/agt.2008.1806Keywords:
Product space, Dense subspace, Continuous factorization, Continuous extensions of mapsAbstract
Let {Xi : i ∈ I} be a set of sets, XJ :=Пi∈J Xi when Ø â‰ 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 â—¦ 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 .
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