### On the continuity of factorizations

#### Abstract

Let {X_{i} : i ∈ I} be a set of sets, X_{J} :=П_{i}_{∈}_{J }X_{i} when Ø ≠ J ⊆ I; Y be a subset of X_{I} , Z be a set, and f : Y → Z. Then f is said to depend on J if p, q ∈ Y , p_{J} = q_{J} ⇒ f(p) = f(q); in this case, f_{J} : π_{J} [Y ] → Z is well-defined by the rule f = fJ ◦ π_{J}|_{Y}

When the X_{i} and Z are spaces and f : Y → Z is continuous with Y dense in X_{I} , several natural questions arise:

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

(b) if it does, when is f_{J} continuous?

(c) if f_{J} is continuous, when does it extend to continuous f_{J} : X_{J} → Z?

(d) if f_{J} so extends, when does f extend to continuous f : X_{I} → Z?

(e) if f depends on some J ⊆ I and f extends to continuous f : X_{I} → 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 : X_{I} → Z that depends on J if and only if f_{J} is continuous and has a continuous extension f_{J} : X_{J} → 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 f_{J} is continuous, and f extends continuously over [0, 1]^{k}.

Example 2. There are a Tychonoff space X_{I}, dense Y ⊆ X_{I}, f ∈ C(Y ), and J ∈ [I]^{<ω} such that f depends on J, π_{J} [Y ] is C-embedded in X_{J} , and f does not extend continuously over X_{I} .

#### Keywords

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PDF#### References

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