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Monotone insertions of continuous functions - University of ...

Monotone insertions of continuous functions - University of ...

12 CHRIS GOOD AND IAN

12 CHRIS GOOD AND IAN STARES intersection of the two sets. Now using Theorem 4.5, let G f : X → [0, 1] be defined by G f = Ψ B(f) (g f ) and define { (Gf (x), η) if x ∈ U(f, η) Ψ κ (f) = 0 otherwise. Since G f is continuous on X and G f (x) = 0 for all x ∉ U(f) and since (U(f, η)) η

MONOTONE INSERTION OF CONTINUOUS FUNCTIONS 13 (3) for all (some) κ, there is an operator Φ κ : UL κ (X) → C(X, J(κ)) such that g ≤ Φ κ (g, h) ≤ h and g(x) < Φ κ (g, h)(x) < h(x) whenever g(x) < h(x) and such that if g ≤ g ′ and h ≤ h ′ then Φ κ (g, h) ≤ Φ κ (g ′ , h ′ ). Proof. The proof of (1) implies (2) ((1) implies (3)) is essentially the same as the proof of (1) implies (2) in Theorem 4.3. The only change is to use the operator Φ from Theorem 1.6 (Theorem 1.5) rather than from Theorem 1.4, when constructing Φ κ . To prove the converses of these two implications proceed as in the proof of (3) implies (1) in Theorem 4.3. It can be shown that X has an operator Φ satisfying the conditions in Theorem 1.6 (resp. Theorem 1.5). □ We can now use this result to prove the stronger analogue of Theorem 4.4 as follows. Theorem 4.9. The following are equivalent for a space X (1) X is stratifiable, (2) for all (some) κ, and closed subspace A of X there is an operator Φ κ : UL κ

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