Universal Functions and Generalized Classes of Functions

On Ohta–Sakai's properties of a topological space

“…Then The property DL(Δ 0 2 -measurable, C(X)) will be denoted simply as JR-property. 11 Thus J.E. Jayne and C.A.…”

“…The equivalence of DL(U, C(X)) and DL( U, C(X)) is by Lemma 4.5. Finally, if X has DL( U, C(X)) then Y has DL(U, C(Y )) for any Y ⊆ X since every lower semicontinuous function on Y with values in [0,1] can be extended to X (see [11], Proposition 1.1).…”

“…Let G = n∈ω G n such that G n is open in X and G n+1 ⊆ G n for any n ∈ ω. For a sequence f n ; n ∈ ω of lower semicontinuous functions on G such that f n : G → [0, 1], n ∈ ω and f n → 0 there is a sequence h n ; n ∈ ω of lower semicontinuous functions on X such that h n |G = f n (see [11], Proposition 1.1). We define a sequence h n ; n ∈ ω of lower semicontinuous functions on X by…”

“…Let us assume that f n ; n ∈ ω is a sequence of lower semicontinuous functions on A ⊆ X such that f n : A → [0, 1], n ∈ ω and f n → 0 on A. There is a sequence f n ; n ∈ ω of lower semicontinuous functions on X such that f n |A = f n (see [11], Proposition 1.1). The set of all points of X which the sequence f n ; n ∈ ω converges in is Borel and so Δ 0 2 set by Corollary 5.2.…”

“…It can be stated for separable metrizable space using Hilbert cube ω [0, 1] and Proposition 1.1 in[11].…”