Sequences of semicontinuous functions accompanying continuous functions

A space X is said to have property (USC) (resp. (LSC)) if whenever { f n : n ∈ ω } is a sequence of upper (resp. lower) semicontinuous functions from X into the closed unit interval [ 0 , 1 ] converging pointwise to the constant function 0 with the value 0, there is a sequence { g n : n ∈ ω } of continuous functions from X into [ 0 , 1 ] such that f n ⩽ g n ( n ∈ ω ) and { g n : n ∈ ω } converges pointwise to 0. In this paper, we study spaces having these properties and related ones. In particular, we show that (a) for a subset X of the real line, X has property (USC) if and only if it is a σ-set; (b) if X is a space of non-measurable cardinal and has property (LSC), then it is discrete. Our research comes of Scheepersʹ conjecture on properties S 1 ( Γ , Γ ) and wQN.