Sequences of semicontinuous functions accompanying continuous functions

AbstractA space X is said to have property (USC) (resp. (LSC)) if whenever {fn: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 {gn:n∈ω} of continuous functions from X into [0,1] such that fn⩽gn (n∈ω) and {gn: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 Scheepers' conjecture on properties S1(Γ,Γ) and wQN