On Ohta–Sakai's properties of a topological space

We investigate modifications of properties USCsUSCs and LSCsLSCs introduced by H. Ohta and M. Sakai [30]. Our property wED(U,C(X)) holds in any S1(Γ,Γ)S1(Γ,Γ)-space and property wED(L,C(X)) holds in perfectly normal QN-space. We present their covering characterizations and hereditary properties. Our main result is that a topological space X   is an S1(Γ,Γ)S1(Γ,Γ)-space if and only if X   is both, a wQN-space and possesses wED(U,C(X)). Property wED(L,C(X)) is related to the condition “to be a discrete limit of a sequence of continuous functions” which is briefly studied in the paper as well. Moreover, for perfectly normal space we show that original property USCsUSCs is hereditary for Δ20 subsets.