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We investigate modifications of properties USC s and LSC s introduced by H. Ohta and M. Sakai [30]. Our property wED(U , C(X)) holds in any S 1 (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 S 1 (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 USCs is hereditary for Δ 0 2 subsets.

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We show that Miller partition forcing preserves selective independent families and P-points, which implies the consistency of $\mbox {cof}(\mathcal {N})=\mathfrak {a}=\mathfrak {u}=\mathfrak {i}<\mathfrak {a}_T=\omega _2$ . In addition, we show that Shelah’s poset for destroying the maximality of a given maximal ideal preserves tight mad families and so we establish the consistency of $\mbox {cof}(\mathcal {N})=\mathfrak {a}=\mathfrak {i}=\omega _1<\mathfrak {u}=\mathfrak {a}_T=\omega _2$ .