On Ohta–Sakai's properties of a topological space

Abstract

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.

On a Lindenbaum Composition Theorem

“…However, it seems that the question which topological spaces possess Jayne-Rogers property is an open problem. In Section 5, extending the result in [16], we show that every lower semicontinuous function on a perfectly normal topological space X is a discrete limit of continuous functions if and only if X has Jayne-Rogers property and X is a σ-set. Moreover, in the same section, we are investigating possible candidates for convergence like characterization of lower and upper Δ 0 2 -measurable functions.…”

“…Let us recall that a topological space X is a σ-set if Π 0 2 (X) = Σ 0 2 (X). 6 For more, see Proposition 4.3 in [16]. 7 The equality MΔ 0…”

“…Inspired by Question 1, the property of topological space "any function from family F is a discrete limit of a sequence of functions from family G" is studied in [16]. 8 Naturally, for an arbitrary topological space X we have…”

“…Moreover, both properties are hereditary and a topological space with any of them is a σ-set, see Corollary 4.6 and Proposition 4.7 in [16]. Let us recall that a topological space X is a σ-set if Π 0 2 (X) = Σ 0 2 (X).…”

“…8 The property is denoted as DL(F , G). For instance, X has DL MΔ 0 2 , C(X) if and only if B Part (b) of Theorem 4.1 in [16] has been proved for separable metric spaces. However, the only constraint in a proof for a more general topological space was Lindenbaum's Theorem which was known to be valid in separable metric spaces.…”