Sequences of semicontinuous functions accompanying continuous functions

Abstract

A 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 S 1 (Γ, Γ) and wQN.

On Ohta–Sakai's properties of a topological space

“…We shall quickly recall preliminary definitions, for more detail we advise the reader to consult [3,6,30]. Most of the definitions can be found also in [4].…”

“…Few authors investigated sufficient conditions for a wQN-space to be an S 1 (Γ, Γ)-space, namely γγ co -space and σ-set 6 by J. Haleš [18], (γ, γ)-shrinkable space by M. Sakai [34], condition "every open γ-cover of X is shrinkable" by L. Bukovský and J. Haleš [6] and properties USC, 7 USC s by H. Ohta and M. Sakai [30]. However, these notions can be divided into two groups, since by [34] and [30] we have …”

“…Basic set-theoretical and topological terminology follows mainly [4] and [16]. See [3,6,30] and our second section or footnotes for preliminary definitions. For more about cardinal invariants see e.g.…”

“…Now, we mention just the property USC s introduced by H. Ohta and M. Sakai [30]: A topological space X has a property USC s if for any sequence f n ; n ∈ ω of upper semicontinuous functions with values in [0, 1] converging to zero, there is a sequence g m ; m ∈ ω of continuous functions converging to zero and an increasing sequence {n m } ∞ m=0 of natural numbers such that E-mail address: jaroslav.supina@upjs.sk. 1 The work on this research has been partially supported by the grant 1/0002/12 of Slovak Grant Agency VEGA and by f n m ≤ g m for any m ∈ ω.…”

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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.

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