The Baire property in hit-and-miss hypertopologies

Abstract

Recently, some techniques have been developed for the study of the Baire property in hyperspaces. These techniques have been applied to solve a long-standing open problem of McCoy in 1975 and a recent open problem of Zsilinszky. In this paper, we extend and apply these techniques further to investigate the Baire property of hyperspaces equipped with the general hit-and-miss topology.

Baire spaces and hyperspace topologies revisited

“…In the present paper we will continue in this research by exhibiting some common features of the plethora of studied hyperspaces as far as their Baireness is concerned. Indeed, following the unified exposition of hyperspace topologies introduced in [20], we prove Baireness of these general hyperspaces in one theorem, thereby generalizing and extending several results about specific hyperspaces spread in the literature ( [1], [14], [19], [23], [21], [3], [4], [9]). …”

“…Note that τ is a "pinched-cube" topology as defined in [3], [23]; indeed, we just need to take ∆ = {M (S) c : S ∈ D}.…”

“…Then for B ∈ ∆ and ε, η > 0, {A ∈ CL(X) : D t (A, B) > ε} = (B c ) + and {A ∈ CL(X) : e t (A, X) < η} = CL(X). Thus τ * = τ + is the general hit-and-miss topology on CL(X) (see [17], [1], [8], [19], [3]). Choosing ∆ = CL(X) we get the most studied hit-and-miss topology, the socalled Vietoris topology τ V (cf.…”

“…Although it would be possible to establish Baireness of (CL(X), τ * ) using the so-called Banach-Mazur game, as done in [14], [19], or [3], we have chosen a different method, which makes the proofs more transparent and actually improves on results of the above mentioned papers.…”

Three open problems on the Wijsman topology

“…For each of these problems, the author will give a brief survey on its background and then also some up-to-date partial solutions. The reader can find more details of these problems and some of their associated questions from the listed references, particularly from [5], [7], [8], [9] and [11], respectively. Note that the author has no intention to give a comprehensive and up-to-date survey on the Wijsman topology.…”