Bornologies and filters applied to selection principles and function spaces

Abstract

We extend known results of selection principles in Cp-theory to the context of spaces of the form C B (X), where B is a bornology on X. Particularly, by using the filter approach of Jordan to Cp-theory, we show that γ-productive spaces are productive with a larger class of γ-like spaces.2010 Mathematics Subject Classification. Primary 54D20; Secondary 54G99, 54A10.

Pointless proofs of the Menger and Rothberger games

“…In an attempt to extend some results regarding selective variations of tightness in the context of C p -theory [2] to the realm of topological groups, the following two important theorems about open covers appeared to be relevant:…”

Selection principles in hyperspaces

“…This chapter is an extended version of [10]. In the previous section, Theorem I.11 illustrates an important situation we shall be paying attention to in this chapter: when does a property P of C p (X) translate itself as a topological property Q of X?…”

“…However, we do not restrict the use of the term "hyperspace" to the above cases. Instead of presenting a formal definition broad enough to fulfill our needs 10 , we shall illustrate what we understand by hyperspaces in the following discussion.…”

“…Hence, := {B γ : γ < ω 1 } is a family of nonempty open sets of Y , where B γ := A∈G γ B A,γ . If | | ≤ ℵ 0 , then we may obtain 10 distinct γ, γ ′ < ω 1 such that B γ = B γ ′ , from which it follows that…”

“…Remark 1.44 (A "Newton-Leibniz" situation). Following the announcement of [10], Boaz Tsaban brought to our attention that a result similar to Corollary 1.39 also appears in the (thus far, unpublished) 17 MSc thesis of his student Nadav Samet [72].…”