Tightness games with bounded finite selections

Selection principles in hyperspaces

“…In the subsequent sections (Sections 1.3 and 1.4) we deal with filters and their applications in C ℬ -theory, by adapting the C p -theory results of Jordan [39]. But first, in the next section we discuss generalizations of the selection principles S 1 and S fin as well as of the games G 1 and G fin , motivated by the work of Aurichi, Bella and Dias [6]. 2 The equivalence 2 ⇔ 3 is due to Sakai [71] in case • = 1, while for • =fin it follows from Theorem I.11 (due to Arhangel'skii) together with a result of Just et al [44].…”

“…They analyzed selection variations of tightness in a point y of a Tychonoff space Y , by using the family Ω y in S f (Ω y , Ω y ), for f : ω → [2, ℵ 0 ). More recently, Aurichi, Bella and Dias [6] defined game variations of the selection principle S ψ , but still in the tightness context. We shall now present our definition of the game G ψ -it is slightly different from the one presented in [6], in order to include both the games G 1 and G fin .…”

“…1. If f is bounded, then a) (García-Ferreira and Tamariz-Mascarúa [27]) the selection principle S f (Ω y , Ω y ) is equivalent to S 1 (Ω y , Ω y ); b) (Aurichi, Bella and Dias [6]) the games G f (Ω y , Ω y ) and G k−1 (Ω y , Ω y ) are equivalent, where k = lim sup n∈ω f (n) ∈ ω. 3 2.…”

“…In fact, García-Ferreira and Tamariz-Mascarúa [27] provide examples showing that the principles S 1 (Ω y , Ω y ), S Id (Ω y , Ω y ) and S fin (Ω y , Ω y ) are not equivalent. Also, Aurichi, Bella and Dias [6] showed that for each k ∈ ω ∖ {0}, the games Proof. We present a proof, for the convenience of the reader, adapted from [27].…”

“…In order to analyze Menger's conjecture, Hurewicz formulated an intermediate property between σ -compactness and S fin ( , ). In modern terminology, Hurewicz's property can be expressed as U fin ( , Γ), where Γ denotes the family of all point-cofinite 6 open coverings: an open covering of a topological space X is point-cofinite if for all x ∈ X, the set {U ∈ : x ∈ U} is cofinite in .…”