Sequence selection properties in Cp(X) with the double ideals.

Recently Bukovský, Das and Šupina [Ideal quasi-normal convergence and related notions, Colloq. Math. 146 (2017), 265–281] started the study of sequence selection properties (𝓘, 𝓙-α1) and (𝓘, 𝓙-α4) of Cp(X) using the double ideals, where 𝓘 and 𝓙 are the proper admissible ideals of ω, which are motivated by Arkhangeľskii local αi-properties [The frequency spectrum of a topological space and the classification of spaces, Dokl. Akad. Nauk SSSR 13 (1972), 1185–1189]. In this paper, we obtain some characterizations of (𝓘, 𝓙-α1) and (𝓘, 𝓙-α4) properties of Cp(X) in the terms of covering properties and selection principles. Under certain conditions on ideals 𝓘 and 𝓙, we identify the minimal cardinalities of a space X for which Cp(X) does not have (𝓘, 𝓙-α1) and (𝓘, 𝓙-α4) properties. [ABSTRACT FROM AUTHOR]