Open AccessRemarks on R-separability of Pixley-Roy hyperspaces
Author(s) -
LI Zu-quan
Publication year - 2022
Publication title -
filomat
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.449
H-Index - 34
eISSN - 2406-0933
pISSN - 0354-5180
DOI - 10.2298/fil2203881l
Subject(s) - hyperspace , mathematics , separable space , countable set , second countable space , polish space , space (punctuation) , topology (electrical circuits) , combinatorics , discrete mathematics , pure mathematics , mathematical analysis , linguistics , philosophy
Let PR(X) denote the hyperspace of nonempty finite subsets of a topological space X with the Pixley-Roy topology. In this paper, motivated by [4], we introduced cf-covers and rcf-covers of X to establish the R-selective separability and the M-selective separability in PR(X) under the Pixley-Roy topology. We proved that the following statements are equivalent for a space X: (1) PR(X) is R-separable (resp., M-separable); (2) X satisfies S1(Crcf, Crcf) (resp., Sfin(Crcf,Crcf)); (3) X is countable and each co-finite subset of X satisfies S1(Ccf, Ccf) (resp.,Sfin(Ccf,Ccf)); (4)Xis countable and PR(X) has countable strong fan tightness (resp., PR(X) has countable fan tightness).