$ω$ -cover and related spaces on the vietoris hyperspace $\mathcal F(X)$ - Nguyen Xuan Truc; Nguyen Thi Hong Phuc; Ong Van Tuyen; Luong Quoc Tuyen (Tạp chí Khoa học và Công nghệ)
Recently, Tuyen et al. [1] showed that a space has a $σ$-(P)-strong network consisting of cs-covers (resp., $cs^*$-covers) if and only if the hyperspace $\mathcal F(X)$ does, where is one of the following properties: point finite, point countable, compact finite, compact-countable, locally finite, locally countable. Moreover, they we also proved that is a Cauchy sn-symmetric space with a $σ$-(P)-property $cs^*$-network (resp., cs-network, sn-network.) if and only if so is $\mathcal F(X)$ (see [1]). In this paper, we study the concepts of $ω$-cover and certain spaces defined by $ω$-covers on the hyperspace of finite subsets of a space X endowed with the Vietoris topology. We prove that is an $ω$-Lindelöf (resp., $ω$-Menger, $ω$-Rothberger) space if and only if so is $\mathcal F(X)$.