$\omega$ -cover and related spaces on the vietoris hyperspace $\mathcal F(X)$

Recently, Tuyen et al. [1] showed that a space has a $\sigma$-(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 $\sigma$-(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 $\omega$-cover and certain spaces defined by $\omega$-covers on the hyperspace of finite subsets of a space X endowed with the Vietoris topology. We prove that is an $\omega$-Lindelöf (resp., $\omega$-Menger, $\omega$-Rothberger) space if and only if so is $\mathcal F(X)$.