Characterizations of weaker forms of the Rothberger and Menger properties in hyperspaces

Abstract

In this paper, we introduce the notions of almost $\pi_{\Delta}(\Lambda)$-network and weakly $\pi_{\Delta}(\Lambda)$-network to characterize the properties almost Rothberger (Menger) and weakly Rothberger (Menger), respectively, in the hyperspaces $CL(X)$, $\mathbb{K}(X)$, $\mathbb{F}(X)$ and $\mathbb{CS}(X)$, endowed with the hit-and-miss topology. Also, we introduce the concepts of groupable $c_{\Delta}(\Lambda)$-cover and weakly $(\Delta,\Lambda)$-groupable cover of $X$ to give equivalences of the selection principles $\mathbf{S}_{1}(\mathscr{D},\mathscr{D}^{gp})$,  $\mathbf{S}_{\textsf{fin}}(\mathscr{D},\mathscr{D}^{gp})$, $\mathbf{S}_{1}(\mathscr{D},\mathscr{D}^{wgp})$ and  $\mathbf{S}_{\textsf{fin}}(\mathscr{D},\mathscr{D}^{wgp})$ in the same hyperspaces.