Characterizations of weakly star-type Rothberger and Menger properties in hyperspaces

Abstract

In this paper, we introduce the selection principles $\mathbf{wSS}_{1}^{*}\left(\Pi_{\Delta}(\Lambda), \mathbb{C}_{\Delta}(\Lambda)\right)$, $\mathbf{wSS}_{\textsf{fin}}^{*}\left(\Pi_{\Delta}(\Lambda), \mathbb{C}_{\Delta}(\Lambda)\right)$, $\mathbf{wS}_{1}^{*}\left(\Pi_{\Delta}(\Lambda), \mathbb{C}_{\Delta}(\Lambda)\right)$ and $\mathbf{wS}_{\textsf{fin}}^{*}\left(\Pi_{\Delta}(\Lambda), \mathbb{C}_{\Delta}(\Lambda)\right)$ to characterize the properties weakly strong-star Rothberger (Menger) and weakly star-Rothberger (Menger) in the hyperspace $(\Lambda,\tau_{\Delta}^+)$, respectively. Furthermore, we introduce the notions $H(\mathbb{C}_{\Delta}(\Lambda))$ and $\mathbf{I}_{\textsf{fin}}(\mathbb{C}_\Delta(\Lambda),\mathbb{C}_\Delta(\Lambda))$ to characterize, respectively, the H-separability and the principle $\mathbf{U}_{\textsf{fin}}(\mathscr{D}, \mathscr{D})$, in the same hyperspace.