Statistical convergence of order $\alpha$ in topology and its applications to selection principles

The statistical convergence in a topological space is constrained in this study up to order $\alpha$, where $\alpha \in (0, 1)$. A fresh group of open covers (namely $s^{\alpha}$-$\gamma$ covers) and an entirely novel category of denseness (namely $s^{\alpha}$-denseness) are proposed using this notion of $s^{\alpha}$-convergence, which has been used to study various topological aspects of $s^{\alpha}$-density. It has been revealed that the class of $s^{\alpha}$-$\gamma$ coverings falls somewhere between the class of $\gamma$ covers and the class of $s$-$\gamma$ covers. The influence of $s^{\alpha}$-$\gamma$ covers in topological games and selection principles are also investigated.