$ \alpha _i$-properties, selection principles and the ideals

Ko\v{c}inac \cite{KDR} introduced several $ \alpha_i $-properties as a selection principles and there were motivated by Arhangel'skii \cite{AAV} $ \alpha_i $-local properties. In this paper, we identify some classes $ \mathcal{A} $ and $ \mathcal{B} $ of open covers in topological spaces, topological groups, hyperspaces and abstract boundedness for which the Ko\v{c}inac $ \alpha_i (\mathcal{A}, \mathcal{B}) $-properties are closely related and often equivalent to $ S_1 (\mathcal{A}, \mathcal{B}) $, using the notion of an ideal. Further we introduce the ideal form of Hurewicz-bounded topological group and characterize it using these $ \alpha_i $-properties.