On $(\tau_1, \tau_2)$-Star-I Compact space

Abstract

This paper introduces the concept of the $(\tau_1, \tau_2)$-star-$\mathcal{I}$ compact space in an ideal bispace, where an ideal bispace is a quadruple $(X,\tau_1,\tau_2,I)$, $\tau_1, \tau_2$ being topologies defined on the set $X$ and $I$ being an ideal defined on $X$. The structure of $(\tau_1, \tau_2)$-star-$\mathcal{I}$ compactness has been compared with some nearer structures like $(\tau_1, \tau_2)$-$I$ compactness, strongly star-$I$ compactness and countably $I$ compactness etc.  With some counter examples the distinct structures of these topological features has been validated. The Nature of subspaces and nature of functions that preserves the $(\tau_1, \tau_2)$-star-$I$ compactness are revealed. It has been shown that real-valued continuous functions defined on $(\tau_1, \tau_2)$-star-$I_{fin}$ compact spaces are bounded. A finite intersection property like characterization is given for this specific topological feature. Lastly, the relation with weakly $(\tau_1, \tau_2)$-star compactness has been established by means of co-dense ideals.