On Semi α-Lindelöf in Bitopological Spaces

This paper set up a Closure–operator scheme for semi–\(\alpha\)–Lindel\"{o}fness in bitopological spaces to manage covering behavior generated by two interacting topologies. With the Čech–closure hull \(H_{ij}=j\!\operatorname{cl}\,i\!\operatorname{int}\,j\!\operatorname{cl}\,i\!\operatorname{int}\), we reformulate \(ij\)–semi–\(\alpha\)–open sets and obtain operator–level criteria for \(ij\)–semi–\(\alpha\)–Lindel\"{o}fness. We prove a network estimate that bounds \(L^{S_\alpha}_{ij}\) by the size of an \(ij\)–\(S_\alpha\)–network, and a star criterion under \(\rho\)–discrete network decompositions of such networks. Structural consequences include hereditary and transfer over dense subsets, stability under countable sums, and a tube–type product when the second topology is discrete and the first factor is \(i\)–compact. Also, we introduce \(ij\)–\(S_\alpha\)–perfect mappings and show preservation of \(ij\)–\(S_\alpha\)–Lindelöfness with explicit cardinal bounds; images under \(ij\)–\(S_\alpha\)– and \(ij\)–\(S_\alpha^\ast\)–continuous maps are correspondingly controlled. Pairwise invariants are examined via \(\widehat L^{S_\alpha}_{\mathrm{pair}}\), which lies between the one–sided quantities and equals their maximum whenever at least one is infinite.