Compact ICA-topobooleans and the Smirnov compactification theorem

Recently, the concepts of topoframe and topoboolean have been introduced as a generalization of point-free topology, and the relation between topobooleans and complete I-contact algebras (ICAs) has been studied. In this paper, we first introduce the ICA-topoboolean \(B_{\tau (C)}\), in which \(\tau (C)\) is induced from the complete ICA (B, C), and then characterize compact atomic ICA-topobooleans by their point clusters. As an example of the noncompact case, we determine all clusters of \(\big ( \mathcal {P}(\mathbb {R}), C\big )\), an ICA on the Boolean algebra of the power set of real numbers \(\mathbb {R}\). Finally, we generalize the Smirnov compactification theorem from proximity spaces to atomic ICA-topobooleans.