Clustering in Celebrating Professor Themba A. Dube (A TAD Celebration II)

Abstract

This paper is the second in the series celebrating the mathematical works of Professor Themba Dube. In this sequel, we give prominence to Dube's pivotal contributions on pointfree convergence at the unstructured frame level, in the category of locales, and on his noteworthy conceptions on extensions and frame quotients. We distill and draw attention to particular studies of Dube on filters and his novel characterizations of certain conservative pointfree properties by filter and ultrafilter convergence, notably normality, almost realcompactness, and pseudocompactness. We also feature Dube's joint work on convergence and clustering of filters in Loc and coconvergence and coclustering of ideals in the category Frm.

Keywords

• frame
• locale
• katˇetov extension
• fomin extension
• βl
• normal
• pseudocompact
• almost realcompact
• ˇcech-complete
• quotient
• filter
• ultrafilter
• clustering
• convergence
• coconvergence
• coclustering