Čech-completeness in pointfree topology

Abstract

We define Čech-complete frames by means of a filter condition which does not require that such frames be completely regular. We then, among regular frames, give a characterization in terms of ideals from which one sees more easily that a Tychonoff space is Čech-complete iff its frame of open sets is Čech-complete. Extending this notion of Čech-completeness to nearness frames in a natural way (meaning that covers are replaced with uniform covers, and convergent filters – the ones that meet every cover – are replaced with Cauchy filters – the ones that meet every uniform cover), we define controlled nearness frames along the lines that Bentley and Hunsaker define controlled nearness spaces. We show that the subcategory they form is closed under countable coproducts.