Inderasan Naidoo

A Pseudocompact Completely Regular Frame which is not Spatial

Compact regular frames are always spatial. In this note we present a method for constructing non-spatial frames. As an application we show that there is a countably compact (and hence pseudocompact) completely regular frame which is not spatial.

Isocompactness in the Category of Locales

We study isocompactness in Loc defined, exactly as in Top, by requiring that every countably compact closed sublocale be compact. This is a genuine extension of the same-named topological concept since every Boolean (or, even more emphatically, every paracompact) locale is isocompact. A slightly stronger variant is defined by decreeing that the closure of every complemented countably compact sublocale be compact. Dropping the adjective “complemented” yields a formally even stronger property, which we show to be preserved by finite products. Metrizable locales (or, more generally, perfectly normal locales) do not distinguish between the three variants of isocompactness. Each of the stronger variants of isocompactness travels across a proper map of locales, and in the opposite direction if the map is a surjection in Loc.

On completion in the category SSNσFrm

We introduce the category SSNσFrm of super strong nearness σ-frames and show the existence of a completion for a super strong nearness σ-frame unique up to isomorphism by the similar construction presented in [WALTERS-WAYLAND, J. L.: Completeness and Nearly Fine Uniform Frames. PhD Thesis, Univ. Catholique de Louvain, 1996] and [WALTERS-WAYLAND, J. L.: A Shirota Theorem for frames, Appl. Categ. Structures 7 (1999), 271–277]. Completion is also shown to be a coreflection in SSNσFrm. We also engage with the notion of total boundedness for nearness σ-frames and provide a characterization of the Samuel compactification of a nearness σ-frame alternative to the description in [NAIDOO, I.: Samuel compactification and uniform coreflection of nearness σ-frames, Czechoslovak Math. J. 56(131) (2006), 1229–1241].

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.

More on Uniform Paracompactness in Pointfree Topology

Abstract We revisit uniformly paracompact uniform frames and show that, in analogy with their spatial counterparts, they have a characterisation in terms of a “completeness property”. Namely, they are precisely those in which every weakly Cauchy filter clusters. We also give another characterisation in terms of the Čech-Stone compactification of the underlying frame. By tweaking the definition of uniformly paracompact frames, we define uniformly para-Lindelöf frames (analogously to same-named uniform spaces) and characterise them in terms of the Lindelöf coreflection of the underlying frame. This latter characterisation has no spatial counterpart.