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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. [ABSTRACT FROM AUTHOR]

We introduce and study the concept of a uniformly locally connected uniform σ-frame. The uniformly locally connected reflection of a locally connected uniform σ-frame is constructed. [ABSTRACT FROM AUTHOR]

We show that complete strong nearness σ-frames are exactly the cozero parts of complete separable strong Lindelöf nearness frames. We also relate nearness σ-frames and metric σ-frames and show that every metric σ-frame admits an admissible nearness such that it is complete as a metric σ-frame if and only if it is complete in this admissible nearness. [ABSTRACT FROM AUTHOR]

We introduce and study the concepts of connectedness and local connectedness in σ-frames. We also consider the local connectedness of the Stone-Čech compactification of a regular σ-frame. [ABSTRACT FROM AUTHOR]

We show that for uniform frames, with the underlying frame being Boolean, uniform paracompactness and strong Cauchy completeness are equivalent conditions. Certain aspects of uniform paracompactness are also considered. We then introduce the pointfree notion of preparacompactness and show that the completion of a preparacompact uniform frame is strongly Cauchy complete. We also formulate pointfree filter characterizations of Lindelöfness for regular frames in analogy to their classical topological counterparts. [ABSTRACT FROM AUTHOR]