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This paper presents a new treatment of the localic Katětov-Tong interpolation theorem, based on an analysis of special properties of normal frames, which shows that it does not hold in full generality. Besides giving us the conditions under which the localic Katětov-Tong interpolation theorem holds, this approach leads to a especially transparent and succinct proof of it. It is also shown that this pointfree extension of Katětov-Tong theorem still covers the localic versions of Urysohn's Lemma and Tietze's Extension Theorem.

As discovered recently, Li and Wang's 1997 treatment of semicontinuity for frames does not faithfully reflect the classical concept. In this paper we continue our study of semicontinuity in the pointfree setting. We define the pointfree concepts of lower and upper regularizations of frame semicontinuous real functions. We present characterizations of extremally disconnected frames in terms of these regularizations that allow us to reprove, in particular, the insertion and extension type characterizations of extremally disconnected frames due to Y.-M. Li and Z.-H. Li [Algebra Universalis 44 (2000), [271][272][273][274][275][276][277][278][279][280][281] in the right semicontinuity context. It turns out that the proof of the insertion theorem becomes very easy after having established a number of basic results regarding the regularizations. Notably, our extension theorem is a much strengthened version of Li and Li's result and it is proved without making use of the insertion theorem.

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Galois connections were originally expressed in a contravariant form with transformations that reverse (rather than preserve) order. Nowadays its covariant form (as residuated maps) is more often used since it is more convenient; namely compositions of residuated maps are handled more easily. In this paper we show that this is not a serious disadvantage of the contravariant form (at least in the natural context for uniform structures, where we need it), by introducing an operation of composition in the complete lattice Gal(L, L) of all (contravariant) Galois connections in a complete lattice L, that allows us to work with Galois connections in the same way as one usually works with residuated maps. This operation endows Gal(L, L) with a structure of quantale whenever L is a locale, allowing the description of uniform structures in terms of Galois connections.

In the framework of pointfree topology, we discuss the rôle of Weil entourages in the study of structures such as uniformities, quasi-uniformities, nearnesses, quasi-nearnesses, proximities and infinitesimal relations.