Recent Preprints

  1. On hereditary properties of extremally disconnected frames and normal frames. (with Javier Gutiérrez García and Tomasz Kubiak)
    Preprint 18-54 of DMUC, December 2018. pdf file
    Abstract: Hereditary extremal disconnectedness of frames and its equivalent form of complete extremal disconnectedness are the topic of this paper. We study them in parallel with the corresponding normality properties of frames. Among several characterizations, we show that a frame is hereditarily normal [resp., hereditarily extremally disconnected] if and only if all its open \emph{and} dense sublocales [resp., closed \emph{and} dense sublocales] are normal [resp., extremally disconnected]. Some of the presented results are new for the traditional topological spaces. Furthermore, we provide such a general setting that permits us to treat several variants of the concepts under study in a unified way.

  2. Axiom TD and the Simmons sublocale theorem. (with Aleš Pultr)
    Preprint 18-48 of DMUC, November 2018. pdf file
    Abstract: More precisely, we are analyzing some of Simmons, Niefield and Rosen- thal results concerning sublocales induced by subspaces. Simmons was concerned with the question when the coframe of sublocales is Boolean; he recognized the role of the axiom TD for the relation of certain degrees of scatteredness but did not emphasize its role in the relation between sublocales and subspaces. Niefield and Rosenthal avoided discussing this condition altogether. In this paper we show that the role of TD in this question is crucial. Concentration on the properties of TD-spaces and technique of sublocales in this context allows us to present a simple, transparent and choice-free proof of the scatteredness theorem.

  3. Another proof of Banaschewski's surjection theorem (with Dharmanand Baboolal and Aleš Pultr)
    Preprint 18-25 of DMUC, July 2018. pdf file
    Abstract: We present a new proof of Banaschewski's theorem stating that the completion lift of a uniform surjection is a surjection. The new procedure allows to extend the fact (and, similarly, the related theorem on closed uniform sublocales of complete uniform frames) to quasi-uniformities (''not necessarily symmetric uniformities''). Further, we show how a (regular) Cauchy point on a closed uniform sublocale can be extended to a (regular) Cauchy point on the larger (quasi-)uniform frame.

  4. Entourages, density, Cauchy structure, and completion (with Aleš Pultr)
    Preprint 18-19 of DMUC, May 2018. pdf file
    Abstract: We study uniformities and quasi-uniformities (uniformities without the symmetry axiom) in the common language of entourages. The techniques developed allow for a general theory in which uniformities are the symmetric part. In particular we have a natural Cauchy structure independent of symmetry and a very simple general completion procedure (perhaps more transparent and simpler than the usual symmetric one).

  5. Some aspects of (non)functoriality of natural discrete covers of locales (with Richard N. Ball and Aleš Pultr)
    Preprint 18-10 of DMUC, March 2018. pdf file
    Abstract: The frame Sc(L) generated by closed sublocales of a locale L is known to be a natural Boolean (''discrete'') extension of a subfit L; also it is known to be its maximal essential extension. In this paper we first show that it is an essential extension of any L and that the maximal essential extensions of L and Sc(L) are isomorphic. The construction Sc is not functorial; this leads to the question of individual liftings of homomorphisms L→ M to homomorphisms Sc(L)Sc(L). This is trivial for Boolean L and easy for a wide class of spatial L, M. Then, we show that one can lift all h: L → 2 for weakly Hausdorff L (and hence the spectra of L and Sc(L) are naturally isomorphic), and finally present liftings of h: L → M for regular L and arbitrary Boolean M.

  6. Hedgehog frames and a cardinal extension of normality (with Javier Gutiérrez García, Imanol Mozo Carollo and Joanne Walters-Wayland)
    Preprint 18-05 of DMUC, January 2018. pdf file
    Abstract: The hedgehog metric topology is presented here in a pointfree form, by specifying its generators and relations. This allows us to deal with the pointfree version of continuous (metric) hedgehog-valued functions that arises from it. We prove that the countable coproduct of the metric hedgehog frame with κ spines is universal in the class of metric frames of weight κ·ℵ0. We then study κ-collectionwise normality, a cardinal extension of normality, in frames. We prove that this is the necessary and sufficient condition under which Urysohn separation and Tietze extension type results hold for continuous hedgehog-valued functions. We show furthermore that κ-collectionwise normality is hereditary with respect to Fσ-sublocales and invariant under closed maps.

  7. Joins of closed sublocales (with Aleš Pultr and Anna Tozzi)
    Preprint 16-39 of DMUC, August 2016. pdf file
    Abstract: Sublocales that are joins of closed ones constitute a frame SJC(L) embedded as a join-sublattice into the coframe S(L) of sublocales of L. We prove that in the case of subfit L it is a subcolocale of S(L), that it is then a Boolean algebra and in fact precisely the Booleanization of S(L). In case of a T1-space X, SJC(Omega(X)) picks precisely the sublocales corresponding to induced subspaces. In linear L and more generally if L is also a coframe, Omega(X) is both a frame and a coframe, but with trivial exceptions not Boolean and not a subcolocale.