Another proof of Banaschewski's surjection theorem
(with Dharmanand Baboolal and Aleš Pultr)
Preprint 18-25 of DMUC, July 2018.
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.
Entourages, density, Cauchy structure, and completion
(with Aleš Pultr)
Preprint 18-19 of DMUC, May 2018.
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).
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.
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.
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.
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.
Remainders in pointfree topology
(with Maria João Ferreira and Sandra Pinto)
Preprint 17-56 of DMUC, December 2017.
Remainders of subspaces are important e.g. in the realm of compactifications.
Their extension to pointfree topology faces a difficulty: sublocale lattices are more complicated than their topological counterparts. Nevertheless, the co-Heyting structure of sublocale lattices is enough to provide a counterpart to subspace remainders: the sublocale supplements.
In this paper we give an account of their fundamental properties, emphasizing their similarities and differences with classical remainders, and provide several examples and applications to illustrate their scope. In particular, we study their behaviour under image and preimage maps, as well as their preservation by localic maps. We then
use them to characterize nearly realcompact and nearly pseudocompact frames. In addition, we introduce and study hyper-real localic maps.
The other closure and complete sublocales
(with Maria Manuel Clementino and Aleš Pultr)
Preprint 17-40 of DMUC, September 2017.
Sublocales of a locale (frame, generalized space) can
be equivalently represented by frame congruences. In this paper we
discuss, a.o., the sublocales corresponding to complete congruences,
that is, to frame congruences which are closed under arbitrary
meets, and present a "geometric" condition for a sublocale to be
complete. To this end we make use of a certain closure operator
on the coframe of sublocales that allows not only to formulate the
condition but also to analyze certain weak separation properties
akin to subfitness or T1.
Trivially, every open sublocale is complete. We specify a very
wide class of frames, containing all the subfit ones, where there are
no others. In consequence, e.g., in this class of frames, complete
homomorphisms are automatically Heyting.
Joins of closed sublocales
(with Aleš Pultr and Anna Tozzi)
Preprint 16-39 of DMUC, August 2016.
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.