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.
Tensor products and relation quantales
(with Marcel Erné)
Preprint 16-50 of DMUC, December 2016.
Abstract: A classical tensor product A⊗B of complete lattices A and B, consisting
of all down-sets in A×B that are join-closed in either coordinate, is isomorphic to the
complete lattice Gal(A,B) of Galois maps from A to B, turning arbitrary joins into
meets. We introduce more general kinds of tensor products for closure spaces and for
posets. They have the expected universal property for bimorphisms (separately continuous
maps or maps preserving restricted joins in the two components) into complete
lattices. The appropriate ingredient for quantale constructions is here distributivity
at the bottom, a generalization of pseudocomplementedness. We show that the truncated
tensor product of a complete lattice B with itself becomes a quantale with the
closure of the relation product as multiplication iff B is pseudocomplemented, and
the tensor product has a unit element iff B is atomistic. The pseudocomplemented
complete lattices form a semicategory in which the hom-set between two objects is
their tensor product. The largest subcategory of that semicategory has as objects
the atomic boolean complete lattices, which is equivalent to the category of sets and
relations. More general results are obtained for closure spaces and posets.
A Boolean extension of a frame and a representation of discontinuity
(with Aleš Pultr)
Preprint 16-46 of DMUC, November 2016.
Abstract: Point-free modeling of mappings that are not necessarily continuous
has been so far based on the extension of a frame to its frame of sublocales, mimicking
the replacement of a topological space by its discretization. This otherwise
successful procedure has, however, certain disadvantages making it not quite parallel
with the classical theory (see Introduction). We mend it in this paper using a
certain extension Sc(L) of a frame L, which is, a.o., Boolean and idempotent. Doing
this we do not loose the merits of the previous approach. In particular we show
that it yields the desired results in the treatment of semicontinuity. Also, there is
no obstacle to use it as a basis of a point-free theory of rings of real functions; the
“ring of all real functions” F(L) = C(Sc(L)) is now order complete.
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.