The other closure and complete sublocales
(with Maria Manuel Clementino and Aleš Pultr)
Abstract: 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 that respect all joins, and present a "geometric" condition for a sublocale to be complete. To this end we make use of an operator of closure type 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.
A note on density and codensity in normal spaces
(with Javier Gutiérrez García and Tomasz Kubiak)
Abstract: This note points out three elementary observations on density and
codensity in normal spaces: (a) each subspace of a [collectionwise] normal space is [collectionwise] normal
iff each its dense subspace is [collectionwise] normal iff each its open and dense subspace
is [collectionwise] normal; (b) a space is normal iff each its codense subspace is normal;
(c) each bounded continuous real-valued map defined on a codense subset of a
normal space has a continuous extension to the whole space.
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.