On an aspect of scatteredness in the point-free setting

  title={On an aspect of scatteredness in the point-free setting},
  author={R. Ball and J. Picado and A. Pultr},
  journal={Portugaliae Mathematica},
It is well known that a locale is subfit iff each of its open sublocales is a join of closed ones, and fit iff each of its closed sublocales is a meet of open ones. This formulation, however, exaggerates the parallelism between the behavior of fitness and subfitness. For it can be shown that a locale is fit iff each of its sublocales is a meet of closed ones, but it is not the case that a locale is subfit iff each of its sublocales is a join of closed ones. Thus we are led to take up the very… Expand
Sublocales that are joins of closed ones constitute a frame Sc(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 ofExpand
A Boolean extension of a frame and a representation of discontinuity
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 topologicalExpand
Exact Filters and Joins of Closed Sublocales
We prove, for a general frame, that the sublocales that can be represented as joins of closed ones are, somewhat surprisingly, in a natural one-to-one correspondence with the filters closed underExpand
On densely normal locales
Abstract Arhangel'skii has defined a topological space X to be densely normal if it has a dense subspace Y such that any two disjoint closed subsets of X that are closures in X of some closed subsetsExpand
New Aspects of Subfitness in Frames and Spaces
Another necessary and sufficient condition for subfitness presented is the validity of the meet formula for the Heyting operation, which was so far known only under much stronger conditions. Expand
Some aspects of (non) functoriality of natural discrete covers of locales
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. InExpand
Exact and Strongly Exact Filters
A characteristic of the coframe of meets of open sublocales as the dual to the frame of strongly exact filters is presented. Expand
C-connected frame congruences
We discuss the congruences θ that are connected as elements of the (totally disconnected) congruence frame CL, and show that they are in a one-to-one correspondence with the completely prime elementsExpand
Maximal essential extensions in the context of frames
We show that every frame can be essentially embedded in a Boolean frame, and that this embedding is the maximal essential extension of the frame in the sense that it factors uniquely through anyExpand


Spatial sublocales and essential primes
Abstract In this paper, we characterize those locales L such that every sublocale is spatial. The notion of essential prime is introduced and it is shown that every sublocale of L is spatial iffExpand
Pointfree Aspects of the Td Axiom of Classical Topology
Abstract Abstraction from the condition defining To-spaces leads to the following notion in an arbitrary frame L: a filter F in L is called slicing if it is prime and there exist a, b e L such that aExpand
More on Subfitness and Fitness
Another property, prefitness, is shown to also produce fitness by heredity, in this case in a way usable for classical spaces, which results in a transparent characteristics of fit spaces, and the properties are proved to be independent. Expand
Higher order dissolutions and Boolean coreflections of locales
Abstract If a locale has a Boolean coreflection, then this coreflection can be constructed by iterating the dissolution functor A↦Ad until it stabilizes at a Boolean locale. In this paper weExpand
Frames and Locales - Topology without points
This chapter discusses the structure of localic morphisms in Spaces and lattices of open sets, which describes the construction of frames and locales in the real world. Expand
Separation axioms and frame representation of some topological facts
The role of sobriety is also being analyzed in some detail, as it is essential for representing continuous maps as frame homo-morphisms. Expand
Variants of openness
It is established that the coincidence of the algebraic and topological notion of openness is equivalent to the separation axiomTD for the domain space. Expand
The lattice theoretic part of topological separation properties
In this paper we show that for each n∈{2,3,4,5,} the topological separationproperty Tn can be decomposedwhere C, N2,…,Nn are purely lattice theoretic properties with the expected im-plicationsExpand
Éléments de géométrie algébrique
© Publications mathématiques de l’I.H.É.S., 1965, tous droits réservés. L’accès aux archives de la revue « Publications mathématiques de l’I.H.É.S. » (http://Expand
Separation Axioms Between T0 and T1