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Frames and Locales - Topology without points
  • J. Picado, A. Pultr
  • Mathematics, Computer Science
  • Frontiers in mathematics
  • 21 October 2011
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
Frames and Locales. Spectra
Is it worthwhile to try to base topology on the algebra of open sets (and to neglect points)? We hope that this question has been answered in the previous chapter satisfactorily by showing that thereExpand
The quantale of Galois connections
Abstract.Galois connections were originally expressed in a contravariant form with transformations that reverse (rather than preserve) order. Nowadays its covariant form (as residuated maps) is moreExpand
Uniform-type structures on lattice-valued spaces and frames
It is proved that this general framework for uniform structures on very general L-valued spaces captures a significant class of Hutton's uniform spaces when restricted to Girard quantales. Expand
(Sub)fit biframes and non-symmetric nearness
Abstract The non-symmetric (quasi-)nearness and its generalized admissibility are studied both in its biframe and paircovers aspect and in the perspective of entourages. The necessary and sufficientExpand
On the parallel between normality and extremal disconnectedness
Abstract Several familiar results about normal and extremally disconnected (classical or pointfree) spaces shape the idea that the two notions are somehow dual to each other and can therefore beExpand
On an aspect of scatteredness in the point-free setting
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,Expand
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
Lower and upper regularizations of frame semicontinuous real functions
Abstract.As discovered recently, Li and Wang’s 1997 treatment of semicontinuity for frames does not faithfully reflect the classical concept. In this paper we continue our study of semicontinuity inExpand
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