A. Pultr

ON THE COMPLETION OF NEARNESS FRAMES

Abstract Several questions naturally arising from the unique existence of the completion of a nearness frame are investigated. In particular, the classical result that completion is a coreflection for uniform frames is extended to a substantially larger class of nearness frames but at the same time shown not to hold in general, and an analogue of this is established for the mere functoriality of the completion. Further, a natural variant of the notion of completion is studied, leading among other things to a completely new coreflection result.

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 a £ F, b e F, and a is covered by 6. This paper deals with various aspects of these slicing filters. As a first step, we present several results about the original td condition. Next, concerning slicing filters, we show they are completely prime and characterize them in various ways. In addition, we prove for the frames £>X of open subsets of a space X that every slicing filter is an open neighbourhood filter U(x) and X is td iff every U(x) is slicing. Further, for TopD and Prm_D the categories of td spaces and their continuous maps, and all frames and those homomorphisms whose associated spectral maps preserve the completely prime elements, respectively, we show that the usual contravariant functors between Top and Frm induce analogous functors here, providing a dual equivalence between TopD and the subcategory of Prm_D given by the To-spatial frames (not coinciding with the spatial ones). In addition, we show that TopD is mono-coreflective in a suitable subcategory of Top. Finally, we provide a comparison between Jo-separation and sobriety showing they may be viewed, in some sense, as mirror images of each other.

A Note on Reconstruction of Spaces and Maps from Lattice Data

Abstract Reconstruction of topological spaces from the lattices Ω(X) of open sets, and of continuous maps from lattice homomorphisms satisfying additional properties (formulated in lattice terms) is discussed. We focus on the question when and how the filters λ(x) = {U | x ε U ε (X)} can be specified by algebraic means.

Entourages, Covers and Localic Groups

Due to the nature of product in the category of locales, the entourage uniformities in the point-free context only mimic the classical Weil approach while the cover (Tukey type) ones can be viewed as an immediate extension. Nevertheless the resulting categories are concretely isomorphic. We present a transparent construction of this isomorphism, and apply it to the natural uniformities of localic groups. In particular we show that localic group homomorphisms are uniform, thus providing natural forgetful functors from the category of localic groups into any of the two categories of uniform locales.

CONCERNING A CATEGORIAL APPROACH TO TOPOLOGICAL AND ALGEBRAIC THEORIES

This chapter presents a survey of some results concerning the relations of algebraic and topological theories. The method used will be categorial as one may often associate with an intuitive notion of a theory an exact notion of a concrete category; for example, with the theory of topological spaces one associate the category of topological spaces and continuous mappings, with the theory of groups the category of groups etc. One kind of relations among theories can be described by the means of full embeddings. If a category can be fully embedded in , then it corresponds to an intuitive meaning that the theory associated with is more general then the one associated with . The chapter presents two kinds of embeddings: full embedding called representation, and full embedding preserving underlying sets and the actual form of the mappings called realization. It may be shown for example that from some point of view the topological categories and the categories of relation differ less than the categories of algebras and categories of relational systems.

Lindelöf tightness and the Dedekind-MacNeille completion of a regular σ-frame

Abstract Tightness is a notion that arose in an attempt to understand the reverse reflection problem: given a monoreflection of a category onto a subcategory, determine which subobjects of an object in the subcategory reflect to it — those which do are termed tight. Thus tightness can be seen as a strong density property. We present an analysis of λ-tightness, tightness with respect to the localic Lindel¨of reflection. Leading to this analysis, we prove that the normal, or Dedekind-MacNeille, completion of a regular σ-frame A is a frame. Moreover, the embedding of A in its normal completion is the Bruns-Lakser injective hull of A in the category of meet semilattices and semilattice homomorphisms. Since every regular σ-frame is the cozero part of a regular Lindel¨of frame, this result points towards λ-tightness. For any regular Lindel¨of frame L, the normal completion of Coz L embeds in L as the sublocale generated by Coz L. Although this completion is clearly contained in every sublocale having the same cozero part as L, we show by example that its cozero part need not be the same as the cozero part as L. We prove that a sublocale S is λ-tight in L iff S has the same cozero part as L. The aforementioned counterexample shows that the completion of Coz L is not always λ-tight in L; on the other hand, we present a large class of locales for which this is the case.

Homomorphisms of complete distributive lattices

The category of all complete distributive lattices and their complete homomorphisms is universal, and this is also true for the category of all complete distributive lattices whose morphisms preserve complete joins, finite meets and an additional nullary operation. A survey of analogous results on algebraic universality of categories based on finitary distributive (0,1)-lattices is included to motivate further questions about categories based on complete distributive lattices.

Extremal Prime Filters and Universality of Some Categories

Abstract Due to the existence of constants, classical topological categories cannot be universal in the sense of containing each concrete category as a full subcategory. In the point-free case, this obstruction vanishes and the question of universality makes sense again. The main problem, namely that as to whether the category of locales and localic morphisms is universal is still open; we prove, however, the universality of the following categories: - pairs (locale, sublocale) with the localic morphisms preserving the distinguished sublocales, - frames with frame homomorphisms reflecting the maximal prime ideals, - Priestley spaces with f-maps preserving the maximal elements.