Anna Tozzi

TOPOLOGICAL CATEGORIES AND CLOSURE OPERATORS

Abstract It is shown that the category CS of closure spaces is a topological category. For each epireflective subcategory A of a topological category X a functor F A :X → X is defined and used to extend to the general case of topological categories some results given in [4], [5] and [10] for epireflective subcategories of the category Top of topological spaces.

Separation Axioms and Frame Representation of Some Topological Facts

Similarly as the sobriety is essential for representing continuous maps as frame homo-morphisms, also other separation axioms play a basic role in expressing topological phenomena in frame language. In particular,TD is equivalent with the correctness of viewing subspaces as sublocates, or with representability of open or closed maps as open or closed homomorphisms. A weaker separation axiom is equivalent with an algebraic recognizability whether the intersection of a system of open sets remains open or not. The role of sobriety is also being analyzed in some detail.

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.

ON EPIDENSE SUBCATEGORIES OF TOPOLOGICAL CATEGORIES

Dense subcategories were introduced by S. Mardesic for an inverse system approach to (categorical) shape theory. In this paper some internal characterizations of (epi,bi)dense subcategories of a topological category are given. We also show that if K ⊂ A is a bidense subcategory then the “best approximation” of an A-object X by a K-inverse system is obtained by “modifications” of the structure of X.

Ideals in Heyting semilattices and open homomorphisms

Subfitness and its relation to openness and completeness is studied in the context of Heyting semilattices. A formally weaker condition (c-subfitness) is shown to be necessary and sufficient for openness and completeness to coincide. For a large class of spatial frames, c-subfit ≡ subfit.

Generalized reflective cum coreflective classes in Top and Unif

The Herrlichs problem from [8] whether there are nontrivial classes of topological spaces that are both almost reflective or injective and almost coreflective or projective, is investigated in a more general setting using cone and cocone modifications of the classes used in the problem. We look also at the problem for uniform spaces. Typical results: There is no nontrivial multiprojective and orthogonal class of topological spaces; There is a reflective class of uniform spaces that is almost coreflective in Unif.

Some Categorical Aspects of Information Systems and Domains

Categories based on the Vickerss continuous information systems and the related categories of continuous domains (algebraic domains, Scott domains, continuous lattices etc.) are shown to be both Kleisli and Eilenberg–Moore categories of a monad of ideals. Further, the functor of ideals is shown to be a completion in the sense of Brümmer, Giuli and Herrlich.

On factorization structures and epidense hulls

Abstract Characterizations of epidense subcategories of topological categories and of existence of epidense hulls have been described in [2, 3, 4]. In this paper a similar characterization is given in a much more general setting; for example the category need not have products. The relationship between finite factorization structures and existence of epidense hulls is investigated. It is found to be analogous to the relationship between general factorization structures and epireflective hulls.

Regular Monomorphisms of Hausdorff Frames

Regular monomorphisms in the category of Hausdorff frames are characterized by means of a naturally defined closure operator; this is used also to characterize the epimorphisms. Further it is shown that for spatial (strongly) Hausdorff frames the regular monomorphisms do not generally coincide with the quotients, and do not generally compose. Also, an additional property (under which regular monomorphisms do compose) is briefly studied.

A Generalization of Herrlich's Question on Almost Reflective and Coreflective Subclasses of Top and Unif

H. Herrlich asked in Topology Appl.49 (1993), 251–264, whether there are nontrivial classes of topological spaces that are almost reflective and almost coreflective at the same time. This question was dealt with (in Hušek and Tozzi, Appl. Categ. Structures4 (1996), 57–68) in a more general setting than almost reflective and almost coreflective classes. The present paper investigates a modified question: when a nontrivial generalized reflective class of topological or uniform spaces is equivalent to a generalized coreflective class of spaces.