Horst Herrlich

A concept of nearness

Abstract This paper offers solutions for two problems which have attracted many topologists over the years: 1. (1) It provides a natural and reasonably simple concept of “nearness” which unifies various concepts of “topological structures” in the sense that the category Near of all nearness spaces and nearness preserving maps contains the categories (a) of all topological Ro-spaces and continuous maps, (b) of all uniform spaces and uniformly continuous maps (Weil [34], Turkey [33]), (c) of all proximity spaces and δ-maps (Efremovic [9], Smirnov [28,29]), (d) of all contiguity spaces and contiguity maps (Ivanova and Ivanov [17]) as nicely embedded (either bireflective or bicoreflective) full (!) subcategories. 2. (2) It provides a general method by means of which as many T1-extension of a T1-space can be obtained as might be reasonably expected; namely, all strict extensions (in the sense of Banaschewski [3]).

Factorizations, denseness, separation, and relatively compact objects

Abstract Some of the relationships among the topological notions: ‘Hausdorff’, ‘compact’, ‘perfect’, and ‘closed’ are abstracted to a more general categorical setting, where they are shown to remain intact. An investigation is made of factorization structures (especially for single morphisms) and their relationships to strong limit operators and to the Pumplu¨n-Ro¨hrl Galois correspondence between classes of objects and classes of morphisms in any category. Many examples as well as internal characterizations of Galois-closed classes are provided.

Cartesian closed topological hulls

ABsTRAcr. It is shown in this paper that if a concrete category 9 admits embedding as a full finitely productive subcategory of a cartesian closed topological (CCT) category, then W admits such embedding into a smallest CCT category, its CCT hull. This hull is characterized internally by means of density properties and externally by means of a universal property. The problem is posed of whether every topological category has a CCT hull.

Weak Factorization Systems and Topological Functors

Weak factorization systems, important in homotopy theory, are related to injective objects in comma-categories. Our main result is that full functors and topological functors form a weak factorization system in the category of small categories, and that this is not cofibrantly generated. We also present a weak factorization system on the category of posets which is not cofibrantly generated. No such weak factorization systems were known until recently. This answers an open problem posed by M. Hovey.

Topological improvements of categories of structured sets

Abstract Categories of structured sets often fail to have some desirable properties. They may even fail to have any interesting decently behaved full subcategories. But, under some natural assumptions (and disregarding purely set theoretic problems concerning ‘size’), it is always possible to embed them into nicely behaved topological categories, in particular each such category has: (1) a topological hull (= Mac Neille completion), (2) a certesian closed topological hull (= Antoine-cpmpletion), (3) a hereditary topological hull, (4) a concrete quasitopos hull (= Wyler completion). The purpose of this paper is to discuss these hulls and to provide several illuminating examples.

Compactness and the Axiom of Choice

In the absence of the axiom of choice four versions of compactness (A-, B-, C-, and D-compactness) are investigated. 1. C-compact spaces from the eprireflective hull in Haus of C-compact completely spaces. 2. Equivalent are: (a) the axiom of choice, (b) A-compact = D-compactness. (c) B-compactness = D-compactness, (d) C-compactness = D-compactness and complete regularity, (e) products of spaces with finite topologies are A-compact,> (f) products of A-compact spaces are A-compact , (g) products of D-compact spaces are D-compact, (h) powers X k of 2-point discrete spaces are D-compact, (i) finite products of D-compact spaces are D-compact, (j) finite coproducts of D-compact spaces are D-compact, (k) D-compact Hausdorff spaces form an epireflective subcategory of Haus, (l) spaces with finite topologies are D-compact. 3. Equivalent are: (a) the Boolean prime ideal theorem, (b) A-compactness = B-compactness, (c) A-compactness and complete regularity = C-compactness, (d) products of spaces with finite undelying sets are A-compact, (e) products of A-compact Hausdorff spaces are A-compact, (f) powers X k of 2-point discrete spaces are A-compact, (g) A-compact Hausdorff spaces form an epireflective subcategory of Haus. 4. Equivalent are: (a) either the axiom of choice holds or every ultrafilter is fixed, (b) products of B-compact spaces are B-compact. 5. Equivalent are: (a) Dedekind-finite sets are finite, (b) every set carries some D-compact Hausdorff topology, (c) every T 1 has a T 1 — D-compactification, (d) Alexandroff-compactifications of discrete spaces are D-compact.

Some open categorical problems inTop

In this paper, a pendant to a recent survey paper, the authors discuss several open problems in categorical topology. The emphasis is on topology-oriented problems rather than on more general category-oriented ones. In fact, most problems deal with full subconstructs or superconstructs of the constructTop of topological spaces and continuous maps.

Galois connections categorically

Abstract Four levels of Galois connections are exhibited, starting with the classical one and going via concrete Galois connections to Galois adjunctions.

Are there convenient subcategories of Top

Abstract The category Top has several pleasant properties but fails to have other desirable ones. Consequently there have been various attempts to replace Top by more convenient categories; mostly subcategories or supercategories of Top. Whereas several of the supercategories of Top have extremely pleasant properties, all the subcategories of Top investigated so far have some deficiencies. In the present article it is shown why this is so and why the search for more convenient subcategories of Top has to be in vain. As a preparation in Section 1 nine convenience-properties for topological categories are defined and their mutual relations are analyzed. In Section 2 it is shown that every topological subcategory of Top (under some minor, natural assumptions) it-with respect to each of the 9 mentioned properties-as convenient or inconvenient as Top itself.

The Historical Development of Uniform, Proximal, and Nearness Concepts in Topology

For formulating results or definitions from older papers and books we shall mostly use modern terms and notation to avoid misunderstanding.