Topological improvements of categories of structured sets

The compact–open topology for function spaces is usually attributed to R. H. Fox in 1945 [16]; and indeed, there is no earlier publication to attribute it to. But it is clear from Fox's paper that the idea of the compact–open topology, and its notable success in locally compact spaces, were already familiar. The topology of course goes back to Riemann; and to generalize to locally compact spaces needs only a definition or two. The actual contributions of Fox were (1) to formulate the partial result, and the problem of extending it, clearly and categorically; (2) to show that in separable metric spaces there is no extension beyond locally compact spaces; (3) to anticipate, partially and somewhat awkwardly, the idea of changing the category so as to save the functorial equation. (Scholarly reservations: Fox attributes the question to Hurewicz, and doubtless Hurewicz had a share in (1). As for (2), when Fox's paper was published R. Arens was completing a dissertation which gave a more general result [1] – though worse formulated.)

General function spaces, products and continuous lattices

This paper develops a unified theory of function spaces Mj^(Y, Z) with set-open topologies, the sets in question being the continuous images of selected classes of topological spaces ^ We prove that at least five of these function spaces are distinct and have corresponding exponential homeomorphisms θ: M^ (X, MSΛY, Z)) ~ M^ (X XS/Y, Z) for suitably retopologized product spaces I X y F , Singleton spaces are normally identities with respect to these products and so we have determined four distinct monoidal closed structures for the category of all spaces. Conditions for the category of spaces generated by sf, i.e., the coreflective hull of J^, to be cartesian closed and/or convenient are given. One result asserts that the category of sequential spaces is the smallest convenient category.

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Monoidal closed, Cartesian closed and convenient categories of topological spaces

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.

Some open categorical problems in Top

Generalizing results of Herrlich and Nel, the author characterizes by means of smallest proper structures those objects X of an initially structured category for which X x—has a right adjoint, and describes the corresponding function spaces. It is shown that reduction to finally and initially dense classes is possible. The results are applied to epireflective subcategories of the category of limit spaces containing a finite non-indiscrete space, in particular to epireflective subcategories of TOP.

Powers and exponential objects in initially structured categories and applications to categories of limit spaces

Conditions are established under which a given class of objects in a topological category will have a Cartesian closed coreflective hull. The main theorem is used to discover new Cartesian closed topological categories and to unify a diversity of known special results. It also provides a mild criterion for the existence of Cartesian closed topological hulls.

Louis Nel

Papers

Cartesian closed coreflective hulls