A Kuratowski–Mrówka theorem in approach theory

Abstract

In this paper we give a number of arguments why, in approach theory, the notion of compactness which from the intrinsic categorical point of view seems most satisfying is 0-compactness, i.e., measure of compactness equal to zero. It was already known from [R. Lowen, Kuratowski's measure of noncompactness revisited, Quart. J. Math. Oxford 39 (1988) 235-254] that measure of compactness has good properties and good interpretations for both topological and metric approach spaces. Here, introducing notions of closed and proper mappings in approach theory, which satisfy all the intrinsic categorical axioms put forth in [Clementino et al., A functional approach to topology, in: M.C. Pedicchio, W. Tholen (Eds.) Categorical Foundations Special Topics in Order, Topology, Algebra, and Sheaf Theory, Cambridge University Press, 2003], we prove fundamental results concerning these concepts, also linked to 0-compactness, and we give a Kuratowski-Mrówka-type characterization of 0-compactness.

Approach Theory in a Category: A Study of Compactness and Hausdorff Separation

“…One can show in exactly the same way as in the paper [8] that this notion of 0-compactness is productive in Prap. Moreover, using the same techniques as in this paper, one obtains that the pre-approach space (X, δ) is 0-compact iff for any pre-approach space (Z , δ ), the projection p Z :…”

“…One can prove the following result in Prap in exactly the same way as it was done earlier in Ap [8].…”

“…But in the more general approach in [6] there need not be an a priori given closure operator. A concrete example of such a more general situation was worked out in [8] for the construct Ap of approach spaces with contractions, where it was shown that 0-compactness of approach spaces coincides with the categorical notion of compactness related to the class of closed contractions.…”

Tensor Functional Topology on Woronowicz Categories

“…After this idea originated in the seventies in work of Herrlich, Manes and Penon [10], [18], [21] [20], since the nineties the subject received a lot of attention in work of Herrlich, Salicrup, Strecker, Clementino, Giuli, Tholen, Hofmann and others with applications to Birkhoff closure spaces, uniform spaces, topological groups, locales, approach spaces, lax algebras and schemes [12], [4], [22] [5], [6], [13], [16]. In this section, we adapt some of these approaches to a monoidal context, or, more generally, to the context of a category endowed with a relation R in the sense of Definition 4.20.…”

“…In the 90's, on categories endowed with closure operators, a close resemblance with the topological situation was obtained by Clementino, Giuli and Tholen [4], applicable to Birkhoff closure spaces, uniform spaces, topological groups and locales. The idea of using a class of closed morphisms in order to derive both proper and separated morphisms from it was contained in Tholen's [22], and gave rise to the theory of functional topology as developed by Clementino, Giuli and Tholen in the presence of a factorization system [5], which is applicable for instance to approach spaces as demonstrated by Colebunders, Lowen and Wuyts [6]. An approach by Hofmann and Tholen later focussed the attention on the proper maps as primary, and is applicable to general categories of lax algebras [13].…”