Authors: Lowen, R.¹; Wuyts, P.²

Source: Applied Categorical Structures, Volume 19, Number 2, April 2011 , pp. 539-555(17)

Publisher: Springer

Abstract:

In Lowen and Wuyts (Appl Categ Struct 8:235-245, 2000) the authors studied the simultaneously concretely reflective and concretely coreflective subconstructs of the category <Emphasis FontCategory="SansSerif">Ap</Emphasis> of approach spaces. For the sake of shortness we call such subconstructs stable. Using a technique introduced in Herrlich and Lowen (1999) it was possible to explicitly describe such stable subconstructs by a condition on the objects which used certain subsets of [0, ∞ ]. Thus each stable subconstruct <Emphasis FontCategory="SansSerif">Ap</Emphasis> _m described in [9] corresponds to the subset {0} ∪ [m, ∞ ] ⊂ [0, ∞ ] for m ∈ [0, ∞ ]. Although this characterization is correct, Theorem 4.7 in [9] stating that the subconstructs <Emphasis FontCategory="SansSerif">Ap</Emphasis> _m were the only stable subconstructs of <Emphasis FontCategory="SansSerif">Ap</Emphasis> is not. The main results, which together prove that the only stable subconstructs are those where a restriction is put on the range of the distances of the objects, are upheld, but it turns out that not only the sets {0} ∪ [m, ∞ ], but actually each closed subsemigroup of [0, ∞ ] determines a stable subconstruct (albeit again in exactly the same way as characterized in [9]). In the first part of our paper, Sections 1 and 2, we develop the general technique, which is totally different to the one from [3], and in Theorem 2.13 we prove the main result for the case of approach spaces. The technique which we develop is also applicable to other cases. Thus, in Section 3, more precisely in Theorems 3.9 and 3.11, we give the complete solution to the corresponding characterization problem for the constructs pq <Emphasis FontCategory="SansSerif">Met</Emphasis> ^∞ of pseudo-quasi-metric spaces and p <Emphasis FontCategory="SansSerif">Met</Emphasis> ^∞ of pseudometric spaces and in Section 4 we briefly sketch how the technique can be adapted and used to also completely solve the problem in the case of more general types of approach spaces and metric spaces. At the same time, in all cases, we are able to give necessary and sufficient conditions under which two stable subconstructs of one of these topological constructs are concretely isomorphic. It turns out that in all cases there are <EquationSource Format="TEX">$2^{aleph_0}$</EquationSource> non-concretely isomorphic stable subconstructs.

Keywords: Approach space; Pseudo quasi metric space; Topological space; Pretopological space; Reflective; Coreflective; Semigroup; Lattice; 18B99; 54A99; 54B30; 54E35; 54E99

Document Type: Research article

DOI: http://dx.doi.org/10.1007/s10485-008-9184-x

Affiliations: 1: Department of Mathematics and Computer Science, University of Antwerp, Middelheimlaan 1, Antwerp, 2020, Belgium, Email: bob.lowen@ua.ac.be 2: Department of Mathematics and Computer Science, University of Antwerp, Middelheimlaan 1, Antwerp, 2020, Belgium

Publication date: 2011-04-01

Related content

• In this: publication
• By this: publisher
• In this Subject: Mathematics and Statistics
• By this author: Lowen, R. ;  Wuyts, P.

Website © Publishing Technology. Article copyright remains with the publisher, society or author(s) as specified within the article.

• Terms and conditions
• Privacy policy
• Information for advertisers