caa a22 4500 445378972 CHVBK 20180317143011.0 cr unu---uuuuu 170323e20110401xx s 000 0 eng 10.1007/s10485-008-9184-x doi (NATIONALLICENCE)springer-10.1007/s10485-008-9184-x Stable Subconstructs: A Correction and New Results [Elektronische Daten] [R. Lowen, P. Wuyts] 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 Ap 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 Ap 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 Ap m were the only stable subconstructs of Ap 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 Met  ∞  of pseudo-quasi-metric spaces and p Met  ∞  of pseudometric spaces and in Section4 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 $2^{\aleph_0}$ non-concretely isomorphic stable subconstructs. Springer Science+Business Media B.V., 2009 Approach space nationallicence Pseudo quasi metric space nationallicence Topological space nationallicence Pretopological space nationallicence Reflective nationallicence Coreflective nationallicence Semigroup nationallicence Lattice nationallicence Lowen R. Department of Mathematics and Computer Science, University of Antwerp, Middelheimlaan 1, 2020, Antwerp, Belgium aut Wuyts P. Department of Mathematics and Computer Science, University of Antwerp, Middelheimlaan 1, 2020, Antwerp, Belgium aut Applied Categorical Structures Springer Netherlands 19/2(2011-04-01), 539-555 0927-2852 19:2<539 2011 19 10485 https://doi.org/10.1007/s10485-008-9184-x text/html Onlinezugriff via DOI 1 research-article jats NATIONALLICENCE 856 40 https://doi.org/10.1007/s10485-008-9184-x text/html Onlinezugriff via DOI NATIONALLICENCE 700 1- Lowen R. Department of Mathematics and Computer Science, University of Antwerp, Middelheimlaan 1, 2020, Antwerp, Belgium aut NATIONALLICENCE 700 1- Wuyts P. Department of Mathematics and Computer Science, University of Antwerp, Middelheimlaan 1, 2020, Antwerp, Belgium aut NATIONALLICENCE 773 0- Applied Categorical Structures Springer Netherlands 19/2(2011-04-01), 539-555 0927-2852 19:2<539 2011 19 10485 Metadata rights reserved Springer special CC-BY-NC licence nationallicence BK010053 XK010053 XK010000 NATIONALLICENCE NATIONALLICENCE NL-springer