Title 



Stable subconstructs: a correction and new results
 
Author 



 
Abstract 



In Lowen and Wuyts (Appl Categ Struct 8:235245, 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 pseudoquasimetric spaces and p Met ∞ 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 nonconcretely isomorphic stable subconstructs.   
Language 



English
 
Source (journal) 



Applied categorical structures.  Dordrecht, 1993, currens  
Publication 



Dordrecht : 2011
 
ISSN 



09272852
 
Volume/pages 



19:2(2011), p. 539555
 
ISI 



000288251400008
 
Full text (Publishers DOI) 


  
