On classes of T0 spaces admitting completions
DOI:
https://doi.org/10.4995/agt.2003.2016Keywords:
Affine set, T0, Sober and injective space, Compact space, Completion, Zariski closure, topological category, Coreflective subcategoryAbstract
For a given class X of T0 spaces the existence of a subclass C, having the same properties that the class of complete metric spaces has in the class of all metric spaces and non-expansive maps, is investigated. A positive example is the class of all T0 spaces, with C the class of sober T0 spaces, and a negative example is the class of Tychonoff spaces. We prove that X has the previous property (i.e., admits completions) whenever it is the class of T0 spaces of an hereditary coreflective subcategory of a suitable supercategory of the category Top of topological spaces. Two classes of examples are provided.
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References
J. Adamek, H. Herrlich and G. Strecker, Abstract and Concrete Categories (Wiley and Sons Inc., 1990).
S. Baron, Note on epi in T0, Canad. Math. Bull. 11(1968), 503-504. http://dx.doi.org/10.4153/CMB-1968-061-6
L. M. Brown and M. Diker, Ditopological texture spaces and intuitionistic sets, Fuzzy Sets and Systems 98 (1998), 217-224. http://dx.doi.org/10.1016/S0165-0114(97)00358-8
L. M. Brown, R. Erturk and S . Dost, Ditopological texture spaces and fuzzy topology. I. General concepts. Preprint, 2002.
G. C. L. Brümmer and E. Giuli, A categorical concept of completion, Comment. Math. Univ. Carolin. 33 (1992), 131-147.
G. C. L. Brümmer, E. Giuli and H. Herrlich, Epireflections which are completions, Cahiers Topologie Géom. Diff. Catég. 33 (1992), 71-93.
M. M. Clementino, E. Giuli and W. Tholen, Topology in a category: compactness, Portugal. Math. 53 (1996), 397-433.
M. M. Clementino and W. Tholen, Separation versus connectedness, Topology Appl. 75 (1997), 143-179. http://dx.doi.org/10.1016/S0166-8641(96)00087-9
D. Deses, E. Giuli and E. Lowen-Colebunders, On complete objects in the category of T0 closure spaces, Applied Gen. Topology, 4 (2003), 25-34.
D. Dikranjan and E. Giuli, Closure operators. I. Topology Appl. 27 (1987), 129-143. http://dx.doi.org/10.1016/0166-8641(87)90100-3
D. Dikranjan, E. Giuli and A. Tozzi, Topological categories and closure operators, Quaestiones Math. 11 (1988), 323-337. http://dx.doi.org/10.1080/16073606.1988.9632148
D. Dikranjan and W. Tholen Categorical structure of closure operators (Kluwer Academic Publishers, Dordrecht, 1995).
Y. Diers, Affine algebraic sets relative to an algebraic theory , J. Geom. 65 (1999), 54-76. http://dx.doi.org/10.1007/BF01228678
R. Engelking, General Topology, (Heldermann Verlag, Berlin 1988).
E. Giuli, Zariski closure, completeness and compactness, Mathematik-Arbeitspapiere (Univ. Bremen) 54 (2000), 207-216.
E. Giuli and M. Husek, A counterpart of compactness, Boll. Un. Mat. Ital.(7) 11-B (1997), 605-621.
H. Herrlich, Topological functors, Gen. Topology Appl. 4 (1974), 125-142. http://dx.doi.org/10.1016/0016-660X(74)90016-6
Th. Marny, On epireflective subcategories of topological categories, Gen. Topology Appl. 10 (1979), 175-181. http://dx.doi.org/10.1016/0016-660X(79)90006-0
W. Pratt, Chu spaces and their interpretation as concurrent objects (Springer Lecture Notes in Computer Science 1000 1995), 392-405. http://dx.doi.org/10.1007/BFb0015256
G. Preuss, Theory of Topological Structures (D. Reidel Publishing Company, 1988). http://dx.doi.org/10.1007/978-94-009-2859-6
L. Skula, On a reflective subcategory of the category of all topological spaces, Trans. Amer. Math. Soc. 142 (1969), 137-141.
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