Hereditary, Additive and Divisible Classes in Epireflective Subcategories of Top
Tóm tắt
Hereditary coreflective subcategories of an epireflective subcategory A of Top such that I
2 ∉ A (here I
2 is the two-point indiscrete space) were studied in [4]. It was shown that a coreflective subcategory B of A is hereditary (closed under the formation subspaces) if and only if it is closed under the formation of prime factors. The main problem studied in this paper is the question whether this claim remains true if we study the (more general) subcategories of A which are closed under topological sums and quotients in A instead of the coreflective subcategories of A. We show that this is true if A ⊆ Haus or under some reasonable conditions on B. E.g., this holds if B contains either a prime space, or a space which is not locally connected, or a totally disconnected space or a non-discrete Hausdorff space. We touch also other questions related to such subclasses of A. We introduce a method extending the results from the case of non-bireflective subcategories (which was studied in [4]) to arbitrary epireflective subcategories of Top. We also prove some new facts about the lattice of coreflective subcategories of Top and ZD.
Tài liệu tham khảo
Adámek, J., Herrlich, H., Strecker, G.: Abstract and Concrete Categories. Wiley, New York (1990) (Available online at http://katmat.math.uni-bremen.de/acc/acc.pdf)
Arhangel’ski\(\breve{{\text{i}}}\), A.V., Franklin, S.P: Ordinal invariants for topological spaces. Michigan Math. J. 15, 313–320 (1968)
Činčura, J.: Heredity and coreflective subcategories of the category of topological spaces. Appl. Categ. Structures 9, 131–138 (2001)
Činčura, J.: Hereditary coreflective subcategories of categories of topological spaces. Appl. Categ. Structures 13, 329–342 (2005)
Cook, H.: Continua which admit only the identity mapping onto non-degenerate subcontinua. Fund. Math. 60, 241–249 (1967)
Engelking, R.: General Topology. PWN, Warsaw (1977)
Giuli, E., Hušek, M.: A counterpart of compactness. Boll. Unione Mat. Ital., VII. Ser., B. 11(3), 605–621 (1997)
de Groot, J.: Groups represented by homeomorphism groups. I. Math. Ann. 138, 80–102 (1959)
Herrlich, H.: Topologische Reflexionen und Coreflexionen, Lecture Notes in Math 78. Springer, Berlin Heidelberg New York (1968)
Herrlich, H.: Limit-operators and topological coreflections. Trans. Amer. Math. Soc. 146, 203–210 (1969)
Herrlich, H., Hušek, M.: Some open categorical problems in Top. Appl. Categ. Structures 1, 1–19 (1993)
Herrlich, H., Strecker, G.: Category Theory. Heldermann, Berlin (1979)
Hušek, M.: Products of quotients and of k′-spaces. Comment. Math. Univ. Carolin. 12, 61–68 (1971)
Juhász, I.: Cardinal Functions in Topology – Ten Years Later. Math. Centre Tracts, Amsterdam (1980)
Kannan, V.: Pairs of topologies with same family of continuous self-maps. In: Herrlich, H., Preuss, G. (eds.) Categorical Topology, Proceedings of the International Conference, pp. 176–184. Springer, Berlin Heidelberg New York (1979)
Kannan, V.: Ordinal invariants in topology. Mem. Amer. Math. Soc. 245 (1981)
Kannan, V., Rajagopalan, M.: Constructions and applications of rigid spaces. I. Adv. Math. 29, 89–130 (1978)
Lukács, G.: A convenient subcategory of Tych. Appl. Categ. Structures 12, 263–267 (2004)
Marny, T.: On epireflective subcategories of topological categories. Gen. Topol. Appl. 10, 175–181 (1979)
Michael, E.: On k-spaces and k R -spaces and k(X). Pacific J. Math. 47(2), 487–498 (1973)
Nakagawa, R.: Categorial topology. In: Morita, K., Nagata, J.I. (eds.) Topics in General Topology, pp. 563–623. North-Holland, Amsterdam (1989)
Sleziak, M.: Subspaces of pseudoradial spaces. Math. Slovaca 53, 505–514 (2003)
Sleziak, M.: On hereditary coreflective subcategories of Top. Appl. Categ. Structures 12, 301–317 (2004)
Todorčević, S., Uzcátegui, C.: Analytic k-spaces. Topology Appl. 146-147, 511–526 (2005)
Willard, S.: General Topology. Addison-Wesley, Massachussets (1970)