Abstract
In previous papers, two notions of pre-Hausdorff (PreT 2) objects in a topological category were introduced and compared. The main objective of this paper is to show that the full subcategory of PreT 2 objects is a topological category and all of T 0, T 1, and T 2 objects in this topological category are equivalent. Furthermore, the characterizations of pre-Hausdorff objects in the categories of filter convergence spaces, (constant) local filter convergence spaces, and (constant) stack convergence spaces are given and as a consequence, it is shown that these categories are homotopically trivial.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Adamek, J., Herrlich, H., Strecker, G.E.: Abstract and Concrete Categories. Wiley, New York (1990)
Baran, M.: Separation properties. Indian J. Pure Appl. Math. 23, 333–341 (1992)
Baran, M.: Stacks and filters. Doğa Mat. Turk. J. Math. 16, 95–108 (1992)
Baran, M.: The notion of closedness in topological categories. Comment. Math. Univ. Carolin. 34, 383–395 (1993)
Baran, M., Altindis, H.: T 2 -Objects in topological categories. Acta Math. Hungar. 71, 41–48 (1996)
Baran, M.: T 3 and T 4 -Objects in topological categories. Indian J. Pure Appl. Math. 29, 59–69 (1998)
Baran, M.: Completely regular objects and normal objects in topological categories. Acta Math. Hungar. 80, 211–224 (1998)
Dikranjan, D., Tholen, W.: Categorical Structure of Closure Operators. Kluwer Academic, Dordrecht (1995)
Dikranjan, D., Giuli, E.: Epis in categories of convergence spaces. Acta Math. Hungar. 61, 195–201 (1993)
Dikranjan, D., Giuli, E., Tozzi, A.: Topological categories and closure operators. Quaestiones Math. 11, 323–337 (1988)
Herrlich, H., Strecker, G.E.: Category Theory, 2nd ed. Heldermann Verlag, Berlin (1979)
Herrlich, H.: Topological functors. Gen. Topol. Appl. 4, 125–142 (1974)
Johnstone, P.T.: Topos Theory, L.M.S Mathematics Monograph: No. 10. Academic, New York (1977)
Lowen-Colebunders, E.: Function Classes of Cauchy Continuous Maps. Marcel Dekker, New York (1989)
MacLane, S., Moerdijk I: Sheaves in Geometry and Logic. Springer, New York (1992)
Mielke, M.V.: The Interval in algebraic topology. Ill. J. Math. 25, 51–62 (1981)
Mielke, M.V.: Convenient categories for internal singular algebraic topology. Ill. J. Math. 27, 519–534 (1983)
Mielke, M.V.: Separation axioms and geometric realizations. Indian J.Pure Appl. Math. 25, 711–722 (1994)
Mielke, M.V.: Hausdorff separations and decidability. In: Symposium on Categorical Topology, pp. 155–160. University of Cape Town, Rondebosch (1999)
Nel, L.D.: Initially structured categories and cartesian closedness. Canad. J. Math. 27, 1361–1377 (1975)
Preuss, G.: Theory of Topological Structures, An Approach to topological Categories. D. Reidel, Dordrecht (1988)
Schwarz, F.: Connections between convergence and nearness. In: Lecture Notes in Math., vol. 719, pp. 345–354. Springer, New York (1978)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Baran, M. PreT 2 Objects in Topological Categories. Appl Categor Struct 17, 591–602 (2009). https://doi.org/10.1007/s10485-008-9161-4
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10485-008-9161-4