Abstract
In this paper we introduce a new topological-type of structured set called merotopological space. The appropriate morphisms are defined and characterizations of the corresponding initial and final structures are given. The resulting category contains as fully embedded subcategories not only the category of topological spaces and continuous maps but also the category of merotopic spaces and uniformly continuous maps, and, a fortiori, the category of nearness spaces and the category of uniform spaces. A functorial completion is constructed for merotopological spaces using bunches. A problem that has remained long open in the setting of nearness spaces is to find an internal characterization of the epireflective hull of the topological spaces. We solve the analogue of this problem in the setting of merotopological spaces. Applications to the Wyler prime closed filter compactification and to Taimanov's extension theorem are given.
Similar content being viewed by others
References
Adámek, J., Herrlich, H. and Strecker, G. E.: Abstract and Concrete Categories, Wiley, New York, 1990.
Bentley, H. L.: Nearness spaces and extensions of topological spaces, in Studies in Topology, Academic Press, New York, 1975, pp. 47–66.
Bentley, H. L.: Normal nearness spaces, Quaestiones Math. 2 (1977), 23–43 and 513.
Bentley, H. L.: Extensions of maps from dense subspaces, in J. Koslowski and A. Melton (eds), Categorical Perspectives, Birkhäuser, Boston, 2001, pp. 151–176.
Bentley, H. L. and Herrlich, H.: Extensions of topological spaces, in Topology, Proc. Memphis Conf. 1975, Marcel Dekker, 1976, pp. 129–184.
Bentley, H. L. and Herrlich, H.: The reals and the reals, General Topology Appl. 9 (1978), 221–232.
Bentley, H. L. and Herrlich, H.: Completeness for nearness spaces, in Topological Structures II, Part 1, Math. Centre Tracts 115, Amsterdam, 1979, pp. 29–40.
Bentley, H. L. and Herrlich, H.: Completeness is productive, in Categorical Topology, Lecture Notes in Mathematics 719, Springer-Verlag, Berlin, 1979, pp. 13–17.
Bentley, H. L. and Herrlich, H.: Morita-extensions and nearness-completions, Topology Appl. 82 (1998), 59–65.
Bentley, H. L., Herrlich, H. and Hušek, M.: The historical development of uniform, proximal, and nearness concepts in topology, A chapter in C. E. Aull and R. Lowen (eds), Handbook of the History of General Topology, Vol. II, Kluwer Academic Publishers, Dordrecht, 1998, pp. 577–629.
Bentley, H. L. and Herrlich, H.: Characterization of the prime closed filter compactification of a topological space, Quaestiones Math. 25 (2002), 1–23.
Bentley, H. L. and Herrlich, H.: On the strict completion of a nearness space, in Proceedings of the 8th Prague Topological Symposium 1996, Electronically published at: http//at.yorku.ca/p/p/a/a/00.htm, 1999, pp. 11–21. Also published in hard form in Quaestiones Math. 24 (2001), 39–50.
Bentley, H. L., Herrlich, H. and Robertson, W. A.: Convenient categories for topologists, Comment. Math. Univ. Carolin. 17 (1976), 207–227.
Dowker, C. H.: Homology groups of relations, Ann. of Math. 56 (1952), 84–95.
Engelking, R.: General Topology, Heldermann Verlag, Berlin, 1989.
Harris, D.: Structures in Topology, Memoirs of the Mathematical Society # 115, Amer. Math. Soc., Providence, 1971.
Hastings, M.: Epireflective hulls in Near, Ph.D. thesis, University of Toledo, 1975.
Heldermann, N. C.: Concentrated nearness spaces, in H. Herrlich and G. Preuss (eds), Categorical Topology, Proceedings Berlin 1978, Springer-Verlag, Berlin, 1979, pp. 122–136.
Herrlich, H.: A concept of nearness, General Topology Appl. 4 (1974), 191–212.
Herrlich, H.: On the extendibility of continuous functions, General Topology Appl. 4 (1974), 213–215.
Herrlich, H.: Topological structures, in Topological Structures I, Symposium in Honour of J. de Groot 1914–1972, Amsterdam 1973, Mathematisch Centrum, 1974, pp. 59–122.
Herrlich, H.: Topological functors, General Topology Appl. 4 (1974), 125–142.
Herrlich, H.: Categorical topology 1971–1981, in J. Novák (ed.), General Topology and its Relations to Modern Analysis and Algebra V, Proceedings of the Fifth Prague Topological Symposium 1981, Heldermann Verlag, Berlin, 1983, pp. 279–383.
Herrlich, H.: Topologie II: Uniforme Räume, Heldermann Verlag, Berlin, 1988.
Ivanova, V. M. and Ivanov, A. A.: Contiguity spaces and bicompact extensions of topological spaces, Dokl. Akad. Nauk SSSR 127 (1959), 20–22.
Katětov, M.: Allgemeine Stetigkeitsstrukturen, in Proceedings of the International Congress of Mathematics Stockholm 1962, 1963, pp. 473–479.
Katětov, M.: On continuity structures and spaces of mappings, Comment. Math. Univ. Carolin. 6 (1965), 257–278.
Robertson, W.: Convergence as a nearness concept, Ph.D. thesis, Carleton University, 1975.
Taimanov, A. D.: On extension of continuous mappings of topological spaces, Math. Sbornik N.S. 31(73) (1952), 459–463.
Wyler, O: Compact ordered spaces and prime Wallman compactifications, in H. L. Bentley, H. Herrlich, M. Rajagopalan, and H. Wolff (eds), Categorical Topology, Proceedings of the International Conference held at the University of Toledo, Toledo, Ohio, USA August 1–5, 1983, Heldermann Verlag, Berlin, 1984, pp. 618–635.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Bentley, H.L., Herrlich, H. Merotopological Spaces. Applied Categorical Structures 12, 155–180 (2004). https://doi.org/10.1023/B:APCS.0000018143.05864.92
Issue Date:
DOI: https://doi.org/10.1023/B:APCS.0000018143.05864.92