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Keywords:
Čech complete; strongly Čech complete; nearness space
Čech complete; strongly Čech complete; nearness space
Summary:
We study Čech complete and strongly Čech complete topological spaces, as well as extensions of topological spaces having these properties. Since these two types of completeness are defined by means of covering properties, it is quite natural that they should have a convenient formulation in the setting of nearness spaces and that in that setting these formulations should lead to new insights and results. Our objective here is to give an internal characterization of (and to study) those nearness structures which are induced by topological extensions of the two above mentioned types.
References:
[Be75] Bentley H.L.: Nearness spaces and extensions of topological spaces. Studies in Topology Edited by Nick M. Stavrakas and Keith R. Allen Academic Press New York (1975), 47-66. MR 0380738 | Zbl 0314.54013
[Be76] Bentley H.L.: The role of nearness spaces in topology. Categorical Topology Edited by E. Binz and H. Herrlich, Lecture Notes in Math. 540 Springer Verlag Berlin (1976), 1-22. MR 0448306 | Zbl 0339.54002
[Be77] Bentley H.L.: Normal nearness spaces. Quaestiones Math. 2 (1977), 23-43. MR 0474215 | Zbl 0369.54013
[Be91] Bentley H.L.: Paracompact spaces. Topology and its Appl. 39 (1991), 283-297. MR 1110572 | Zbl 0733.54021
[BH76] Bentley H.L., Herrlich H.: Extensions of topological spaces. Topology Edited by Stanley P. Franklin and Barbara V. Smith Thomas, Lect. Notes in Pure and Appl. Math. 24 Marcel Dekker New York (1976), 129-184. MR 0433392 | Zbl 0345.54027
[BHO89] Bentley H.L., Herrlich H., Ori R.G.: Zero sets and complete regularity for nearness spaces. Categorical Topology Edited by Jiří Adámek and Saunders MacLane, World Scientific Singapore (1989), 446-461. MR 1047917
[BHR76] Bentley H.L., Herrlich H., Robertson W.A.: Convenient categories for topologists. Comm. Math. Univ. Carolinae 17 (1976), 207-227. MR 0425890 | Zbl 0327.54001
[BLC91] Bentley H.L., Lowen-Colebunders E.: Completely regular spaces. Comm. Math. Univ. Carolinae 32 (1991), 129-153. MR 1118297 | Zbl 0763.54016
[Br77] Brandenburg H.: On a class of nearness spaces and the epireflective hull of developable topological spaces. Proc. of the Int. Topology Symp., Belgrade, 1977.
[Ca79] Carlson John W.: H-Closed and countably compact extensions. Pacific J. Math. 81 (1979), 317-326. MR 0547602 | Zbl 0425.54011
[Ca80] Carlson John W.: Developable spaces and nearness structures. Proc. Amer. Math. Soc. 78 (1980), 573-579. MR 0556635 | Zbl 0451.54023
[Ca81] Carlson John W.: Baire space and second category extensions. Topology and its Appl. 12 (1981), 135-140. MR 0612010 | Zbl 0458.54018
[En77] Engelking R.: General Topology. Heldermann Verlag Berlin (1989). MR 1039321 | Zbl 0684.54001
[FL72] Fletcher P., Lindgren W.F.: Orthocompactness and strong Čech completeness in Moore spaces. Duke. Math. Jour. 4 (1972), 753-766. MR 0312472 | Zbl 0251.54013
[Fr60] Frolík Z.: Generalizations of the $G_{\delta}$-property of complete metric spaces. Czech. Math. Jour. 10 (1960), 359-379. MR 0116305
[He74a] Herrlich H.: A concept of nearness. Gen. Topology Appl. 4 (1974), 191-212. MR 0350701 | Zbl 0288.54004
[He74b] Herrlich H.: Topological structures. Topological Structures Math. Centre Tracts 52 (1974), 59-122. MR 0358706 | Zbl 0311.54003
[He83] Herrlich H.: Categorical topology 1971-1981. General Topology and its Relations to Modern Analysis and Algebra V Proceedings of the Fifth Prague Topological Symposium 1981 Heldermann Verlag Berlin (1983), 279-383. MR 0698425 | Zbl 0502.54001
[He88] Herrlich H.: Topologie II: Uniforme Räume. Heldermann Verlag Berlin (1988). MR 0950372 | Zbl 0644.54001
[Ka62] Katětov M.: Allgemeine Stetigkeitsstrukturen. Proc. Inter. Congr. Math. Stockholm 1962 (1963), 473-479. MR 0176431
[Ka65] Katětov M.: On continuity structures and spaces of mappings. Comment. Math. Univ. Carol. 6 (1965), 257-278. MR 0193608
[LC89] Lowen-Colebunders E.: Function Classes of Cauchy Continuous Maps. (1989), Marcel Dekker, Inc. New York. MR 1001563 | Zbl 0697.54001
[Ro75] Robertson W.A.: Convergence as a Nearness Concept. Thesis Carlton University Ottawa (1975). MR 2625906
[Ša43] Šanin N.A.: On separation in topological spaces. Dokl. Akad. Nauk SSSR 38 (1943), 110-113. MR 0008688
[Tu40] Tukey J.W.: Convergence and uniformity in topology. Ann. Math. Studies 2 (1940), Princeton Univ. Press Princeton. MR 0002515 | Zbl 0025.09102
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