# Article

Full entry | PDF   (0.2 MB)
Keywords:
fibrewise topology; continuous map; closed map; paracompact map
Summary:
In this paper we continue with the study of paracompact maps introduced in [1]. We give two external characterizations for paracompact maps including a characterization analogous to The Tamano Theorem in the category \$\Cal TOP\$ (of topological spaces and continuous maps as morphisms). A necessary and sufficient condition for the Tychonoff product of a closed map and a compact map to be closed is also given.
References:
[1] Buhagiar D.: Paracompact maps. Questions Answers Gen. Topology 15 (1997), 2 203-223. MR 1472184 | Zbl 0997.54012
[2] Buhagiar D.: The category \${\mathscr{MAP}}\$. submitted for publication, 1998.
[3] Buhagiar D., Miwa T.: Covering properties on maps. Questions Answers Gen. Topology 16 (1998), 1 53-66. MR 1614753 | Zbl 0997.54033
[4] Buhagiar D., Miwa T., Pasynkov B.A.: On metrizable type \$({MT}\$-\$)\$ maps and spaces. to appear in Topology Appl. MR 1701238 | Zbl 0953.54017
[5] Dieudonné J.: Une généralisation des espaces compacts. J. de Math. Pures et Appl. 23 (1944), 65-76. MR 0013297
[6] Engelking R.: General Topology. revised ed., Heldermann, Berlin, 1989. MR 1039321 | Zbl 0684.54001
[7] James I.M.: Spaces. Bull. London Math. Soc. 18 (1986), 529-559. MR 0859946 | Zbl 0647.05005
[8] James I.M.: Fibrewise Topology. Cambridge Univ. Press, Cambridge, 1989. MR 1010230 | Zbl 1147.55001
[9] Johnstone P.T.: The Gleason cover of a topos II. J. Pure Appl. Algebra 22 (1981), 229-247. MR 0629332 | Zbl 0445.18005
[10] Johnstone P.T.: Wallman compactification of locales. Houston J. Math. 10 (1984), 201-206. MR 0744904 | Zbl 0549.54018
[11] Künzi H.P.A., Pasynkov B.A.: Tychonoff compactifications and R-completions of mappings and rings of continuous functions. Categorical Topology (L'Aquila, 1994), Kluwer Acad. Publ., Dordrecht, 1996, pp.175-201. MR 1412584
[12] Lever D.: Continuous families: Categorical aspects. Cahiers de topologie et géométrie différentielle 24 (1983), 393-432. MR 0749470 | Zbl 0535.18004
[13] Lever D.: Relative topology. Categorical Topology, Proc. Conference Toledo, Ohio, 1983, Heldermann, Berlin, 1984. MR 0785023 | Zbl 0555.18006
[14] Morita K.: Paracompactness and product spaces. Fund. Math. 50 (1962), 223-236. MR 0132525 | Zbl 0099.17401
[15] Pasynkov B.A.: On extension to mappings of certain notions and assertions concerning spaces (in Russian). Mappings and Functors, Izdat. MGU, Moscow, 1984, pp.72-102. MR 0791256
[16] Pasynkov B.A.: Elements of the general topology of continuous maps (in Russian). On Compactness and Completeness Properties of Topological Spaces, ``FAN'' Acad. of Science of the Uzbek. Rep., Tashkent, 1994, pp.50-120.
[17] Preuss G.: Theory of topological structures (an approach to categorical topology). Mathematics and its Applications, D.Reidel Publishing Company, Dordrecht, Holland, 1987. MR 0937052 | Zbl 0649.54001
[18] Tamano H.: On paracompactness. Pacific J. Math. 10 (1960), 1043-1047. MR 0124876 | Zbl 0094.35403
[19] Tamano H.: On compactifications. J. Math. Kyoto Univ. 1 (1962), 161-193. MR 0142096 | Zbl 0106.15601
[20] Ulyanov V.M.: On compact extensions of countable character and absolutes (in Russian). Math. Sb. 98 (1975), 2 223-254. MR 0394534
[21] Vainstein I.A.: On closed maps of metric spaces (in Russian). Doklady Akad. Nauk SSSR 57 (1947), 319-321. MR 0022067
[22] Whyburn G.T.: A unified space for mappings. Trans. Amer. Math. Soc. 74 (1953), 2 344-350. MR 0052762 | Zbl 0053.12303
[23] Whyburn G.T.: Compactification of mappings. Math. Ann. 166 (1966), 1 168-174. MR 0200905 | Zbl 0141.20505

Partner of