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Characterizations of Developable Topological Spaces

Published online by Cambridge University Press:  20 November 2018

J. M. Worrell Jr.  and
H. H. Wicke
J. M. Worrell Jr.
Affiliation:
Sandia Laboratory, Albuquerque
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The class of developable topological spaces, which includes the metrizable spaces, has been fundamentally involved in investigations in point set topology. One example is the remarkable edifice of theorems relating to these spaces constructed by R. L. Moore (13). Another is the role played by the developable property in several metrization theorems, including Alexandroff and Urysohn's original solution of the general metrization problem (1).

This paper presents an anslysis of the concept of developable space in terms of certain more extensive classes of spaces satisfying the first axiom of countability : spaces with a base of countable order and those having what is here called a θ-base. The analysis is given in the characterizations of Theorems 3 and 4 below.

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 17 , 1965 , pp. 820 - 830
Copyright
Copyright © Canadian Mathematical Society 1965

References

1. Alexandroff, P. S. and Urysohn, P., Une condition nécessaire et suffisante pour qu'une classe soit une classe , C. R. Acad. Sci. Paris, 177 (1923), 1274–1276.Google Scholar
2. Alexandroff, P. S. and Urysohn, P., Mémoire sur les espaces topologiques compacts, Verh. Nederl. Akad. Wetensch. Afd. Natuurk. Sect. I, 14, no. 1 (1929), 1–96.Google Scholar
3. Arens, R. and Dugundji, J., Remark on the concept of compactness, Portugal. Math., 9 (1950), 141–143.Google Scholar
4. Arhangel, A.'skiï, Some metrization theorems, Uspehi Mat. Nauk, 18 (1963), no. 5 (113), 139–145 (in Russian).Google Scholar
5. Aronszajn, N., Über die Bogenverknùpfung in topologischen Räumen, Fund. Math., 15 (1930), 228–241.Google Scholar
6. Bing, R. H., Metrization of topological spaces, Can. J. Math., 8 (1951), 175–186.Google Scholar
7. Chittenden, E. W., On the equivalence of écart and voisinage, Trans. Amer. Math. Soc, 18 (1917), 161–166.Google Scholar
8. Fr, M.échet, Sur quelques points du calcul fonctionnel, Rend. Cire. Mat. Palermo, 22 (1906), 1–74.Google Scholar
9. Kelley, J. L., General topology (New York, 1955).Google Scholar
10. König, D., Sur les correspondances multivoques, Fund. Math., 8 (1926), 114–134.Google Scholar
11. Kowalsky, H.-J., Topologische Rdume (Basel, 1961).Google Scholar
12. Michael, E., Point-finite and locally finite coverings, Can. J. Math., 7 (1955), 275–279.Google Scholar
13. Moore, R. L., Foundations of point set theory (Providence, 1962).Google Scholar
14. Nagata, J., On a necessary and sufficient condition of metrizability, J. Inst. Poly tech. Osaka City Univ. Ser. A, 1 (1950), 93–100.Google Scholar
15. Ponomarev, V., Axioms of countability and continuous mappings, Bull. Acad. Polon. Sci. Sér. Sci. Math. Astr. Phys., 8 (1960), 127–133 (in Russian).Google Scholar
16. Smirnov, Yu. M., On metrization of topological spaces, Uspehi Mat. Nauk, 6 (1951), no. 6 (46), 100–111 (Amer. Math. Soc. Transi, no. 91).Google Scholar
17. Urysohn, P., Zum Metrisationsproblem, Math. Ann., 94 (1925), 309–315.Google Scholar