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A Metrization Theorem for 2-Manifolds

Published online by Cambridge University Press:  20 November 2018

Paul A. Vincent
Paul A. Vincent*
Affiliation:
University of Moncton, Moncton, N.B., Canada E1A 3E9
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There are few known metrization theorems for manifolds (locally Euclidean, connected, Hausdorff space). It is well known that for manifolds metrizability, second countability, Lindelöf's condition, σ-compactness and paracompactness are equivalent. Although these conditions imply separability, the latter does not imply any of the former (see Example 2.2), as is often believed. A common source of metrization for a covering manifold is that lifted from the base manifold [8; p. 181].

For 2-manifolds, the presence of a complex analytic structure gives us a metrization theorem; it has been shown [1] that such manifolds are topologically characterized as those which are orientable and second countable.

Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 19 , Issue 1 , 01 March 1976 , pp. 95 - 104
Copyright
Copyright © Canadian Mathematical Society 1976

Footnotes

(1)

This paper represents a chapter of the author's dissertation, submitted to the University of Rochester in partial fulfillment of the requirements for the degree of Doctor of Philosophy. The author wishes to express his gratitude to Professor Gail S. Young who directed his dissertation.

References

1. Ahlfors, L. V. and Sario, L., Riemann Surfaces, Princeton Univ. Press, Princeton, N.J., 1960.Google Scholar
2. Calabi, E. and Rosenlicht, M., Complex analytic manifolds without countable base, Proc. Amer. Math. Soc. 4 (1953), 335–340.Google Scholar
3. Cannon, R. J. Quasiconformal structures and the metrization of 2-manifolds, Trans. Amer. Math. Soc, Vol. 135(1969), 95–103.Google Scholar
4. Jenkins, J., Univalent Functions and Conformai Mapping, Springer-Verlag, Berlin, 1958.Google Scholar
5. Jenkins, J. and Morse, M., The existence of pseudoconjugates on Riemann surfaces, Fund. Math. 39 (1952) 269–287.Google Scholar
6. Jenkins, J. and Morse, M., Topological methods on Riemann surfaces, Annals of Math. Studies, No. 30, 111–139. Princeton, 1953.Google Scholar
7. Jenkins, J. and Morse, M., Conjugate nets on an open Riemann surface, Lectures on Functions of a Complex Variable, ed. Kaplan, W. et al., 123–185. The University of Michigan Press, 1955.Google Scholar
8. Massey, W., Algebraic Topology: An Introduction, Harcourt, Brace and World, Inc., New York, 1967.Google Scholar
9. Stoilow, S., Principles Topologiques de la Théorie des Fonctions Analytiques, Paris, Gauthiers-Villars, 2nd ed., 1956.Google Scholar
10. Vincent, P., Families of Generalized Continua on 2-manifolds, Ph.D. Thesis, University of Rochester, 1973.Google Scholar