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Metrization of Symmetric Spaces

Published online by Cambridge University Press:  20 November 2018

P. W. Harley III  and
G. D. Faulkner
P. W. Harley III
Affiliation:
University of South Carolina, Columbia, South Carolina
G. D. Faulkner
Affiliation:
University of South Carolina, Columbia, South Carolina
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A distance function d on a set X is a function X × X → [0, ∞ ) satisfying (1) d(x, y) = 0 if and only if x = y, and (2) d(x, y) = d(y, x). Such a function determines a topology T on X by agreeing that U is an open set if it contains an ∈-sphere N(p; ∈)( = {x: d(p, x) < ∈﹜} about each of its points. Equivalently, F is closed if and only if d(x, F) > 0 for each x ∈ X — F. A topological space is symmetrizable via a distance function d if its topology is determined by d as above, and semi-metrizahle via d if x ∈ Ā is equivalent to d(x, A) = 0.

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 27 , Issue 5 , 01 October 1975 , pp. 986 - 990
Copyright
Copyright © Canadian Mathematical Society 1975

References

1. Arhangelskii, A. V., Mappings and spaces, Russian Math. Surveys 21 (1966), 115–162.Google Scholar
2. Hodel, R. E., Some results in metrization theory, 1950–1972, V.P.I. Conference, April, 1973.Google Scholar
3. Jones, F. B., Metrization, Amer. Math. Soc. Monthly 73 (1966), 571–576.Google Scholar
4. Martin, H. Y., Metrization of symmetric spaces and regular maps, Proc. Amer. Math. Soc. 35 (1972), 269–274.Google Scholar