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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

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