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A Geometric Approach to the Heine-Borel Theorem

Published online by Cambridge University Press:  20 November 2018

R. B. Killgrove ,
Jason Frand ,
William Giles  and
Henry Bray
R. B. Killgrove
Affiliation:
California State College, Los Angeles, California
Jason Frand
Affiliation:
California State College, Los Angeles, California
William Giles
Affiliation:
California State College, Los Angeles, California
Henry Bray
Affiliation:
San Diego State College, San Diego, California
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In a topological plane with strong enough topological properties one can use [6] open triangular regions to define a base for the topology. Similarly, one can use these regions to define boundedness of a set. In this setting we show that in the absolute plane geometry, the holding of the Heine-Borel theorem is equivalent to every four points being contained in some such region and that this second condition is equivalent to the parallel postulate. Thus we give two new conditions equivalent to the parallel postulate.

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 23 , Issue 5 , October 1971 , pp. 845 - 848
Copyright
Copyright © Canadian Mathematical Society 1971

References

1. Borsuk, K. and Szmielew, W., Foundations of geometry (North Holland, Amsterdam, 1960).Google Scholar
2. Coxeter, H. S. M., Non-Euclidean geometry (University of Toronto Press, Toronto, 1961).Google Scholar
3. Forder, H. G., The foundations of Euclidean geometry (Dover, New York, 1958).Google Scholar
4. Hilbert, D., The foundations of geometry (Open Court, Chicago, 1902).Google Scholar
5. Kelly, J. L., General topology (Van Nostrand, Princeton, 1955).Google Scholar
6. Salzmann, H., Topological planes, Advances in Mathematics Vol. 2, pp. 1–60 (Academic Press, New York, 1967).Google Scholar
7. Salzmann, H., Geometries on surfaces, Pacific J. Math. 29, 397–402.Google Scholar