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On the Non-Cutpoint Existence Theorem

Published online by Cambridge University Press:  20 November 2018

L. E. Ward
L. E. Ward*
Affiliation:
University of Oregon Eugene, Oregon
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The theorem of the title asserts that every non-degenerate continuum (that is, every compact connected Hausdorff space containing more than one point) contains at least two non-cutpoints. This is a fundamental result in set - theoretic topology and several standard proofs, each varying from the others to some extent, have been published. (See, for example, [1], [4] and [5]). The author has presented a less standard proof in [3] where the non-cutpoint existence theorem was obtained as a corollary to a result on partially ordered spaces. In this note a refinement of that argument is offered which seems to the author to be the simplest proof extant. To facilitate its exposition, the notion of a weak partially ordered space is introduced and the cutpoint partial order of connected spaces is reviewed.

Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 11 , Issue 2 , June 1968 , pp. 213 - 216
Copyright
Copyright © Canadian Mathematical Society 1968

References

1. Moore, R. L., Foundations of point-set theory. (Rev. ed., New York, 1962) 38.Google Scholar
2. Wallace, A.D., A fixed point theorem. Bull. Amer. Math. Soc. vol. 51 (1945) 413-416.Google Scholar
3. Ward, L. E., Partially ordered topological spaces. Proc. Amer. Math. Soc, vol. 5 (1954) 144-161.Google Scholar
4. Whyburn, G.T., Analytic topology. (New York, 1942)54.Google Scholar
5. Wilder, R.L., Topology of manifolds. (New York, 1949)37.Google Scholar