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On the existence of rigid arcs

Published online by Cambridge University Press:  09 April 2009

B. J. Pearson
B. J. Pearson
Affiliation:
University of Missouri, Kansas City, Missouri 64110, U.S.A.
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By an arc we mean a nondegenerate simply ordered set which is compact and connected in its order topology. A space X is rigid if the only homeomorphism of X onto itself is the identity map. Examples of rigid totally disconnected compact ordered spaces may be found in [1], [2], and [3]. It is the purpose of this note to prove the existence of an arc such that no two of its subarcs are homeomorphic. The proof makes use of arcs of large cardinality and the technique of inverse limit spaces.

Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 18 , Issue 3 , November 1974 , pp. 306 - 309
Copyright
Copyright © Australian Mathematical Society 1974

References

[1]de Groot, J. and Maurice, M. A., ‘On the existence of rigid compact ordered spaces’, Proc. Amer. Math. Soc. 19 (1968), 844–846.CrossRefGoogle Scholar
[2]Jonsson, B., ‘A Boolean algebra without proper automorphisms’, Proc. Amer. Math. Soc 2 (1951), 766–770.CrossRefGoogle Scholar
[3]Rieger, L., ‘Some remarks on automorphisms of Boolean algebras’, Fund. Math. 38 (1951), 209–216.CrossRefGoogle Scholar