Hostname: page-component-586b7cd67f-t7czq Total loading time: 0 Render date: 2024-11-25T22:24:13.345Z Has data issue: false hasContentIssue false
  • English
  • Français

Stable Homeomorphisms of the Pseudo-Arc

Published online by Cambridge University Press:  20 November 2018

Wayne Lewis
Wayne Lewis*
Affiliation:
Texas Tech University, Lubbock, Texas
Rights & Permissions [Opens in a new window]

Extract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

Noting that certain restrictions are placed on homeomorphisms of the pseudo-arc, since it is hereditarily indecomposable, in 1955 [4] R. H. Bing asked if the identity is the only stable homeomorphism of the pseudo-arc. In this paper we prove the following theorem.

THEOREM. Let U be an open subset of the pseudo-arc P. Let p and q be distinct points of P such that the subcontinuum M irreducible between p and q does not intersect cl(U). Then there exists a homeomorphism h : P → P with h (p) = q andh│U = 1U.

1. Definitions. A chain C is a collection of open sets C = {C(i))}i≧n such that C(i) ∩ C(j) ≠ ∅ if and only if |i – j| ≦ 1, cl|(C(i)) ∩ cl(C(j)) ≠ ∅ if and only if |i – j| ≦ 1, C(0) – C(l) ≠ ∅, and C(n) – C(n – 1) ≠ ∅. Each C(i) is called a link of C.

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 31 , Issue 2 , 01 April 1979 , pp. 363 - 374
Copyright
Copyright © Canadian Mathematical Society 1979

References

1. Bing, R. H., A homogeneous indecomposable plane continuum, Duke Math. J.. 15 (1948), 729 742.Google Scholar
2. Bing, R. H., Snake-like continua, Duke Math. J.. 18 (1951), 653 663.Google Scholar
3. Bing, R. H., Concerning hereditarily indecomposable continua, Pac J. Math.. 1 (1951), 43 51.Google Scholar
4. Bing, R. H., The pseudo-arc, Summary of Lectures and Seminars, Summer Institute on Set-Theoretic Topology, Madison (1955), A.M.S., 72 75.Google Scholar
5. Brechner, B., On stable homeomorphisms and imbeddings of the pseudo-arc, III. J. Math. 22 (1978), 630 661.Google Scholar
6. Brown, M. and Gluck, H., Stable structures on manifolds, Ann. of Math., 79 (1), (1964), 1 17.Google Scholar
7. Fearnley, L., Hereditarily indecomposable circularly chainable continua, Ph.D. Thesis, Univ. of London, 1970.Google Scholar
8. Hamilton, O. H., A fixed point theorem for pseudo-arcs and certain other metric continua, Proc. Amer. Math. Soc. 2 (1951), 173 174.Google Scholar
9. Lehner, G. R., Extending homeomorphisms on the pseudo-arc, Trans. Amer. Math. Soc. 98 (1961), 369 394.Google Scholar
10. Moise, E. E., An indecomposable plane continuum which is homeomorphic to each of its nondegenerate sub-continua, Trans. Amer. Math. Soc. 63 (1948), 581 594.Google Scholar
11. Moise, E. E., A note on the pseudo-arc, Trans. Amer. Math. Soc. 64 (1949), 57 58.Google Scholar
12. Rogers, J. T., Jr., The pseudo-circle is not homogeneous, Trans. Amer. Math. Soc. 148 (1970), 417 428.Google Scholar