Most maps of the pseudo-arc are homeomorphisms
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- by Wayne Lewis PDF
- Proc. Amer. Math. Soc. 91 (1984), 147-154 Request permission
Abstract:
We prove the following results. (1) If $M(P)$ is the space of maps of the pseudo-arc into itself with the sup metric, then the subset $\hat H(P)$ of maps of the pseudo-arc into itself which are homeomorphisms onto their images is a dense ${G_\delta }$ in $M(P)$. (2) Every homeomorphism of the pseudo-arc onto itself is a product of $\in$-homeomorphisms. (3) There exists a nonidentity homeomorphism of the pseudo-arc with an infinite sequence of $p$th roots. (4) Every map between chainable continua can be lifted to a homeomorphism of pseudo-arcs.References
- R. H. Bing, A homogeneous indecomposable plane continuum, Duke Math. J. 15 (1948), 729–742. MR 27144
- R. H. Bing, Concerning hereditarily indecomposable continua, Pacific J. Math. 1 (1951), 43–51. MR 43451
- R. H. Bing, Snake-like continua, Duke Math. J. 18 (1951), 653–663. MR 43450
- Beverly L. Brechner, On the dimensions of certain spaces of homeomorphisms, Trans. Amer. Math. Soc. 121 (1966), 516–548. MR 187208, DOI 10.1090/S0002-9947-1966-0187208-2 —, Homeomorphism groups of chainable and homogeneous continua, Topology Proc. 8 (1983).
- H. Cook, Continua which admit only the identity mapping onto non-degenerate subcontinua, Fund. Math. 60 (1967), 241–249. MR 220249, DOI 10.4064/fm-60-3-241-249
- Lawrence Fearnley, Characterizations of the continuous images of the pseudo-arc, Trans. Amer. Math. Soc. 111 (1964), 380–399. MR 163293, DOI 10.1090/S0002-9947-1964-0163293-7
- A. Lelek, On weakly chainable continua, Fund. Math. 51 (1962/63), 271–282. MR 143182, DOI 10.4064/fm-51-3-271-282
- Wayne Lewis, Stable homeomorphisms of the pseudo-arc, Canadian J. Math. 31 (1979), no. 2, 363–374. MR 528817, DOI 10.4153/CJM-1979-041-1
- Wayne Lewis, Periodic homeomorphisms of chainable continua, Fund. Math. 117 (1983), no. 1, 81–84. MR 712216, DOI 10.4064/fm-117-1-81-84
- Wayne Lewis, Pseudo-arcs and connectedness in homeomorphism groups, Proc. Amer. Math. Soc. 87 (1983), no. 4, 745–748. MR 687655, DOI 10.1090/S0002-9939-1983-0687655-1
- Edwin E. Moise, An indecomposable plane continuum which is homeomorphic to each of its nondegenerate subcontinua, Trans. Amer. Math. Soc. 63 (1948), 581–594. MR 25733, DOI 10.1090/S0002-9947-1948-0025733-4 J. Toledo, Finite and compact actions on chainable and tree-like continua, Ph.D. Dissertation, Univ. of Florida, 1982.
Additional Information
- © Copyright 1984 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 91 (1984), 147-154
- MSC: Primary 54F20; Secondary 54H15
- DOI: https://doi.org/10.1090/S0002-9939-1984-0735582-4
- MathSciNet review: 735582