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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



Inverse limits on graphs and monotone mappings

Author: J. W. Rogers
Journal: Trans. Amer. Math. Soc. 176 (1973), 215-225
MSC: Primary 54F20; Secondary 54B25
MathSciNet review: 0324670
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Abstract: In 1935, Knaster gave an example of an irreducible continuum (i.e. compact connected metric space) K which can be mapped onto an arc so that each point-preimage is an arc. The continuum K is chainable (or arc-like). In this paper it is shown that every one-dimensional continuum M is a continuous image, with arcs as point-preimages, of some one-dimensional continuum $ M'$. Moreover, if M is G-like, for some collection G of graphs, then $ M'$ can be chosen to be G-like. A corollary is that every chainable continuum is a continuous image, with arcs as point-inverses, of a chainable (and hence, by a theorem of Bing, planar) continuum. These investigations give rise to the study of certain special types of inverse limit sequences on graphs.

References [Enhancements On Off] (What's this?)

  • [1] R. H. Bing, Snake-like continua, Duke Math. J. 18 (1951), 653-663. MR 13, 265. MR 0043450 (13:265a)
  • [2] M. Brown, Some applications of an approximation theorem for inverse limits, Proc. Amer. Math. Soc. 11 (1960), 478-483. MR 22 #5959. MR 0115157 (22:5959)
  • [3] S. Eilenberg and N. E. Steenrod, Foundations of algebraic topology, Princeton Univ. Press, Princeton, N. J., 1952. MR 14, 398. MR 0050886 (14:398b)
  • [4] B. Knaster, Une continu irreductible à décomposition continue en tranches, Fund. Math. 25 (1935), 568-577.
  • [5] S. Mardešić and J. Segal, $ \varepsilon $-mappings onto polyhedra, Trans. Amer. Math. Soc. 109 (1963), 146-164. MR 28 #1592. MR 0158367 (28:1592)

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Keywords: Monotone map, decomposition into arcs, simplicial inverse limit, G-like continuum
Article copyright: © Copyright 1973 American Mathematical Society

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