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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.


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DOI: https://doi.org/10.1090/S0002-9947-1973-0324670-X
Keywords: Monotone map, decomposition into arcs, simplicial inverse limit, G-like continuum
Article copyright: © Copyright 1973 American Mathematical Society