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

DOI:
https://doi.org/10.1090/S0002-9947-1973-0324670-X

MathSciNet review:
0324670

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Abstract | References | Similar Articles | Additional Information

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 . Moreover, if *M* is *G*-like, for some collection *G* of graphs, then 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.

**[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, -*mappings onto polyhedra*, Trans. Amer. Math. Soc.**109**(1963), 146-164. MR**28**#1592. MR**0158367 (28:1592)**

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Additional Information

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