ArticleOriginal scientific text
Title
Exactly two-to-one maps from continua onto some tree-like continua
Authors 1, 2, 1
Affiliations
- Institute of Mathematics, Silesian University, Bankowa 14, 40-007 Katowice, Poland
- Mathematics Department, Parker Hall, Auburn University, Alabama 36849-5310, U.S.A.
Abstract
It is known that no dendrite (Gottschalk 1947) and no hereditarily indecomposable tree-like continuum (J. Heath 1991) can be the image of a continuum under an exactly 2-to-1 (continuous) map. This paper enlarges the class of tree-like continua satisfying this property, namely to include those tree-like continua whose nondegenerate proper subcontinua are arcs. This includes all Knaster continua and Ingram continua. The conjecture that all tree-like continua have this property, stated by S. Nadler Jr. and L. E. Ward Jr. (1983), is still neither confirmed nor rejected.
Keywords
Knaster continua, Ingram continua, 2-to-1 map
Bibliography
- D. Fox, k-to-1 continuous transformations, Dissertation, Univ. of California at Riverside, 1973.
- W. H. Gottschalk, On k-to-1 transformations, Bull. Amer. Math. Soc. 53 (1947), 168-169.
- J. Grispolakis and E. D. Tymchatyn, Continua which are images of weakly confluent mappings only (I), Houston J. Math. 5 (1979), 483-501.
- J. Heath, Tree-like continua and exactly k-to-1 functions, Proc. Amer. Math. Soc. 105 (1989), 765-772.
- J. Heath, 2-to-1 maps with hereditarily indecomposable images, ibid. 113 (1991), 839-846.
- W. T. Ingram, An atriodic tree-like continuum with positive span, Fund. Math. 77 (1972), 99-107.
- T. Maćkowiak, Semiconfluent mappings and their invariants, ibid. 79 (1973), 251-264.
- S. B. Nadler, Jr. and L. E. Ward, Jr., Concerning exactly (n,1) images of continua, Proc. Amer. Math. Soc. 87 (1983), 351-354.
Additional information
http://matwbn.icm.edu.pl/ksiazki/fm/fm141/fm14136.pdf
Pages:
269-276
Main language of publication
English
Received
1991-11-26
Accepted
1992-05-19
Published
1992
Exact and natural sciences