Exactly two-to-one maps from continua onto some tree-like continua

Wojciech Dębski; J. Heath; J. Mioduszewski

Fundamenta Mathematicae (1992)

  • Volume: 141, Issue: 3, page 269-276
  • ISSN: 0016-2736

Abstract

top
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.

How to cite

top

Dębski, Wojciech, Heath, J., and Mioduszewski, J.. "Exactly two-to-one maps from continua onto some tree-like continua." Fundamenta Mathematicae 141.3 (1992): 269-276. <http://eudml.org/doc/211965>.

@article{Dębski1992,
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.},
author = {Dębski, Wojciech, Heath, J., Mioduszewski, J.},
journal = {Fundamenta Mathematicae},
keywords = {Knaster continua; Ingram continua; 2-to-1 map; two-to-one maps; tree-like continua},
language = {eng},
number = {3},
pages = {269-276},
title = {Exactly two-to-one maps from continua onto some tree-like continua},
url = {http://eudml.org/doc/211965},
volume = {141},
year = {1992},
}

TY - JOUR
AU - Dębski, Wojciech
AU - Heath, J.
AU - Mioduszewski, J.
TI - Exactly two-to-one maps from continua onto some tree-like continua
JO - Fundamenta Mathematicae
PY - 1992
VL - 141
IS - 3
SP - 269
EP - 276
AB - 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.
LA - eng
KW - Knaster continua; Ingram continua; 2-to-1 map; two-to-one maps; tree-like continua
UR - http://eudml.org/doc/211965
ER -

References

top
  1. [1] D. Fox, k-to-1 continuous transformations, Dissertation, Univ. of California at Riverside, 1973. 
  2. [2] W. H. Gottschalk, On k-to-1 transformations, Bull. Amer. Math. Soc. 53 (1947), 168-169. Zbl0040.25402
  3. [3] J. Grispolakis and E. D. Tymchatyn, Continua which are images of weakly confluent mappings only (I), Houston J. Math. 5 (1979), 483-501. Zbl0412.54039
  4. [4] J. Heath, Tree-like continua and exactly k-to-1 functions, Proc. Amer. Math. Soc. 105 (1989), 765-772. Zbl0664.54008
  5. [5] J. Heath, 2-to-1 maps with hereditarily indecomposable images, ibid. 113 (1991), 839-846. Zbl0738.54012
  6. [6] W. T. Ingram, An atriodic tree-like continuum with positive span, Fund. Math. 77 (1972), 99-107. Zbl0244.54023
  7. [7] T. Maćkowiak, Semiconfluent mappings and their invariants, ibid. 79 (1973), 251-264. Zbl0261.54011
  8. [8] S. B. Nadler, Jr. and L. E. Ward, Jr., Concerning exactly (n,1) images of continua, Proc. Amer. Math. Soc. 87 (1983), 351-354. Zbl0503.54018

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

                
Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.