Conditions which ensure that a simple map does not raise dimension

W. Dębski; J. Mioduszewski

Colloquium Mathematicae (1992)

  • Volume: 63, Issue: 2, page 173-185
  • ISSN: 0010-1354

Abstract

top
The present paper deals with those continuous maps from compacta into metric spaces which assume each value at most twice. Such maps are called here, after Borsuk and Molski (1958) and as in our previous paper (1990), simple. We investigate the possibility of decomposing a simple map into essential and elementary factors, and the so-called splitting property of simple maps which raise dimension. The aim is to get insight into the structure of those compacta which have the property that simple maps from them do not raise dimension. In what follows a map means a continuous map, unless explicitly stated otherwise. A space is, except in some general lemmas, understood to be metrizable. A compactum means a compact metric space.

How to cite

top

Dębski, W., and Mioduszewski, J.. "Conditions which ensure that a simple map does not raise dimension." Colloquium Mathematicae 63.2 (1992): 173-185. <http://eudml.org/doc/210143>.

@article{Dębski1992,
abstract = {The present paper deals with those continuous maps from compacta into metric spaces which assume each value at most twice. Such maps are called here, after Borsuk and Molski (1958) and as in our previous paper (1990), simple. We investigate the possibility of decomposing a simple map into essential and elementary factors, and the so-called splitting property of simple maps which raise dimension. The aim is to get insight into the structure of those compacta which have the property that simple maps from them do not raise dimension. In what follows a map means a continuous map, unless explicitly stated otherwise. A space is, except in some general lemmas, understood to be metrizable. A compactum means a compact metric space.},
author = {Dębski, W., Mioduszewski, J.},
journal = {Colloquium Mathematicae},
keywords = {2-to-1 map; dimension-raising map; dendrite; pair of twins; simple map},
language = {eng},
number = {2},
pages = {173-185},
title = {Conditions which ensure that a simple map does not raise dimension},
url = {http://eudml.org/doc/210143},
volume = {63},
year = {1992},
}

TY - JOUR
AU - Dębski, W.
AU - Mioduszewski, J.
TI - Conditions which ensure that a simple map does not raise dimension
JO - Colloquium Mathematicae
PY - 1992
VL - 63
IS - 2
SP - 173
EP - 185
AB - The present paper deals with those continuous maps from compacta into metric spaces which assume each value at most twice. Such maps are called here, after Borsuk and Molski (1958) and as in our previous paper (1990), simple. We investigate the possibility of decomposing a simple map into essential and elementary factors, and the so-called splitting property of simple maps which raise dimension. The aim is to get insight into the structure of those compacta which have the property that simple maps from them do not raise dimension. In what follows a map means a continuous map, unless explicitly stated otherwise. A space is, except in some general lemmas, understood to be metrizable. A compactum means a compact metric space.
LA - eng
KW - 2-to-1 map; dimension-raising map; dendrite; pair of twins; simple map
UR - http://eudml.org/doc/210143
ER -

References

top
  1. R. D. Anderson and G. Choquet, A plane continuum no two of whose nondegenerate subcontinua are homeomorphic: an application of inverse limits, Proc. Amer. Math. Soc. 10 (1959), 347-353. Zbl0093.36501
  2. J. J. Andrews, A chainable continuum no two of whose nondegenerate subcontinua are homeomorphic, ibid. 12 (1961), 333-334. Zbl0129.38605
  3. K. Borsuk and R. Molski, On a class of continuous mappings, Fund. Math. 45 (1958), 84-98. Zbl0081.38803
  4. W. Dębski and J. Mioduszewski, Simple plane images of the Sierpiński triangular curve are nowhere dense, Colloq. Math. 59 (1990), 125-140. Zbl0735.54023
  5. H. Freudenthal, Über dimensionserhöhende stetige Abbildungen, Sitzungsber. Preuss. Akad. Wiss. Phys.-Math. Kl. 1932, 34-38. 
  6. H. Hahn, Über die Abbildung einer Strecke auf ein Quadrat, Ann. Mat. Ser. III 21 (1913), 33-35. Zbl44.0560.02
  7. W. Hurewicz, Über dimensionserhöhende stetige Abbildungen, J. Reine Angew. Math. 169 (1933), 71-78. 
  8. W. Hurewicz and H. Wallman, Dimension Theory, Princeton Univ. Press, Princeton 1941. Zbl67.1092.03
  9. Ya. M. Kazhdan, On continuous mappings which increase dimension, Dokl. Akad. Nauk SSSR 67 (1949), 19-22 (in Russian). 
  10. C. Kuratowski, Topologie II, PWN, Warszawa-Wrocław 1950. 
  11. A. Lelek, On Peano functions, Prace Mat. 7 (1962), 127-140 (in Polish). Zbl0126.18403
  12. S. Mazurkiewicz, Sur les points multiples des courbes qui remplissent une aire plane, Prace Mat.-Fiz. 26 (1915), 113-120 (in Polish); French transl. in: S. Mazurkiewicz, Travaux de topologie, PWN, Warszawa 1969, 48-56. 
  13. K. Sieklucki, A generalization of a theorem of S. Mazurkiewicz concerning Peano functions, Prace Mat. (Comment. Math.) 12 (1969), 251-253. Zbl0235.54038
  14. G. T. Whyburn, Analytic Topology, Amer. Math. Soc. Colloq. Publ. 28, New York 1942. Zbl0061.39301

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.