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.
@article{bwmeta1.element.bwnjournal-article-cmv63i2p173bwm,
author = {W. D\k ebski and J. Mioduszewski},
title = {Conditions which ensure that a simple map does not raise dimension},
journal = {Colloquium Mathematicae},
volume = {63},
year = {1992},
pages = {173-185},
zbl = {0757.54012},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-cmv63i2p173bwm}
}
Dębski, W.; Mioduszewski, J. Conditions which ensure that a simple map does not raise dimension. Colloquium Mathematicae, Tome 63 (1992) pp. 173-185. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-cmv63i2p173bwm/
[000] 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. | Zbl 0093.36501
[001] J. J. Andrews, A chainable continuum no two of whose nondegenerate subcontinua are homeomorphic, ibid. 12 (1961), 333-334. | Zbl 0129.38605
[002] K. Borsuk and R. Molski, On a class of continuous mappings, Fund. Math. 45 (1958), 84-98. | Zbl 0081.38803
[003] W. Dębski and J. Mioduszewski, Simple plane images of the Sierpiński triangular curve are nowhere dense, Colloq. Math. 59 (1990), 125-140. | Zbl 0735.54023
[004] H. Freudenthal, Über dimensionserhöhende stetige Abbildungen, Sitzungsber. Preuss. Akad. Wiss. Phys.-Math. Kl. 1932, 34-38.
[005] H. Hahn, Über die Abbildung einer Strecke auf ein Quadrat, Ann. Mat. Ser. III 21 (1913), 33-35. | Zbl 44.0560.02
[006] W. Hurewicz, Über dimensionserhöhende stetige Abbildungen, J. Reine Angew. Math. 169 (1933), 71-78.
[007] W. Hurewicz and H. Wallman, Dimension Theory, Princeton Univ. Press, Princeton 1941. | Zbl 67.1092.03
[008] Ya. M. Kazhdan, On continuous mappings which increase dimension, Dokl. Akad. Nauk SSSR 67 (1949), 19-22 (in Russian).
[009] C. Kuratowski, Topologie II, PWN, Warszawa-Wrocław 1950.
[010] A. Lelek, On Peano functions, Prace Mat. 7 (1962), 127-140 (in Polish). | Zbl 0126.18403
[011] 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.
[012] K. Sieklucki, A generalization of a theorem of S. Mazurkiewicz concerning Peano functions, Prace Mat. (Comment. Math.) 12 (1969), 251-253. | Zbl 0235.54038
[013] G. T. Whyburn, Analytic Topology, Amer. Math. Soc. Colloq. Publ. 28, New York 1942. | Zbl 0061.39301