We prove that if X is a perfect finite-dimensional compactum, then for almost every continuous surjection of the Cantor set onto X, the set of points of maximal order is uncountable. Moreover, if X is a perfect compactum of positive finite dimension then for a typical parametrization of X on the Cantor set, the set of points of maximal order is homeomorphic to the product of the rationals and the Cantor set.
@article{bwmeta1.element.bwnjournal-article-doi-10_4064-fm174-3-5,
author = {Pawe\l\ Milewski},
title = {On typical parametrizations of finite-dimensional compacta on the Cantor set},
journal = {Fundamenta Mathematicae},
volume = {173},
year = {2002},
pages = {253-261},
zbl = {1010.54024},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-fm174-3-5}
}
Paweł Milewski. On typical parametrizations of finite-dimensional compacta on the Cantor set. Fundamenta Mathematicae, Tome 173 (2002) pp. 253-261. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-fm174-3-5/