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On typical parametrizations of finite-dimensional compacta on the Cantor set

Paweł Milewski — 2002

Fundamenta Mathematicae

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.

Functions Equivalent to Borel Measurable Ones

Andrzej Komisarski;  Henryk Michalewski;  Paweł Milewski — 2010

Bulletin of the Polish Academy of Sciences. Mathematics

Let X and Y be two Polish spaces. Functions f,g: X → Y are called equivalent if there exists a bijection φ from X onto itself such that g∘φ = f. Using a theorem of J. Saint Raymond we characterize functions equivalent to Borel measurable ones. This characterization answers a question asked by M. Morayne and C. Ryll-Nardzewski.

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