Document Type
Article
Publication Date
2014
College/Unit
Eberly College of Arts and Sciences
Department/Program/Center
Mathematics
Abstract
In this paper we investigate for which closed subsets P of the real line R there exists a continuous map from P onto P 2 and, if such a function exists, how smooth can it be. We show that there exists an infinitely many times differentiable function f : R → R 2 which maps an unbounded perfect set P onto P 2 . At the same time, no continuously differentiable function f : R → R 2 can map a compact perfect set onto its square. Finally, we show that a disconnected compact perfect set P admits a continuous function from P onto P 2 if, and only if, P has uncountably many connected components.
Digital Commons Citation
Ciesielski, Krzysztof, "Smooth Peano Functions for Perfect Subsets of the Real Line" (2014). Faculty & Staff Scholarship. 847.
https://researchrepository.wvu.edu/faculty_publications/847