Real Analysis Exchange

On Closed Subsets of $\mathbb{R}$ and of $\mathbb{R}^2$ Admitting Peano Functions

Krzysztof Chris Ciesielski and Jakub Jasinski

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In this note we describe closed subsets of the real line $P\subset {\mathbb R}$ for which there exists a continuous function from $P$ onto $P^2$, called a Peano function. Our characterization of those sets is based on the number of connected components of $P$. We also include a few remarks on compact subsets of $\mathbb{R}^2$ admitting Peano functions, expressed in terms of connectedness and local connectedness.

Article information

Real Anal. Exchange, Volume 40, Number 2 (2015), 309-318.

First available in Project Euclid: 4 April 2017

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 26A30: Singular functions, Cantor functions, functions with other special properties
Secondary: 26B05: Continuity and differentiation questions

Peano curve space filling curve


Ciesielski, Krzysztof Chris; Jasinski, Jakub. On Closed Subsets of $\mathbb{R}$ and of $\mathbb{R}^2$ Admitting Peano Functions. Real Anal. Exchange 40 (2015), no. 2, 309--318.

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