On closed subsets of \(\mathbb{R}\) and of \(\mathbb{R}^2\) admitting Peano functions (Q1704438)

Suppose that \(X\) is a topological space and \(f\) is a continuous map from \(X\) onto \(X^2\). Then \(f\) is called a Peano function for \(X\). By \(\kappa(X)\) we denote the number or cardinality of connected components of \(X\). Generalizing a previous result, the authors proved Theorem 2.1. A closed non-empty subset \(P\) of \(\mathbb R\) admits a Peano function if and only if one of the following conditions holds: {\parindent12mm \begin{itemize}\item[(1)] \(\kappa(P)=1\); \item[(2.1)] \(\kappa(P)=\omega\), \(P\) is countable and unbounded; \item[(2.2)] for every \(n<\omega\), \(\kappa(P\setminus(-n,n))=\omega\) and \(P\setminus(-n,n)\) is uncountable; \item[(3.1)] \(\kappa(P)=\mathfrak{c}\) and \(P\) is bounded; \item[(3.2)] \(\kappa(P\setminus(-n,n))=\mathfrak{c}\) for every \(n<\omega\).\end{itemize}}