Constructing space-filling curves of compact connected manifolds
It is well known that every \(n\)-dimensional cube \([0,1]^n\) is a continuous image of the interval \([0,1]\), the first example was given by Peano in 1890. Here is a generalization: a metrizable space \(X\) is a continuous image of \([0,1]\) if and only if \(X\) is compact, connected, and locally compact. In the paper under review the authors construct \textit{explicitly} a continuous function \(f\) from \([0,1]\) onto \([0,1]^n\), \(n=1,2,\ldots,\omega\), which maps a compact set \(A\) of Hausdorff dimension \(r\), \(0\leqslant r \leqslant 1\), onto \([0,1]^n\). A subset of \(A\) where \(f\) is one-to-one is dense in \(A\) and its image is dense in \([0,1]^n\) -- provided \(0\leqslant r \leqslant \log 2^n/\log(2^n+2)\) (so that \(0\leqslant r \leqslant 1\) for \(n=\omega\)). The same statements are true for an arbitrary compact connected manifold \(M\) of dimension \(n=1,2,\ldots,\omega\) (which means that \(M\) is connected and there exist compact subsets \(U_1,\ldots,U_k\) such that \(M\) is covered by the interiors of \(U_1,\ldots,U_k\) and each of them is homeomorphic to \([0,1]^n\) ). In conclusion two examples are considered: \(M=[0,1]^3\) and \(M\) is an ellipsoid (a surface) in \(\mathbb R^3\).
scientific article; zbMATH DE number 2082269
scientific article