### Abstract

Let M be a compact connected (topological) manifold of finite- or infinite-dimension n. Let 0 ≤ r ≤ 1 be arbitrary but fixed. We construct in this paper a space-filling curve f from [0, 1] onto M, under which M is the image of a compact set A of Hausdorff dimension r. Moreover, the restriction of f to A is one-to-one over the image of a dense subset provided that 0 ≤ r ≤ log 2^{n}/log(2^{n} + 2). The proof is based on the special case where M is the Hilbert cube [0, 1]^{ω}.

Original language | English |
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Pages (from-to) | 1871-1881 |

Number of pages | 11 |

Journal | Computers and Mathematics with Applications |

Volume | 45 |

Issue number | 12 |

DOIs | |

Publication status | Published - 01 Jun 2003 |

Externally published | Yes |

### Keywords

- Hausdorff dimensions
- Hilbert cube manifolds
- Space-filling curves

### ASJC Scopus subject areas

- Modelling and Simulation
- Computational Theory and Mathematics
- Computational Mathematics

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## Cite this

Lin, Y. F., & Wong, N. C. (2003). Constructing space-filling curves of compact connected manifolds.

*Computers and Mathematics with Applications*,*45*(12), 1871-1881. https://doi.org/10.1016/S0898-1221(03)90008-3