# Peano curve

A *Peano curve ^{}* or

*space-filling curve*is a continuous mapping of a closed interval

^{}onto a square.

Such mappings, introduced by Peano in 1890, played an
important role in the development of topology^{} as a counterexample
to the naive ideas of dimension^{} — while it is
true that one cannot map a space onto a space of higher dimension using a
smooth map^{}, this is no longer true if one only requires continuity as opposed to
smoothness. The Peano curve and similar examples led to a rethinking of the foundations
of topology and analysis^{}, and underscored the importance of formulating
topological notions in a rigorous fashion.

However, still, a space-filling curve cannot ever be one-to-one; otherwise invariance of dimension would be violated.

Many space-filling curves may be obtained as the limit of a sequence, $\u27e8{h}_{n}\mid n\in \mathbb{N}\u27e9$, of continuous functions^{} ${h}_{n}:[0,1]\to [0,1]\times [0,1]$. Figure 1 (\PMlinktofilesource codehilbert.cc), showing the ranges of the first few approximations to Hilbert’s space-filling curve, illustrates a common case in which each successive approximation is obtained by applying a recursive procedure to its predecessor.

Title | Peano curve |
---|---|

Canonical name | PeanoCurve |

Date of creation | 2013-03-22 16:32:29 |

Last modified on | 2013-03-22 16:32:29 |

Owner | stevecheng (10074) |

Last modified by | stevecheng (10074) |

Numerical id | 13 |

Author | stevecheng (10074) |

Entry type | Definition |

Classification | msc 28A80 |

Synonym | space-filling curve |

Synonym | space filling curve |