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, , of continuous functions . 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.
|Date of creation||2013-03-22 16:32:29|
|Last modified on||2013-03-22 16:32:29|
|Last modified by||stevecheng (10074)|
|Synonym||space filling curve|