Talk:Space-filling curve

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Contents 1 A curve filling a countably dimensional hypercube 2 Homemomorphism 3 Self-intersection 4 Any space-filling curve must be densely self-intersecting

[edit] A curve filling a countably dimensional hypercube

You should describe such a curve. Also, you should make a link to a description of the discontinuous mapping of the unit interval onto the unit square, discovered by Georg Cantor. -- Leocat 14:13, 25 October 2006 (UTC)

[edit] Homemomorphism

so this space-filling curve is a continuous bijection, right? Why isn't it a homemomorphism (it certainly can't be). Because its inverse isn't continuous? That must be it. Lethe | Talk 07:42, Mar 3, 2005 (UTC)

   Space filling curves aren't one to one, so aren't bijections.(Balthamos 23:35, 11 June 2006 (UTC))

[edit] Self-intersection

Wasn't one of the attributes of the Peano curve the property of non self-intersection?

WLD 13:14, 23 Apr 2005 (UTC)

[edit] Any space-filling curve must be densely self-intersecting

There cannot be any non-self-intersecting (i.e. injective) continuous curve filling up the unit square, because that will make the curve a homeomorphism from the unit interval onto the unit square (using the fact that any continuous bijection from a compact space onto a Hausdorff space is a homeomorphism), but the unit-square (which has no cut-point) is not homeomorphic to the unit interval (all points of which, except the endpoints, are cut-points).

A.D.