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Does there exist a continuous curve $C$ on $\mathbb{R}^2$ that crosses itself at every point on $C$?

For two pieces of curve to cross and not just touch, each must have some length either side of the other. $\mathbb{R}$ cannot be partitioned into uncountably many disjoint intervals of non-zero length, therefore if any non-zero length of the curve does not cross itself there must be points along that length that the curve does not cross at.

If such a curve does not exist, is it possible for the set of points where $C$ crosses itself to merely be dense in $C$?

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Parts of the curve cross and don't merely intersect.
i.e. (idiosyncratic emergency definition that may not make sense) for every point $p$ on the curve there exists $\epsilon$ and two connected pieces of curve intersecting at p such that $\forall \delta<\epsilon$, four points can be placed on the circumference of the circle of radius $\delta$ centred at $p$ to divide the circle into four arcs such that each of the two opposite pairs of arcs on the circle contains all the intersections of one of the pieces of curve (but not the same piece for both pairs) and each of the four arcs contains at least one such point of intersection.

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    $\begingroup$ You should think about a way to make your crossing requirement more topologically precise. Intuitively, it is clear what you mean, but I fear there are some pathological examples (of the kind that come up in discussions of the Jordan Curve Theorem) that have points where it is not clear if they "just touch" or "have some length on either side." $\endgroup$ – Potato Jul 13 '13 at 7:43
  • $\begingroup$ Isn't the Hilbert's space-filling curve an example with a dense set of intersection points? $\endgroup$ – Hagen von Eitzen Jul 13 '13 at 9:33
  • $\begingroup$ @HagenvonEitzen It is not clear to me that the space filling curve intersects (or even that it is not injective). Could you elaborate, please? $\endgroup$ – Daniel Robert-Nicoud Jul 13 '13 at 10:25
  • $\begingroup$ If you define "crosses itself" by "it is never injective", then a trivial example is the constant path. $\endgroup$ – Daniel Robert-Nicoud Jul 13 '13 at 10:26
  • $\begingroup$ Do curves of the type $\gamma(\mathbb{R})$, $\gamma(t)=(2\cos(t)+\cos(\sqrt{2}t),2\sin(t)+\sin(\sqrt{2}t))$, qualify as "crosses itself in a dense subset"? $\endgroup$ – Dominik Jul 13 '13 at 11:15

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