Cantor curve

A metrizable one-dimensional continuum. A Cantor curve originally referred to a plane nowhere-dense continuum, and it was the first (but not intrinsic) characterization of one-dimensional closed connected subsets of the plane considered by G. Cantor. A Cantor curve contains a nowhere-dense subcontinuum if and only if the closure of the set of its branching points is one-dimensional. If, on the other hand, a Cantor curve does not contain a nowhere-dense subcontinuum, then all its points have finite branch index. A Cantor curve without branching points is either a simple arc or a simple closed line. The set of end points of a Cantor curve, i.e. the set of points of index 1, is zero-dimensional, but can be everywhere dense. If all points of a Cantor curve have the same finite branch index, then the Cantor curve is a simple closed line. The universal Cantor curve (the Menger curve) can be constructed; this is a Cantor curve that contains a topological image of every Cantor curve.

References

[1]	P.S. Urysohn, "Works on topology and other areas of mathematics" , 2 , Moscow-Leningrad (1951) (In Russian)
[2]	K. Menger, "Kurventheorie" , Teubner (1932)

Comments

Not every metrizable one-dimensional continuum can be imbedded in the plane. For instance, the $ 1 $- skeleton of a $ 4 $- simplex is such a space [a1].

References

[a1]	R. Engelking, "Dimension theory" , North-Holland & PWN (1978)