If the universe has a two dimensional Möbius topology then it has infinite length but zero area. This spacetime structure is classified as a Sierpinski carpet. It is a 2D analog of the 1D Cantor set. See the following links: http://mathworld.wolfram.com/SierpinskiCarpet.html , http://en.wikipedia.org/wiki/Sierpinski_carpet , http://en.wikipedia.org/wiki/Sierpinski_carpet , http://mathworld.wolfram.com/CantorSet.html . In comparison with a Menger sponge, it shows how topological dimensional transformations must necessarily passes the domain points of zero and infinity. This suggests that the one dimensional Möbius topology as represented by a Cantor set would have infinite points but zero length.

The fractal structure of the Cantor set is a proper topological description for a quantum theory of the spacetime continuum, except for the fact that the Lebesgue measure is zero which suggests zero metric tensor for the spacetime continuum. See links:http://en.wikipedia.org/wiki/Lebesgue_measure , http://mathworld.wolfram.com/LebesgueMeasure.html . However, at the local infinitesimal region of the spacetime continuum, the two end points of zero length can be replaced by a pair of complementary trivectors representing two spacetime vertices. Four pairs of complementary trivectors then form a higher dimensional spacetime vertex as a directional invariance topological structure for the quantum theory of the spacetime continuum.