Approximating a diagonal slope through 9 degree steps. Why does it not work?

[Rydiro]

That is a fair point in this instance, though I was only speaking to the fact that (most) 1D topological fractals have infinite perimeters confined to a finite area (when embedded in the plane). Typically, topological dimension is the default assumption when speaking about "dimensions", though it of course depends on context (which I should've taken into account when responding.)

That is a way to tell if two spaces are homeomorphic (i.e., they share all of their topological properties, including dimension), though it's important to note that the mapping must have a continuous inverse as well in order to be a homeomorphism.

Honestly, I never heared about a topological dimension (just about things being homeomorphic instead). And I heared two semesters in topology.