The dimension of a continuum usually means its topological dimension.

It was first described by while exploring the concept of topological dimension.

The topological dimension of the Menger sponge is one, the same as any curve.

The topological dimension tells you how many different little balls connect a given point to other points in the space, generically.

But the topological dimension doesn't tell you anything about volumes.

A curve that is almost space-filling can still have topological dimension one, even if it fills up most of the area of a region.

It is also possible for the isoperimetric dimension to be larger than the topological dimension.

Both are curves with topological dimension of 1, so one might hope to be able to measure their length or slope, as with ordinary lines.

The topological dimension of a discrete space is equal to 0.

An algebraic curve likewise has topological dimension two; in other words, it is a surface.