What is it to "enumerate a space" (TPTB_need_war)? The cardinality of the set of all real numbers is Aleph-1 - uncountably infinite. Any continuous, nonzero displacement consists of (at least, in the context of Euclidean spaces) uncountably infinite, zero-dimensional points (i.e., 0-spaces). That those points correspond to a particular distance is only for the establishment of a particular line segment as the multiplicative identity of position within the (again, Euclidean) space.

(Do you - that one having made, and continuing to support, a claim about all spaces - care to elaborate on the relevance of your assertion to the Hilbert space - a space?)