What `shape' is space-time?

Some examples from the mathematics of shape are presented that question some of the almost hidden assumptions behind results on limiting behaviour of finitary approximations to space-time. These are presented so as to focus attention on the observational problem of refinement and suggest the necessity for an alternative theory of `fractafolds’ against which the limiting theory of $C^{\mathrm{\infty }}$-differential manifolds usually underlying (quantum) general relativity can be measured.