Small singular regions of spacetime

We prove that every open connected region of relativistic spacetime $(M,\textbf{g})$ that encloses a $b$-incomplete half-curve has an open connected subregion that encloses a $b$-incomplete half-curve and is also 'small' in the following sense: it is the image, under the bundle projection map, of some open region in the (connected) orthonormal frame bundle $O^+M$ over that spacetime which is bounded, and whose closure is Cauchy incomplete, with respect to any 'natural' distance function on $O^+M$. As a corollary, it follows that every $b$-incomplete half-curve can be covered by a sequence of singular regions which are images of a sequence of bounded subsets of $O^+M$ whose diameter, with respect to any 'natural' distance function on $O^+M$, tends to zero. We discuss to what extent these results can be interpreted in favour of the claim that singular structure in classical general relativity is 'localizable'.