  We show how to make precise the vague idea that for compact metric spaces
that are close together for Gromov-Hausdorff distance, suitable vector bundles
on one metric space will have counterpart vector bundles on the other. Our
approach employs the Lipschitz constants of projection-valued functions that
determine vector bundles. We develop some computational techniques, and we
illustrate our ideas with simple specific examples involving vector bundles on
the circle, the two-torus, the two-sphere, and finite metric spaces. Our topic
is motivated by statements concerning "monopole bundles" over matrix algebras
in the literature of theoretical high-energy physics.
