  We study the subsets of metric spaces that are negligible for the infimal
length of connecting curves; such sets are called metrically removable. In
particular, we show that every totally disconnected set with finite Hausdorff
measure of codimension 1 is metrically removable, which answers a question
raised by Hakobyan and Herron. The metrically removable sets are shown to be
related to other classes of "thin" sets that appeared in the literature. They
are also related to the removability problems for classes of holomorphic
functions with restrictions on the derivative.
