A compact metric space T
has Lebesgue covering dimension at most n if for each positive 𝜀 the space T has an
𝜀-cover of order at most n. We show that if T is a compact subset of Euclidean
n-space and T has an 𝜀-cover of order at most n − 2, then any two points whose
distance from T is greater than 𝜀 can be joined by a path bounded away from T.
This refines, and provides a constructive proof for, the theorem that the
complement of an (n − 2)-dimensional compact subset of Euclidean n-space is
connected.
