On the dimension of almost n-dimensional spaces

Oversteegen and Tymchatyn proved that homeomorphism groups of positive dimensional Menger compacta are 1-dimensional by proving that almost 0-dimensional spaces are at most 1-dimensional. These homeomorphism groups are almost 0-dimensional and at least 1-dimensional by classical results of Brechner and Bestvina. In this note we prove that almost n-dimensional spaces for n ≥ 1 are n-dimensional. As a corollary we answer in the affirmative an old question of R. Duda by proving that every hereditarily locally connected, non-degenerate, separable, metric space is 1-dimensional.