On the dimension of almost $n$-dimensional spaces

Abstract:

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 $n \geq 1$ 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.