Homogeneity in powers of subspaces of the real line

Working in ZFC, we prove that for every zero-dimensional subspace $S$ of the real line, the Tychonoff power ${}^\omega S$ is homogeneous ($\omega$ denotes the nonnegative integers). It then follows as a corollary that ${}^\omega S$ is homogeneous whenever $S$ is a separable zero-dimensional metrizable space. The question of homogeneity in powers of this type was first raised by Ben Fitzpatrick, and was subsequently popularized by Gary Gruenhage and Hao-xuan Zhou.