  In this paper, we approach the question if some of the separation axioms are
equivalent in the class of asymmetric normed spaces. In particular, we make a
remark on a known theorem which states that every $T_1$ asymmetric normed space
with compact closed unit ball must be finite-dimensional. We also explore the
product structure of these spaces and characterize the topological (covering)
dimension of all finite-dimensional asymmetric normed spaces.
