Separation axioms and covering dimension of asymmetric normed spaces

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.