SEPARATION AXIOMS AND COVERING DIMENSION OF ASYMMETRIC NORMED SPACES

摘要

It is well known that every asymmetric normed space is a T-0 paratopological group. Since all T-i axioms (i = 0, 1, 2, 3) are pairwise non-equivalent in the class of paratopological groups, it is natural to ask if some of these axioms are equivalent in the class of asymmetric normed spaces. In this paper, we will consider this question. We will also show some topological properties of asymmetric normed spaces that are closely related with the axioms T-1 and T-2 (among others). In particular, we will make a remark on [14, Theorem 13], which states that every T-1 asymmetric normed space with compact closed unit ball must be finite-dimensional (as a vector space). We will show that when the asymmetric normed space is finite-dimensional, the topological structure and the covering dimension of the space can be described in terms of certain algebraic properties. In particular, we will characterize the covering dimension of every finite-dimensional asymmetric normed space.