Separation axioms and covering dimension of asymmetric normed spaces

Abstract

It is well known that every asymmetric normed space is a T0 paratopological group. Since all Ti 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 T1 and T2 (among others). In particular, we will make a remark on [14, Theorem 13], which states that every T1 asymmetric normed space with compact closed unit ball must be nite-dimensional (as a vector space). We will show that when the asymmetric normed space is nite- 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 nite-dimensional asymmetric normed space.

Mathematics Subject Classication (2010): 22A30, 46A19, 52A21, 54D10, 54F45, 54H11.

Key words: Asymmetric norm, right bounded, covering dimension, separation axioms, paratopological group.