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 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.

In this note, we study the geometric structure of compact convex sets in 2-dimensional asymmetric normed lattices. We prove that every q-compact convex set is strongly q-compact and we give a complete geometric description of the compact convex set with non empty interior in (ℝ2, q), where q is an asymmetric lattice norm.

Denote by $${\cal K}_0^n$$K 0 n the family of all closed convex sets A ⊂ ℝⁿ containing the origin 0 ∈ ℝⁿ. For $$A \in {\cal K}_0^n$$A ∈ K 0 n , its polar set is denoted by A°. In this paper, we investigate the topological nature of the polar mapping A → A° on $$({\cal K}_0^n,{d_{AW}})$$( K 0 n , d A W ) , where d_AW denotes the Attouch–Wets metric. We prove that $$({\cal K}_0^n,{d_{AW}})$$( K 0 n , d A W ) is homeomorphic to the Hilbert cube $$Q = \prod\nolimits_{i = 1}^\infty {[ - 1,1]} $$Q = ∏ i = 1 ∞ [ − 1 , 1 ] and the polar mapping is topologically conjugate with the standard based-free involution σ: Q → Q, defined by σ(x) = −x for all x ∈ Q. We also prove that among the inclusion-reversing involutions on $${\cal K}_0^n$$K 0 n (also called dualities), those and only those with a unique fixed point are topologically conjugate with the polar mapping, and they can be characterized as all the maps $$f:{\cal K}_0^n \to {\cal K}_0^n$$f : K 0 n → K 0 n of the form f(A) = T(A°), with T a positive-definite linear isomorphism of ℝⁿ.