Separation axioms and covering dimension of asymmetric normed spaces

Abstract

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 nonequivalent 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 [16, 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.2010 Mathematics Subject Classification. 22A30, 46A19, 52A21, 54D10 , 54F45, 54H11,

$\theta$-metric function in the problem of minimization of functionals

“…In particular, as ψ one can consider any asymmetric norm ∥ • | (assuming that it is continuous). Note that continuity of ∥ • | is equivalent to saying that the ball B(0, 1) is closed (see [18]).…”

Existence, uniqueness, and stability of best and near-best approximations

“…The theory of asymmetric normed spaces and their applications is in active development at the present time. For example, various topological and functionalanalytic topics are considered in [17], [18], and [24], optimal location problems (with asymmetric norms) are studied, for instance, in [25], [47], and [43] (in such problems, an important role is also played by Chebyshev centres and Chebyshev nets relative to the asymmetric norms), and problems related to principal component analysis in statistics (one of the most popular methods of compact representation of data) are dealt with in [56]. For other applications, also see [17].…”

Continuous selections of set-valued mappings and approximation in asymmetric and semilinear spaces

“…An important example of such a space is the space of all closed bounded sets equipped with the Hausdorff metric. For numerous results on general properties of asymmetric spaces, and various problems of geometric approximation theory, see, for example, [1]- [16].…”