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Separation axioms and covering dimension of asymmetric normed spaces. (English) Zbl 1479.46006

Summary: 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 [L. M. García-Raffi, Topology Appl. 153, No. 5–6, 844–853 (2005; Zbl 1101.46017), 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.

MSC:

46A19 Other “topological” linear spaces (convergence spaces, ranked spaces, spaces with a metric taking values in an ordered structure more general than \(\mathbb{R}\), etc.)
52A21 Convexity and finite-dimensional Banach spaces (including special norms, zonoids, etc.) (aspects of convex geometry)
54D10 Lower separation axioms (\(T_0\)–\(T_3\), etc.)
54F45 Dimension theory in general topology
54H11 Topological groups (topological aspects)
22A30 Other topological algebraic systems and their representations

Citations:

Zbl 1101.46017
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References:

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