No 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,

Let X be a topological space and CL(X) be the family of all nonempty closed subsets of X.In this paper we discuss the problem of when a continuous map between topological spaces induces a continuous function between their respective hyperspaces. As a main result we characterize the continuity of the induced function in the case of the Fell and Attouch-Wets hyperspaces. Additionally we explore the problem of whether a continuous action of a topological group G on a topological space X induces a continuous action on CL(X). In particular we give sufficient conditions on the topology of G to guarantee that the induced action on CL(X) is continuous, provided that CL(X) is equipped with the Hausdorff or the Attouch-Wets metric topology.