Some Special Sets in an Exponential Vector Space

Abstract

In this paper, we have studied 'absorbing' and 'balanced' sets in an Exponential Vector Space (\emph{evs} in short) over the field $\mathbb K$ of real or complex. These sets play pivotal role to describe several aspects of a topological evs. We have characterised a local base at the additive identity in terms of balanced and absorbing sets in a topological evs over the field $\mathbb K$. Also, we have found a sufficient condition under which an evs can be topologised to form a topological evs. Next, we have introduced the concept of 'bounded sets' in a topological evs over the field $\mathbb K$ and characterised them with the help of balanced sets. Also we have shown that compactness implies boundedness of a set in a topological evs. In the last section we have introduced the concept of `radial' evs which characterises an evs over the field $\mathbb K$ up to order-isomorphism. Also, we have shown that every topological evs is radial. Further, it has been shown that "the usual subspace topology is the finest topology with respect to which $[0,\infty)$ forms a topological evs over the field $\mathbb K$".