Some Special Sets in an Exponential Vector Space

TL;DR

This paper explores the structure of exponential vector spaces, introducing concepts like absorbing, balanced, bounded, and radial sets, and characterizes their roles in topological properties and classifications of these spaces.

Contribution

It provides new characterizations of topological exponential vector spaces, introduces bounded and radial sets, and establishes conditions for topologizing such spaces.

Findings

Characterized local bases using balanced and absorbing sets.

Established conditions for topologizing exponential vector spaces.

Proved that compactness implies boundedness in these spaces.

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 $\mathbb K$".