Group structures of a function spaces with the set-open topology

TL;DR

This paper investigates conditions on a family lambda that ensure the function space C(X,R^alpha) with the lambda-open topology forms various algebraic structures such as semitopological groups, topological groups, and topological vector spaces.

Contribution

It characterizes properties of lambda that guarantee the function space's algebraic and topological structures under the lambda-open topology.

Findings

Identifies conditions for C(X,R^alpha) to be a semitopological group

Establishes when C(X,R^alpha) forms a topological group or vector space

Provides a framework for understanding algebraic structures in function spaces

Abstract

In this paper, we find at the properties of the family lambda which imply that the function space C(X,R^alpha) with the lambda-open topology is a semitopological group (paratopological group, topological group, topological vector space and other algebraic structures) under the usual operations of addition and multiplication (and multiplication by scalars).