A Reverse topology and translated topology of semi-linear topological spaces

Reverse topology and translated topology

Abstract

On topological semi-linear spaces there are multiple ways to defne distinct convergences starting from the basic semi-linear topology, such as translated convergence, convergence in di¤erence, reverse convergence. Translated convergence comes from translated topology and it has many good properties. The aim of this paper is to show that reverse convergence is also topological. New connection properties between these convergences have also been obtained. We study this problem in general framework or using neighborhoods of the origin which are totally bounded by nets. Fi- nally they are examined in the case of semi-metrizable semi-linear spaces. An important tool of our research is the concept of translated Cauchy net, through which we study the "completeness" of topological semi-linear spaces.