On a topological method in semi-ordered linear spaces

In Banach spaces, we always obtain a continuous linear func- tional as the limit of a weakly converging sequence of continuous linear functionals.And this property is based on a fact that a complete metric space is of second category.In continuous semi- ordered linear spaces, bounded (continuous or universally conti- nuous) linear tunctionals have the same property.To investigate the relation o these two cases, first in 1 we will define a kind of topology in abstract spaces by which we obtain a topological space having the property akin to that of second category one under some condition.In 2 spplying it to semi-ordered linear spaces we will show that we can discuss the problem mentioned above by the topological method.We shall make use of notations ia the books of H. Nskano ).1. Cell-topology.