We consider two partitions over the space of linear semi-infinite programming parameters with a fixed index set and bounded coefficients (the functions of the constraints are bounded). The first one is the primal-dual partition inspired by consistency and boundedness of the optimal value of the linear semi-infinite optimization problems. The second one is a refinement of the primal-dual partition that arises considering the boundedness of the optimal set. These two partitions have been studied in the continuous case, this is, the set of indices is a compact infinite compact Hausdorff topological space and the functions defining the constraints are continuous. In this work, we present an extension of this case. We study same topological properties of the cells generated by the primal-dual partitions and characterize their interior. Through examples, we show that the results characterizing the sets of the partitions in the continuous case are neither necessary nor sufficient in both refinements. In addition, a sufficient condition for the boundedness of the optimal set of the dual problem has been presented.