Matrix-Closed Weak Topologies and the Intersection of the Line-Open Topologies in Rn

Abstract

In the paper published by the authors in the International Journal of Research and Scientific Innovation (IJRSI), in September 2024, titled Hyperplane-Open Weak Topologies in Rn, a general procedure for constructing hyperplane-open weak topology on Rn was revealed. The process led to, among other things, the formulation of matrix-open weak topology on the Cartesian plane. In the present work, we are to construct a matrix-closed topology on the Cartesian plane. Also, in the first work we constructed horizontal and vertical line-open weak topologies of the plane and showed that the usual topology (of the plane) is actually weaker than the intersection of these two topologies. Then we made a conjecture that the usual topology is actually equal to this intersection; and our reviewers opined that a definite proof (or disproof) of that conjecture was necessarily important, in order to answer an unsettled question regarding the entire work. This task is what we have in addition undertaken in the present work: We have supplied here a definite proof that the usual topology of R2 and indeed of Rn is actually the intersection of the horizontal line and the vertical line-open weak topologies of R2 (or Rn).