Chapter 5 - Topological structure on quotient space

Abstract

To investigate the geometric structure of the set of matrices and its quotient set it is necessary to pose a proper topology on such sets. Using conventional topology on the set of matrices, the product topology and quotient topology are proposed for quotient space. Next, an inner product is proposed for matrix space. A metric follows immediately. Then they are extended to quotient space, which makes it into a metric space. Some useful properties are investigated. Then metric topology deduced by metric is naturally provided, and compared with quotient topology and product topology.