THE FUNCTON SPACE TOPOLOGY AS PRODUCT TOPOLOGY ON PRODUCT SPACE

Abstract

Function space is one of the fundamental areas of research in functional analysis. We need to explore some potential topologies in the family of functions F from some arbitrary set X into another set Y with a view to investigate the properties of the various topologies generated on the family of functions F by subsets of X, such as compactness, separability and completeness. From the result of the research, we established the nature or characteristics of the following function space topologies, namely: Product topology, Point-open-topology  , Topology of point-wise-convergence , Compact-open topology , Topology of uniform convergence and the Seminorm topology  . Comparing the function space topologies, we established that  . This shows that  is the strongest. But the three topologies coincide when X is finite. The product topology  is equally, the Seminorm topology  of point-wise –convergence on F. If the function space has the topology with the base of the form  , we call this topology, the Seminorm topology of uniform convergence on F. Uses were made of the definition of the defining subbase and base of a topology. The concept of compactness of a set, and the composite mapping were used to establish results. We have also made use of the definition and properties of seminorm to establish results. It was also established that the function space topologies are Hausdorff as each one separates points of X. Uses were made of the Separation Axioms. Not alone, it was established that the function space topologies are locally convex if 𝐹+ contains zero function. This work contributes to knowledge by having established that “every function space topology is a product topology”.