Generalized Open Spheres in Generalized Metric Spaces PDF

Keywords

spheres, mathematics, metric spaces

Abstract

We shall define size functions and sizable spaces. These spaces are generalizations of metric spaces. We shall discuss some of their properties and then we discuss the metrizable space they induce under certain conditions on the sizable space. We shall also define open spheres and open balls and discuss the topologies they induce. Among the results we obtain, we have the following: 1) Every countably compact sizable space must be separable, and hence, it must have at most countably many discrete points. 2) Every countably compact sizable space induces a compact Hausdorff metrizable space which is weaker than the original topology. We shall also discuss the metrizability of countably compact spaces as an application of our concepts