The Stereographic Projection in Topological Modules

Abstract

The stereographic projection is constructed in topological modules. Let A be an additively symmetric closed subset of a topological R-module M such that 0∈int(A). If there exists a continuous functional m*:M→R in the dual module M*, an invertible s∈U(R) and an element a in the topological boundary bd(A) of A in such a way that m*−1({s})∩int(A)=⌀, a∈m*−1({s})∩bd(A), and s+m*bd(A)∖{−a}⊆U(R), then the following function b↦−a+2s(m*(b)+s)−1(b+a), from bd(A)∖{−a} to (m*)−1({s}), is a well-defined stereographic projection (also continuous if multiplicative inversion is continuous on R). Finally, we provide sufficient conditions for the previous stereographic projection to become a homeomorphism.