The Stereographic Projection in Topological Modules - Francisco Javier García-Pacheco (Axioms, 2023)
The stereographic projection is constructed in topological modules. Let A be an additively symmetric closed subset of a topological R-module M such that <semantics>0∈int(A)</semantics>. If there exists a continuous functional <semantics>m*:M→R</semantics> in the dual module <semantics>M*</semantics>, an invertible <semantics>s∈U(R)</semantics> and an element a in the topological boundary <semantics>bd(A)</semantics> of A in such a way that <semantics><mfenced separators="" open="(" close=")">m*</mfenced>−1({s})∩int(A)=⌀</semantics>, <semantics>a∈<mfenced separators="" open="(" close=")">m*</mfenced>−1({s})∩bd(A)</semantics>, and <semantics>s+m*<mfenced separators="" open="(" close=")">bd(A)\{−a}</mfenced>⊆U(R)</semantics>, then the following function <semantics>b↦−a+2s(m*(b)+s)−1(b+a)</semantics>, from <semantics>bd(A)\{−a}</semantics> to <semantics>(m*)−1({s})</semantics>, 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.