The Stereographic Projection in Topological Modules

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 is an element of int(A). If there exists a continuous functional m* : M -> R in the dual module M*, an invertible s is an element of U(R) and an element a in the topological boundary bd(A) of A in such a way that (m*)(-1)({s}) boolean AND int(A) = phi, a is an element of (m*)(-1)({s}) boolean AND bd(A), and s + m*(bd(A) \ {-a}) subset of U(R), then the following function b proves -> -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.