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- Title
The Stereographic Projection in Topological Modules.
- Authors
García-Pacheco, Francisco Javier
- 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 + 2 s (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.
- Subjects
SPHERICAL projection; HOMEOMORPHISMS
- Publication
Axioms (2075-1680), 2023, Vol 12, Issue 2, p225
- ISSN
2075-1680
- Publication type
Article
- DOI
10.3390/axioms12020225