| Summary: | The stereographic projection is constructed in topological modules. Let <i>A</i> be an additively symmetric closed subset of a topological <i>R</i>-module <i>M</i> such that <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mn>0</mn><mo>∈</mo><mi>int</mi><mo>(</mo><mi>A</mi><mo>)</mo></mrow></semantics></math></inline-formula>. If there exists a continuous functional <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msup><mi>m</mi><mo>*</mo></msup><mo>:</mo><mi>M</mi><mo>→</mo><mi>R</mi></mrow></semantics></math></inline-formula> in the dual module <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msup><mi>M</mi><mo>*</mo></msup></semantics></math></inline-formula>, an invertible <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>s</mi><mo>∈</mo><mi mathvariant="script">U</mi><mo>(</mo><mi>R</mi><mo>)</mo></mrow></semantics></math></inline-formula> and an element <i>a</i> in the topological boundary <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>bd</mi><mo>(</mo><mi>A</mi><mo>)</mo></mrow></semantics></math></inline-formula> of <i>A</i> in such a way that <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msup><mfenced separators="" open="(" close=")"><msup><mi>m</mi><mo>*</mo></msup></mfenced><mrow><mo>−</mo><mn>1</mn></mrow></msup><mrow><mo>(</mo><mrow><mo>{</mo><mi>s</mi><mo>}</mo></mrow><mo>)</mo></mrow><mo>∩</mo><mi>int</mi><mrow><mo>(</mo><mi>A</mi><mo>)</mo></mrow><mo>=</mo><mo>⌀</mo></mrow></semantics></math></inline-formula>, <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>a</mi><mo>∈</mo><msup><mfenced separators="" open="(" close=")"><msup><mi>m</mi><mo>*</mo></msup></mfenced><mrow><mo>−</mo><mn>1</mn></mrow></msup><mrow><mo>(</mo><mrow><mo>{</mo><mi>s</mi><mo>}</mo></mrow><mo>)</mo></mrow><mo>∩</mo><mi>bd</mi><mrow><mo>(</mo><mi>A</mi><mo>)</mo></mrow></mrow></semantics></math></inline-formula>, and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>s</mi><mo>+</mo><msup><mi>m</mi><mo>*</mo></msup><mfenced separators="" open="(" close=")"><mi>bd</mi><mo>(</mo><mi>A</mi><mo>)</mo><mo>\</mo><mo>{</mo><mo>−</mo><mi>a</mi><mo>}</mo></mfenced><mo>⊆</mo><mi mathvariant="script">U</mi><mrow><mo>(</mo><mi>R</mi><mo>)</mo></mrow></mrow></semantics></math></inline-formula>, then the following function <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>b</mi><mo>↦</mo><mo>−</mo><mi>a</mi><mo>+</mo><mn>2</mn><mi>s</mi><msup><mrow><mo>(</mo><msup><mi>m</mi><mo>*</mo></msup><mrow><mo>(</mo><mi>b</mi><mo>)</mo></mrow><mo>+</mo><mi>s</mi><mo>)</mo></mrow><mrow><mo>−</mo><mn>1</mn></mrow></msup><mrow><mo>(</mo><mi>b</mi><mo>+</mo><mi>a</mi><mo>)</mo></mrow></mrow></semantics></math></inline-formula>, from <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>bd</mi><mo>(</mo><mi>A</mi><mo>)</mo><mo>\</mo><mo>{</mo><mo>−</mo><mi>a</mi><mo>}</mo></mrow></semantics></math></inline-formula> to <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msup><mrow><mo>(</mo><msup><mi>m</mi><mo>*</mo></msup><mo>)</mo></mrow><mrow><mo>−</mo><mn>1</mn></mrow></msup><mrow><mo>(</mo><mrow><mo>{</mo><mi>s</mi><mo>}</mo></mrow><mo>)</mo></mrow></mrow></semantics></math></inline-formula>, is a well-defined stereographic projection (also continuous if multiplicative inversion is continuous on <i>R</i>). Finally, we provide sufficient conditions for the previous stereographic projection to become a homeomorphism.
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