❆❜str❛❝t ✕ ❆ss❡ss✐♥❣ s♦❢t✇❛r❡ q✉❛❧✐t2 ❛ttr✐❜✉t❡s ✭s✉❝❤ ❛s ♣❡r❢♦r♠❛♥❝❡✱ r❡❧✐❛❜✐❧✐t2✱ ❛♥❞ s❡❝✉r✐t2✮ ❢r♦♠ s♦✉r❝❡ ❝♦❞❡ ✐s ♦❢ t❤❡ ✉t♠♦st ✐♠♣♦rt❛♥❝❡✳ ❚❤❡ ♣❡r❢♦r♠❛♥❝❡ ♦❢ ❛ s♦❢t✇❛r❡ s2st❡♠ ❝❛♥ ❜❡ ✐♠♣r♦✈❡❞ ❜2 ✐ts ♣❛r❛❧❧❡❧ ❛♥❞ ❞✐str✐❜✉t❡❞ ❡1❡❝✉t✐♦♥✳ ❚❤❡ ❛✐♠ ♦❢ t❤❡ ♣❛r❛❧❧❡❧ ❛♥❞ ❞✐str✐❜✉t❡❞ ❡1❡❝✉t✐♦♥ ✐s t♦ s♣❡❡❞ ✉♣ ❜2 ♣r♦✈✐❞✐♥❣ t❤❡ ♠❛1✐♠✉♠ ♣♦ss✐❜❧❡ ❝♦♥❝✉rr❡♥❝2 ✐♥ ❡1❡❝✉t✐♥❣ t❤❡ ❞✐str✐❜✉t❡❞ s❡❣♠❡♥ts✳ ■t ✐s ❛...

❆❜str❛❝t ✕ ❆ss❡ss✐♥❣ s♦❢t✇❛r❡ q✉❛❧✐t2 ❛ttr✐❜✉t❡s ✭s✉❝❤ ❛s ♣❡r❢♦r♠❛♥❝❡✱ r❡❧✐❛❜✐❧✐t2✱ ❛♥❞ s❡❝✉r✐t2✮ ❢r♦♠ s♦✉r❝❡ ❝♦❞❡ ✐s ♦❢ t❤❡ ✉t♠♦st ✐♠♣♦rt❛♥❝❡✳ ❚❤❡ ♣❡r❢♦r♠❛♥❝❡ ♦❢ ❛ s♦❢t✇❛r❡ s2st❡♠ ❝❛♥ ❜❡ ✐♠♣r♦✈❡❞ ❜2 ✐ts ♣❛r❛❧❧❡❧ ❛♥❞ ❞✐str✐❜✉t❡❞ ❡1❡❝✉t✐♦♥✳ ❚❤❡ ❛✐♠ ♦❢ t❤❡ ♣❛r❛❧❧❡❧ ❛♥❞ ❞✐str✐❜✉t❡❞ ❡1❡❝✉t✐♦♥ ✐s t♦ s♣❡❡❞ ✉♣ ❜2 ♣r♦✈✐❞✐♥❣ t❤❡ ♠❛1✐♠✉♠ ♣♦ss✐❜❧❡ ❝♦♥❝✉rr❡♥❝2 ✐♥ ❡1❡❝✉t✐♥❣ t❤❡ ❞✐str✐❜✉t❡❞ s❡❣♠❡♥ts✳ ■t ✐s ❛...

❆❜str❛❝t ✕ ❆ss❡ss✐♥❣ s♦❢t✇❛r❡ q✉❛❧✐t2 ❛ttr✐❜✉t❡s ✭s✉❝❤ ❛s ♣❡r❢♦r♠❛♥❝❡✱ r❡❧✐❛❜✐❧✐t2✱ ❛♥❞ s❡❝✉r✐t2✮ ❢r♦♠ s♦✉r❝❡ ❝♦❞❡ ✐s ♦❢ t❤❡ ✉t♠♦st ✐♠♣♦rt❛♥❝❡✳ ❚❤❡ ♣❡r❢♦r♠❛♥❝❡ ♦❢ ❛ s♦❢t✇❛r❡ s2st❡♠ ❝❛♥ ❜❡ ✐♠♣r♦✈❡❞ ❜2 ✐ts ♣❛r❛❧❧❡❧ ❛♥❞ ❞✐str✐❜✉t❡❞ ❡1❡❝✉t✐♦♥✳ ❚❤❡ ❛✐♠ ♦❢ t❤❡ ♣❛r❛❧❧❡❧ ❛♥❞ ❞✐str✐❜✉t❡❞ ❡1❡❝✉t✐♦♥ ✐s t♦ s♣❡❡❞ ✉♣ ❜2 ♣r♦✈✐❞✐♥❣ t❤❡ ♠❛1✐♠✉♠ ♣♦ss✐❜❧❡ ❝♦♥❝✉rr❡♥❝2 ✐♥ ❡1❡❝✉t✐♥❣ t❤❡ ❞✐str✐❜✉t❡❞ s❡❣♠❡♥ts✳ ■t ✐s ❛...

❆❜str❛❝t ✕ ❆ss❡ss✐♥❣ s♦❢t✇❛r❡ q✉❛❧✐t2 ❛ttr✐❜✉t❡s ✭s✉❝❤ ❛s ♣❡r❢♦r♠❛♥❝❡✱ r❡❧✐❛❜✐❧✐t2✱ ❛♥❞ s❡❝✉r✐t2✮ ❢r♦♠ s♦✉r❝❡ ❝♦❞❡ ✐s ♦❢ t❤❡ ✉t♠♦st ✐♠♣♦rt❛♥❝❡✳ ❚❤❡ ♣❡r❢♦r♠❛♥❝❡ ♦❢ ❛ s♦❢t✇❛r❡ s2st❡♠ ❝❛♥ ❜❡ ✐♠♣r♦✈❡❞ ❜2 ✐ts ♣❛r❛❧❧❡❧ ❛♥❞ ❞✐str✐❜✉t❡❞ ❡1❡❝✉t✐♦♥✳ ❚❤❡ ❛✐♠ ♦❢ t❤❡ ♣❛r❛❧❧❡❧ ❛♥❞ ❞✐str✐❜✉t❡❞ ❡1❡❝✉t✐♦♥ ✐s t♦ s♣❡❡❞ ✉♣ ❜2 ♣r♦✈✐❞✐♥❣ t❤❡ ♠❛1✐♠✉♠ ♣♦ss✐❜❧❡ ❝♦♥❝✉rr❡♥❝2 ✐♥ ❡1❡❝✉t✐♥❣ t❤❡ ❞✐str✐❜✉t❡❞ s❡❣♠❡♥ts✳ ■t ✐s ❛...

The stereographic projection is constructed in topological modules. Let A be an additively symmetric closed subset of a R-module M such that 0∈int(A). If there exists continuous functional m*:M→R the dual module M*, invertible s∈U(R) and element boundary bd(A) way m*−1({s})∩int(A)=⌀, a∈m*−1({s})∩bd(A), s+m*bd(A)\{−a}⊆U(R), then following function b↦−a+2s(m*(b)+s)−1(b+a), from bd(A)\{−a} to (m*)...

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