Planes and Spheres as Topological Manifolds. Stereographic Projection

The goal of this article is to show some examples of topological manifolds: planes and spheres in Euclidean space. In doing it, the article introduces the stereographic projection [25].Via del Pero 102, 54038 Montignoso, ItalyGrzegorz Bancerek. Cardinal numbers. Formalized Mathematics, 1(2):377-382, 1990.Grzegorz Bancerek. The fundamental properties of natural numbers. Formalized Mathematics, 1(1):41-46, 1990.Grzegorz Bancerek. König's theorem. Formalized Mathematics, 1(3):589-593, 1990.Grzegorz Bancerek. The ordinal numbers. Formalized Mathematics, 1(1):91-96, 1990.Grzegorz Bancerek. Monoids. Formalized Mathematics, 3(2):213-225, 1992.Grzegorz Bancerek and Krzysztof Hryniewiecki. Segments of natural numbers and finite sequences. Formalized Mathematics, 1(1):107-114, 1990.Czesław Byliński. Binary operations. Formalized Mathematics, 1(1):175-180, 1990.Czesław Byliński. Finite sequences and tuples of elements of a non-empty sets. Formalized Mathematics, 1(3):529-536, 1990.Czesław Byliński. Functions and their basic properties. Formalized Mathematics, 1(1):55-65, 1990.Czesław Byliński. Functions from a set to a set. Formalized Mathematics, 1(1):153-164, 1990.Czesław Byliński. Partial functions. Formalized Mathematics, 1(2):357-367, 1990.Czesław Byliński. Some basic properties of sets. Formalized Mathematics, 1(1):47-53, 1990.Czesław Byliński. The sum and product of finite sequences of real numbers. Formalized Mathematics, 1(4):661-668, 1990.Agata Darmochwał. Families of subsets, subspaces and mappings in topological spaces. Formalized Mathematics, 1(2):257-261, 1990.Agata Darmochwał. Finite sets. Formalized Mathematics, 1(1):165-167, 1990.Agata Darmochwał. The Euclidean space. Formalized Mathematics, 2(4):599-603, 1991.Agata Darmochwał and Yatsuka Nakamura. Metric spaces as topological spaces - fundamental concepts. Formalized Mathematics, 2(4):605-608, 1991.Krzysztof Hryniewiecki. Basic properties of real numbers. Formalized Mathematics, 1(1):35-40, 1990.Katarzyna Jankowska. Matrices. Abelian group of matrices. Formalized Mathematics, 2(4):475-480, 1991.Stanisława Kanas, Adam Lecko, and Mariusz Startek. Metric spaces. Formalized Mathematics, 1(3):607-610, 1990.Artur Korniłowicz and Yasunari Shidama. Intersections of intervals and balls in En/T Formalized Mathematics, 12(3):301-306, 2004.Artur Korniłowicz and Yasunari Shidama. Some properties of circles on the plane. Formalized Mathematics, 13(1):117-124, 2005.Jarosław Kotowicz. Real sequences and basic operations on them. Formalized Mathematics, 1(2):269-272, 1990.Eugeniusz Kusak, Wojciech Leończuk, and Michał Muzalewski. Abelian groups, fields and vector spaces. Formalized Mathematics, 1(2):335-342, 1990.John M. Lee. Introduction to Topological Manifolds. Springer-Verlag, New York Berlin Heidelberg, 2000.Robert Milewski. Bases of continuous lattices. Formalized Mathematics, 7(2):285-294, 1998.Yatsuka Nakamura, Artur Korniłowicz, Nagato Oya, and Yasunari Shidama. The real vector spaces of finite sequences are finite dimensional. Formalized Mathematics, 17(1):1-9, 2009, doi:10.2478/v10037-009-0001-2.Henryk Oryszczyszyn and Krzysztof Prażmowski. Real functions spaces. Formalized Mathematics, 1(3):555-561, 1990.Beata Padlewska. Families of sets. Formalized Mathematics, 1(1):147-152, 1990.Beata Padlewska. Locally connected spaces. Formalized Mathematics, 2(1):93-96, 1991.Beata Padlewska and Agata Darmochwał. Topological spaces and continuous functions. Formalized Mathematics, 1(1):223-230, 1990.Karol Pąk. Basic properties of metrizable topological spaces. Formalized Mathematics, 17(3):201-205, 2009, doi: 10.2478/v10037-009-0024-8.Marco Riccardi. The definition of topological manifolds. Formalized Mathematics, 19(1):41-44, 2011, doi: 10.2478/v10037-011-0007-4.Andrzej Trybulec. Binary operations applied to functions. Formalized Mathematics, 1(2):329-334, 1990.Andrzej Trybulec. On the sets inhabited by numbers. Formalized Mathematics, 11(4):341-347, 2003.Wojciech A. Trybulec. Basis of real linear space. Formalized Mathematics, 1(5):847-850, 1990.Wojciech A. Trybulec. Linear combinations in real linear space. Formalized Mathematics, 1(3):581-588, 1990.Wojciech A. Trybulec. Subspaces and cosets of subspaces in real linear space. Formalized Mathematics, 1(2):297-301, 1990.Wojciech A. Trybulec. Vectors in real linear space. Formalized Mathematics, 1(2):291-296, 1990.Edmund Woronowicz. Relations and their basic properties. Formalized Mathematics, 1(1):73-83, 1990.Edmund Woronowicz. Relations defined on sets. Formalized Mathematics, 1(1):181-186, 1990.Mariusz Żynel and Adam Guzowski. T0 topological spaces. Formalized Mathematics, 5(1):75-77, 1996

Fundamental Group of n-sphere for n ≥ 2

Triviality of fundamental groups of spheres of dimension greater than 1 is proven, [17].This work has been supported by the Polish Ministry of Science and Higher Education project “Managing a Large Repository of Computer-verified Mathematical Knowledge” (N N519 385136).Riccardi Marco - Via del Pero 102, 54038 Montignoso, ItalyKorniłowicz Artur - Institute of Informatics, University of Białystok, Sosnowa 64, 15-887 Białystok, PolandGrzegorz Bancerek. The fundamental properties of natural numbers. Formalized Mathematics, 1(1):41-46, 1990.Grzegorz Bancerek. The ordinal numbers. Formalized Mathematics, 1(1):91-96, 1990.Grzegorz Bancerek and Krzysztof Hryniewiecki. Segments of natural numbers and finite sequences. Formalized Mathematics, 1(1):107-114, 1990.Czesław Bylinski. Binary operations. Formalized Mathematics, 1(1):175-180, 1990.Czesław Bylinski. Functions and their basic properties. Formalized Mathematics, 1(1):55-65, 1990.Czesław Bylinski. Functions from a set to a set. Formalized Mathematics, 1(1):153-164, 1990.Czesław Bylinski. Partial functions. Formalized Mathematics, 1(2):357-367, 1990.Czesław Bylinski. Some basic properties of sets. Formalized Mathematics, 1(1):47-53, 1990.Agata Darmochwał. The Euclidean space. Formalized Mathematics, 2(4):599-603, 1991.Agata Darmochwał and Yatsuka Nakamura. Metric spaces as topological spaces - fundamental concepts. Formalized Mathematics, 2(4):605-608, 1991.Adam Grabowski. Introduction to the homotopy theory. Formalized Mathematics, 6(4):449-454, 1997.Adam Grabowski and Artur Korniłowicz. Algebraic properties of homotopies. Formalized Mathematics, 12(3):251-260, 2004.Artur Korniłowicz. The fundamental group of convex subspaces of EnT. Formalized Mathematics, 12(3):295-299, 2004.Artur Korniłowicz. On the isomorphism of fundamental groups. Formalized Mathematics, 12(3):391-396, 2004.Artur Korniłowicz and Yasunari Shidama. Intersections of intervals and balls in EnT. Formalized Mathematics, 12(3):301-306, 2004.Artur Korniłowicz, Yasunari Shidama, and Adam Grabowski. The fundamental group. Formalized Mathematics, 12(3):261-268, 2004.John M. Lee. Introduction to Topological Manifolds. Springer-Verlag, New York Berlin Heidelberg, 2000.Beata Padlewska and Agata Darmochwał. Topological spaces and continuous functions. Formalized Mathematics, 1(1):223-230, 1990.Konrad Raczkowski and Paweł Sadowski. Equivalence relations and classes of abstraction. Formalized Mathematics, 1(3):441-444, 1990.Konrad Raczkowski and Paweł Sadowski. Topological properties of subsets in real numbers. Formalized Mathematics, 1(4):777-780, 1990.Marco Riccardi. The definition of topological manifolds. Formalized Mathematics, 19(1):41-44, 2011, doi: 10.2478/v10037-011-0007-4.Marco Riccardi. Planes and spheres as topological manifolds. Stereographic projection. Formalized Mathematics, 20(1):41-45, 2012, doi: 10.2478/v10037-012-0006-0.Zinaida Trybulec. Properties of subsets. Formalized Mathematics, 1(1):67-71, 1990.Edmund Woronowicz. Relations and their basic properties. Formalized Mathematics, 1(1):73-83, 1990.Edmund Woronowicz. Relations defined on sets. Formalized Mathematics, 1(1):181-186, 1990