The Borsuk-Ulam Theorem

• Autorzy: Artur Korniłowicz , Marco Riccardi
• Języki publikacji: EN

Abstrakty

EN

The Borsuk-Ulam theorem about antipodals is proven, [18, pp. 32-33].

• Wydawca: De Gruyter Open
• Czasopismo: Formalized Mathematics
• Rocznik: 2012
• Tom: 20
• Numer: 2
• Strony: 105-112

Daty

wydano

2012-12-01

online

2013-02-02

Twórcy

autor

Artur Korniłowicz

• Institute of Informatics, University of Białystok, Sosnowa 64, 15-887 Białystok, Poland

autor

Marco Riccardi

• Via del Pero 102, 54038 Montignoso, Italy

Bibliografia

• [1] Grzegorz Bancerek. The fundamental properties of natural numbers. Formalized Mathematics, 1(1):41-46, 1990.
• [2] Grzegorz Bancerek. König’s theorem. Formalized Mathematics, 1(3):589-593, 1990.
• [3] Grzegorz Bancerek. The ordinal numbers. Formalized Mathematics, 1(1):91-96, 1990.
• [4] Grzegorz Bancerek and Krzysztof Hryniewiecki. Segments of natural numbers and finite sequences. Formalized Mathematics, 1(1):107-114, 1990.
• [5] Czesław Bylinski. The complex numbers. Formalized Mathematics, 1(3):507-513, 1990.
• [6] Czesław Bylinski. Functions and their basic properties. Formalized Mathematics, 1(1):55-65, 1990.
• [7] Czesław Bylinski. Functions from a set to a set. Formalized Mathematics, 1(1):153-164, 1990.
• [8] Czesław Bylinski. The modification of a function by a function and the iteration of the composition of a function. Formalized Mathematics, 1(3):521-527, 1990.
• [9] Czesław Bylinski. Partial functions. Formalized Mathematics, 1(2):357-367, 1990.
• [10] Czesław Bylinski. Some basic properties of sets. Formalized Mathematics, 1(1):47-53, 1990.
• [11] Czesław Bylinski. The sum and product of finite sequences of real numbers. Formalized Mathematics, 1(4):661-668, 1990.
• [12] Agata Darmochwał. Families of subsets, subspaces and mappings in topological spaces. Formalized Mathematics, 1(2):257-261, 1990.
• [13] Agata Darmochwał. The Euclidean space. Formalized Mathematics, 2(4):599-603, 1991.
• [14] Agata Darmochwał and Yatsuka Nakamura. Metric spaces as topological spaces - fundamental concepts. Formalized Mathematics, 2(4):605-608, 1991.
• [15] Adam Grabowski. Introduction to the homotopy theory. Formalized Mathematics, 6(4):449-454, 1997.
• [16] Adam Grabowski. On the subcontinua of a real line. Formalized Mathematics, 11(3):313-322, 2003.
• [17] Jarosław Gryko. Injective spaces. Formalized Mathematics, 7(1):57-62, 1998.
• [18] Allen Hatcher. Algebraic Topology. Cambridge University Press, 2002.
• [19] Krzysztof Hryniewiecki. Basic properties of real numbers. Formalized Mathematics, 1(1):35-40, 1990.
• [20] Kanchun, Hiroshi Yamazaki, and Yatsuka Nakamura. Cross products and tripple vector products in 3-dimensional Euclidean space. Formalized Mathematics, 11(4):381-383, 2003.
• [21] Artur Korniłowicz. Arithmetic operations on functions from sets into functional sets. Formalized Mathematics, 17(1):43-60, 2009, doi:10.2478/v10037-009-0005-y.[Crossref]
• [22] Artur Korniłowicz. On the continuity of some functions. Formalized Mathematics, 18(3):175-183, 2010, doi: 10.2478/v10037-010-0020-z.[Crossref]
• [23] Artur Korniłowicz and Yasunari Shidama. Intersections of intervals and balls in En T. Formalized Mathematics, 12(3):301-306, 2004.
• [24] Artur Korniłowicz and Yasunari Shidama. Some properties of circles on the plane. FormalizedMathematics, 13(1):117-124, 2005.
• [25] Artur Korniłowicz, Yasunari Shidama, and Adam Grabowski. The fundamental group. Formalized Mathematics, 12(3):261-268, 2004.
• [26] Akihiro Kubo and Yatsuka Nakamura. Angle and triangle in Euclidian topological space. Formalized Mathematics, 11(3):281-287, 2003.
• [27] Adam Naumowicz and Grzegorz Bancerek. Homeomorphisms of Jordan curves. FormalizedMathematics, 13(4):477-480, 2005.
• [28] Beata Padlewska. Connected spaces. Formalized Mathematics, 1(1):239-244, 1990.
• [29] Beata Padlewska and Agata Darmochwał. Topological spaces and continuous functions. Formalized Mathematics, 1(1):223-230, 1990.
• [30] Konrad Raczkowski and Paweł Sadowski. Real function continuity. Formalized Mathematics, 1(4):787-791, 1990.
• [31] Konrad Raczkowski and Paweł Sadowski. Topological properties of subsets in real numbers. Formalized Mathematics, 1(4):777-780, 1990.
• [32] Marco Riccardi and Artur Korniłowicz. Fundamental group of n-sphere for n ≥2. FormalizedMathematics, 20(2):97-104, 2012, doi: 10.2478/v10037-012-0013-1.[Crossref]
• [33] Piotr Rudnicki and Andrzej Trybulec. Abian’s fixed point theorem. Formalized Mathematics, 6(3):335-338, 1997.
• [34] Andrzej Trybulec. Binary operations applied to functions. Formalized Mathematics, 1(2):329-334, 1990.
• [35] Andrzej Trybulec. Enumerated sets. Formalized Mathematics, 1(1):25-34, 1990.
• [36] Andrzej Trybulec. A Borsuk theorem on homotopy types. Formalized Mathematics, 2(4):535-545, 1991.
• [37] Andrzej Trybulec and Czesław Bylinski. Some properties of real numbers. FormalizedMathematics, 1(3):445-449, 1990.
• [38] Michał J. Trybulec. Integers. Formalized Mathematics, 1(3):501-505, 1990.
• [39] Wojciech A. Trybulec. Vectors in real linear space. Formalized Mathematics, 1(2):291-296, 1990.
• [40] Zinaida Trybulec. Properties of subsets. Formalized Mathematics, 1(1):67-71, 1990.
• [41] Edmund Woronowicz. Relations and their basic properties. Formalized Mathematics, 1(1):73-83, 1990.
• [42] Edmund Woronowicz. Relations defined on sets. Formalized Mathematics, 1(1):181-186, 1990.
• [43] Yuguang Yang and Yasunari Shidama. Trigonometric functions and existence of circle ratio. Formalized Mathematics, 7(2):255-263, 1998.

Identyfikatory

DOI

10.2478/v10037-012-0014-0