Brouwer Fixed Point Theorem for Simplexes

• Autorzy: Karol Pąk
• Języki publikacji: EN

Abstrakty

EN

In this article we prove the Brouwer fixed point theorem for an arbitrary simplex which is the convex hull of its n + 1 affinely indepedent vertices of εn. First we introduce the Lebesgue number, which for an arbitrary open cover of a compact metric space M is a positive real number so that any ball of about such radius must be completely contained in a member of the cover. Then we introduce the notion of a bounded simplicial complex and the diameter of a bounded simplicial complex. We also prove the estimation of diameter decrease which is connected with the barycentric subdivision. Finally, we prove the Brouwer fixed point theorem and compute the small inductive dimension of εn. This article is based on [16].

• Wydawca: De Gruyter Open
• Czasopismo: Formalized Mathematics
• Rocznik: 2011
• Tom: 19
• Numer: 3
• Strony: 145-150

Daty

wydano

2011-01-01

online

2012-04-26

Twórcy

autor

Karol Pąk

• Institute of Informatics, University of Białystok, Poland

Bibliografia

• Grzegorz 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. 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.
• Grzegorz Bancerek and Yasunari Shidama. Introduction to matroids. Formalized Mathematics, 16(4):325-332, 2008, doi:10.2478/v10037-008-0040-0.[Crossref]
• Leszek Borys. Paracompact and metrizable spaces. Formalized Mathematics, 2(4):481-485, 1991.
• Czesław Byliński. Binary operations. Formalized Mathematics, 1(1):175-180, 1990.
• Czesław Byliński. The complex numbers. Formalized Mathematics, 1(3):507-513, 1990.
• Czesław Byliński. Functions and their basic properties. Formalized Mathematics, 1(1):55-65, 1990.
• Czesław Byliński. Some basic properties of sets. Formalized Mathematics, 1(1):47-53, 1990.
• Agata Darmochwał. Compact spaces. Formalized Mathematics, 1(2):383-386, 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.
• Alicia de la Cruz. Totally bounded metric spaces. Formalized Mathematics, 2(4):559-562, 1991.
• Roman Duda. Wprowadzenie do topologii. PWN, 1986.
• Noboru Endou, Takashi Mitsuishi, and Yasunari Shidama. Convex sets and convex combinations. Formalized Mathematics, 11(1):53-58, 2003.
• Krzysztof Hryniewiecki. Basic properties of real numbers. Formalized Mathematics, 1(1):35-40, 1990.
• Stanisława Kanas, Adam Lecko, and Mariusz Startek. Metric spaces. Formalized Mathematics, 1(3):607-610, 1990.
• Artur Korniłowicz. The correspondence between n-dimensional Euclidean space and the product of n real lines. Formalized Mathematics, 18(1):81-85, 2010, doi: 10.2478/v10037-010-0011-0.[Crossref]
• Yatsuka Nakamura, Andrzej Trybulec, and Czesław Byliński. Bounded domains and unbounded domains. Formalized Mathematics, 8(1):1-13, 1999.
• Adam Naumowicz. On Segre's product of partial line spaces. Formalized Mathematics, 9(2):383-390, 2001.
• Beata Padlewska. Families of sets. Formalized Mathematics, 1(1):147-152, 1990.
• Beata Padlewska and Agata Darmochwał. Topological spaces and continuous functions. Formalized Mathematics, 1(1):223-230, 1990.
• Karol Pąk. Small inductive dimension of topological spaces. Formalized Mathematics, 17(3):207-212, 2009, doi: 10.2478/v10037-009-0025-7.[Crossref]
• Karol Pąk. Affine independence in vector spaces. Formalized Mathematics, 18(1):87-93, 2010, doi: 10.2478/v10037-010-0012-z.[Crossref]
• Karol Pąk. Abstract simplicial complexes. Formalized Mathematics, 18(1):95-106, 2010, doi: 10.2478/v10037-010-0013-y.[Crossref]
• Karol Pąk. Sperner's lemma. Formalized Mathematics, 18(4):189-196, 2010, doi: 10.2478/v10037-010-0022-x.[Crossref]
• Karol Pąk. Continuity of barycentric coordinates in Euclidean topological spaces. Formalized Mathematics, 19(3):139-144, 2011, doi: 10.2478/v10037-011-0022-5.[Crossref]
• Andrzej Trybulec. Domains and their Cartesian products. Formalized Mathematics, 1(1):115-122, 1990.
• Andrzej Trybulec. A Borsuk theorem on homotopy types. Formalized Mathematics, 2(4):535-545, 1991.
• Michał J. Trybulec. Integers. Formalized Mathematics, 1(3):501-505, 1990.
• Wojciech A. Trybulec. Vectors in real linear space. Formalized Mathematics, 1(2):291-296, 1990.
• Zinaida Trybulec. Properties of subsets. Formalized Mathematics, 1(1):67-71, 1990.
• Edmund Woronowicz. Relations defined on sets. Formalized Mathematics, 1(1):181-186, 1990.

Identyfikatory

DOI

10.2478/v10037-011-0023-4