Sperner's Lemma

• Autorzy: Karol Pąk
• Języki publikacji: EN

Abstrakty

EN

In this article we introduce and prove properties of simplicial complexes in real linear spaces which are necessary to formulate Sperner's lemma. The lemma states that for a function ƒ, which for an arbitrary vertex υ of the barycentric subdivision B of simplex K assigns some vertex from a face of K which contains υ, we can find a simplex S of B which satisfies ƒ(S) = K (see [10]).

• Wydawca: De Gruyter Open
• Czasopismo: Formalized Mathematics
• Rocznik: 2010
• Tom: 18
• Numer: 4
• Strony: 189-196

Daty

wydano

2010-01-01

online

2011-01-05

Twórcy

autor

Karol Pąk

• Institute of Informatics, University of Białystok, Poland

Bibliografia

• [1] Broderick Arneson and Piotr Rudnicki. Recognizing chordal graphs: Lex BFS and MCS. Formalized Mathematics, 14(4):187-205, 2006, doi:10.2478/v10037-006-0022-z.[Crossref]
• [2] Grzegorz Bancerek. Cardinal numbers. Formalized Mathematics, 1(2):377-382, 1990.
• [3] Grzegorz Bancerek. The fundamental properties of natural numbers. Formalized Mathematics, 1(1):41-46, 1990.
• [4] Grzegorz Bancerek. The ordinal numbers. Formalized Mathematics, 1(1):91-96, 1990.
• [5] Grzegorz Bancerek and Yasunari Shidama. Introduction to matroids. Formalized Mathematics, 16(4):325-332, 2008, doi:10.2478/v10037-008-0040-0.[Crossref]
• [6] Czesław Byliński. Functions and their basic properties. Formalized Mathematics, 1(1):55-65, 1990.
• [7] Czesław Byliński. Functions from a set to a set. Formalized Mathematics, 1(1):153-164, 1990.
• [8] Czesław Byliński. Some basic properties of sets. Formalized Mathematics, 1(1):47-53, 1990.
• [9] Agata Darmochwał. Finite sets. Formalized Mathematics, 1(1):165-167, 1990.
• [10] Roman Duda. Wprowadzenie do topologii. PWN, 1986.
• [11] Noboru Endou, Takashi Mitsuishi, and Yasunari Shidama. Convex sets and convex combinations. Formalized Mathematics, 11(1):53-58, 2003.
• [12] Krzysztof Hryniewiecki. Basic properties of real numbers. Formalized Mathematics, 1(1):35-40, 1990.
• [13] Adam Naumowicz. On Segre's product of partial line spaces. Formalized Mathematics, 9(2):383-390, 2001.
• [14] Beata Padlewska. Families of sets. Formalized Mathematics, 1(1):147-152, 1990.
• [15] Beata Padlewska and Agata Darmochwał. Topological spaces and continuous functions. Formalized Mathematics, 1(1):223-230, 1990.
• [16] Karol Pąk. Affine independence in vector spaces. Formalized Mathematics, 18(1):87-93, 2010, doi: 10.2478/v10037-010-0012-z.[Crossref]
• [17] Karol Pąk. Abstract simplicial complexes. Formalized Mathematics, 18(1):95-106, 2010, doi: 10.2478/v10037-010-0013-y.[Crossref]
• [18] Karol Pąk. The geometric interior in real linear spaces. Formalized Mathematics, 18(3):185-188, 2010, doi: 10.2478/v10037-010-0021-y.[Crossref]
• [19] Andrzej Trybulec. Domains and their Cartesian products. Formalized Mathematics, 1(1):115-122, 1990.
• [20] Wojciech A. Trybulec. Vectors in real linear space. Formalized Mathematics, 1(2):291-296, 1990.
• [21] Zinaida Trybulec. Properties of subsets. Formalized Mathematics, 1(1):67-71, 1990.
• [22] Edmund Woronowicz. Relations and their basic properties. Formalized Mathematics, 1(1):73-83, 1990.
• [23] Edmund Woronowicz. Relations defined on sets. Formalized Mathematics, 1(1):181-186, 1990.

Identyfikatory

DOI

10.2478/v10037-010-0022-x