PL EN

Preferencje
Język
Widoczny [Schowaj] Abstrakt
Liczba wyników
Czasopismo

## Formalized Mathematics

2014 | 22 | 1 | 21-28
Tytuł artykułu

### Brouwer Invariance of Domain Theorem

Autorzy
Treść / Zawartość
Warianty tytułu
Języki publikacji
EN
Abstrakty
EN
In this article we focus on a special case of the Brouwer invariance of domain theorem. Let us A, B be a subsets of εn, and f : A → B be a homeomorphic. We prove that, if A is closed then f transform the boundary of A to the boundary of B; and if B is closed then f transform the interior of A to the interior of B. These two cases are sufficient to prove the topological invariance of dimension, which is used to prove basic properties of the n-dimensional manifolds, and also to prove basic properties of the boundary and the interior of manifolds, e.g. the boundary of an n-dimension manifold with boundary is an (n − 1)-dimension manifold. This article is based on [18]; [21] and [20] can also serve as reference books.
Słowa kluczowe
EN
Wydawca
Czasopismo
Rocznik
Tom
Numer
Strony
21-28
Opis fizyczny
Twórcy
autor
• Institute of Informatics University of Białystok Sosnowa 64, 15-887 Białystok Poland
Bibliografia
• [1] Grzegorz Bancerek. Cardinal numbers. Formalized Mathematics, 1(2):377-382, 1990.
• [2] Grzegorz Bancerek. The fundamental properties of natural numbers. Formalized Mathematics, 1(1):41-46, 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] Leszek Borys. Paracompact and metrizable spaces. Formalized Mathematics, 2(4):481-485, 1991.
• [6] Czesław Bylinski. Finite sequences and tuples of elements of a non-empty sets. Formalized Mathematics, 1(3):529-536, 1990.
• [7] Czesław Bylinski. Functions and their basic properties. Formalized Mathematics, 1(1): 55-65, 1990.
• [8] Czesław Bylinski. Functions from a set to a set. Formalized Mathematics, 1(1):153-164, 1990.
• [9] Czesław Bylinski. Partial functions. Formalized Mathematics, 1(2):357-367, 1990.
• [10] Czesław Bylinski. Introduction to real linear topological spaces. Formalized Mathematics, 13(1):99-107, 2005.
• [11] Czesław Bylinski. The sum and product of finite sequences of real numbers. Formalized Mathematics, 1(4):661-668, 1990.
• [12] Czesław Bylinski. Some basic properties of sets. Formalized Mathematics, 1(1):47-53, 1990.
• [13] Czesław Bylinski and Piotr Rudnicki. Bounding boxes for compact sets in E2. Formalized Mathematics, 6(3):427-440, 1997.
• [14] Agata Darmochwał. The Euclidean space. Formalized Mathematics, 2(4):599-603, 1991.
• [15] Agata Darmochwał. Finite sets. Formalized Mathematics, 1(1):165-167, 1990.
• [16] Agata Darmochwał. Families of subsets, subspaces and mappings in topological spaces. Formalized Mathematics, 1(2):257-261, 1990.
• [17] Agata Darmochwał and Yatsuka Nakamura. Metric spaces as topological spaces - fundamental concepts. Formalized Mathematics, 2(4):605-608, 1991.
• [18] Roman Duda. Wprowadzenie do topologii. PWN, 1986.
• [19] Noboru Endou and Yasunari Shidama. Completeness of the real Euclidean space. Formalized Mathematics, 13(4):577-580, 2005.
• [20] Ryszard Engelking. Dimension Theory. North-Holland, Amsterdam, 1978.
• [21] Ryszard Engelking. General Topology. Heldermann Verlag, Berlin, 1989.
• [22] Zbigniew Karno. Continuity of mappings over the union of subspaces. Formalized Mathematics, 3(1):1-16, 1992.
• [23] Zbigniew Karno. Separated and weakly separated subspaces of topological spaces. Formalized Mathematics, 2(5):665-674, 1991.
• [24] Artur Korniłowicz. Homeomorphism between [:EiT , EjT :] and Ei+jT . Formalized Mathematics, 8(1):73-76, 1999.
• [25] Artur Korniłowicz. On the continuity of some functions. Formalized Mathematics, 18(3): 175-183, 2010. doi:10.2478/v10037-010-0020-z.[Crossref]
• [26] 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]
• [27] Artur Korniłowicz and Yasunari Shidama. Brouwer fixed point theorem for disks on the plane. Formalized Mathematics, 13(2):333-336, 2005.
• [28] Artur Korniłowicz and Yasunari Shidama. Intersections of intervals and balls in En T. Formalized Mathematics, 12(3):301-306, 2004.
• [29] Artur Korniłowicz and Yasunari Shidama. Some properties of circles on the plane. Formalized Mathematics, 13(1):117-124, 2005.
• [30] Eugeniusz Kusak, Wojciech Leonczuk, and Michał Muzalewski. Abelian groups, fields and vector spaces. Formalized Mathematics, 1(2):335-342, 1990.
• [31] Roman Matuszewski and Yatsuka Nakamura. Projections in n-dimensional Euclidean space to each coordinates. Formalized Mathematics, 6(4):505-509, 1997.
• [32] Robert Milewski. Bases of continuous lattices. Formalized Mathematics, 7(2):285-294, 1998.
• [33] Beata Padlewska and Agata Darmochwał. Topological spaces and continuous functions. Formalized Mathematics, 1(1):223-230, 1990.
• [34] Karol Pak. The rotation group. Formalized Mathematics, 20(1):23-29, 2012. doi:10.2478/v10037-012-0004-2.[Crossref]
• [35] Karol Pak. Small inductive dimension of topological spaces. Formalized Mathematics, 17 (3):207-212, 2009. doi:10.2478/v10037-009-0025-7.[Crossref]
• [36] Karol Pak. Small inductive dimension of topological spaces. Part II. Formalized Mathematics, 17(3):219-222, 2009. doi:10.2478/v10037-009-0027-5.[Crossref]
• [37] Andrzej Trybulec. Binary operations applied to functions. Formalized Mathematics, 1 (2):329-334, 1990.
• [38] Andrzej Trybulec. On the sets inhabited by numbers. Formalized Mathematics, 11(4): 341-347, 2003.
• [39] Andrzej Trybulec and Czesław Bylinski. Some properties of real numbers. Formalized Mathematics, 1(3):445-449, 1990.
• [40] Michał J. Trybulec. Integers. Formalized Mathematics, 1(3):501-505, 1990.
• [41] Wojciech A. Trybulec. Non-contiguous substrings and one-to-one finite sequences. Formalized Mathematics, 1(3):569-573, 1990.
• [42] Wojciech A. Trybulec. Vectors in real linear space. Formalized Mathematics, 1(2):291-296, 1990.
• [43] Zinaida Trybulec. Properties of subsets. Formalized Mathematics, 1(1):67-71, 1990.
• [44] Edmund Woronowicz. Relations and their basic properties. Formalized Mathematics, 1 (1):73-83, 1990.
• [45] Edmund Woronowicz. Relations defined on sets. Formalized Mathematics, 1(1):181-186, 1990.
• [46] Mirosław Wysocki and Agata Darmochwał. Subsets of topological spaces. Formalized Mathematics, 1(1):231-237, 1990.
• [47] Mariusz Zynel and Adam Guzowski. T0 topological spaces. Formalized Mathematics, 5 (1):75-77, 1996.
Typ dokumentu
Bibliografia
Identyfikatory