Linear Transformations of Euclidean Topological Spaces. Part II
Karol Pąk (2011)
Formalized Mathematics
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We prove a number of theorems concerning various notions used in the theory of continuity of barycentric coordinates.
Displaying similar documents to “Continuity of Barycentric Coordinates in Euclidean Topological Spaces”
Linear Transformations of Euclidean Topological Spaces. Part II
Study of -closed sets and the related notions in topological spaces.
Caldas, M., Georgiou, D.N., Jafari, S. (2007)
Bulletin of the Malaysian Mathematical Sciences Society. Second Series
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On the Continuity of Some Functions
Artur Korniłowicz (2010)
Formalized Mathematics
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We prove that basic arithmetic operations preserve continuity of functions.
On splitting of extensions of rings and topological rings.
Abel, M. (2010)
Annals of Functional Analysis (AFA) [electronic only]
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Affine Independence in Vector Spaces
Karol Pąk (2010)
Formalized Mathematics
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In this article we describe the notion of affinely independent subset of a real linear space. First we prove selected theorems concerning operations on linear combinations. Then we introduce affine independence and prove the equivalence of various definitions of this notion. We also introduce the notion of the affine hull, i.e. a subset generated by a set of vectors which is an intersection of all affine sets including the given set. Finally, we introduce and prove selected properties...
Brouwer Invariance of Domain Theorem
Karol Pąk (2014)
Formalized Mathematics
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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...
Are EC-spaces AE(metrizable)?
Carlos Borges (1991)
Colloquium Mathematicae
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Topological Interpretation of Rough Sets
Adam Grabowski (2014)
Formalized Mathematics
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Rough sets, developed by Pawlak, are an important model of incomplete or partially known information. In this article, which is essentially a continuation of [11], we characterize rough sets in terms of topological closure and interior, as the approximations have the properties of the Kuratowski operators. We decided to merge topological spaces with tolerance approximation spaces. As a testbed for our developed approach, we restated the results of Isomichi [13] (formalized in Mizar in...
Uniform boundedness principles for ordered topological vector spaces.
Zhong, Shuhui, Li, Ronglu (2011)
Annals of Functional Analysis (AFA) [electronic only]
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The Correspondence Between n -dimensional Euclidean Space and the Product of n Real Lines
Artur Korniłowicz (2010)
Formalized Mathematics
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In the article we prove that a family of open n-hypercubes is a basis of n-dimensional Euclidean space. The equality of the space and the product of n real lines has been proven.
On the dimension modulo classes of topological spaces and free topological groups.
Bakhia, I. (2003)
Georgian Mathematical Journal
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