Displaying similar documents to “Miscellaneous Facts about Open Functions and Continuous Functions”

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...

Open Mapping Theorem

Hideki Sakurai, Hisayoshi Kunimune, Yasunari Shidama (2008)

Formalized Mathematics

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In this article we formalize one of the most important theorems of linear operator theory the Open Mapping Theorem commonly used in a standard book such as [8] in chapter 2.4.2. It states that a surjective continuous linear operator between Banach spaces is an open map.MML identifier: LOPBAN 6, version: 7.10.01 4.111.1036

Continuity of Barycentric Coordinates in Euclidean Topological Spaces

Karol Pąk (2011)

Formalized Mathematics

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In this paper we present selected properties of barycentric coordinates in the Euclidean topological space. We prove the topological correspondence between a subset of an affine closed space of εn and the set of vectors created from barycentric coordinates of points of this subset.

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...