Brouwer Invariance of Domain Theorem

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.Institute of Informatics University of Białystok Sosnowa 64, 15-887 Białystok PolandGrzegorz Bancerek. Cardinal numbers. Formalized Mathematics, 1(2):377-382, 1990.Grzegorz Bancerek. The fundamental properties of natural numbers. 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Heteroclinic solutions of singular Φ\Phi-Laplacian boundary value problems on infinite time scales

In this paper, we derive sufficient conditions for the existence of heteroclinic solutions to the singular $\Phi$-Laplacian boundary value problem, $$\left[\Phi(y^{\Delta}(t))\right]^{\Delta}=f(t,y(t),y^{\Delta}(t)),~~t\in\mathbb{T}$$ $$y(-\infty)=-1,~~~y(+\infty)=+1,$$ on infinite time scales by using the Brouwer invariance domain theorem. As an application we demonstrate our result with an example

A remark on the invariance of dimension

Combining Kulpa's proof of the cubical Sperner lemma and a dimension theoretic idea of van Mill we give a very short proof of the invariance of dimension, i.e. the statement that cubes [0,1]^n, [0,1]^m are homeomorphic if and only if n=m. This note is adapted from lecture notes for a course on general topology.Comment: latex2e, 8 pages, 1 eps figur