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Brouwer Invariance of Domain Theorem

100%
Pąk K.
Formalized Mathematics
|
2014
|
tom 22
|
nr 1
21-28
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.
2
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Closed subgroups in Banach spaces

80%
D. Ancel F., Dobrowolski T., Grabowski J.
Studia Mathematica
|
1994
|
tom 109
|
nr 3
277-290
EN
We show that zero-dimensional nondiscrete closed subgroups do exist in Banach spaces E. This happens exactly when E contains an isomorphic copy of $c_0$. Other results on subgroups of linear spaces are obtained.
3
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Topological type of weakly closed subgroups in Banach spaces

70%
Dobrowolski T., Grabowski J., Kawamura K.
Studia Mathematica
|
1996
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tom 118
|
nr 1
49-62
EN
The main result says that nondiscrete, weakly closed, containing no nontrivial linear subspaces, additive subgroups in separable reflexive Banach spaces are homeomorphic to the complete Erdős space. Two examples of such subgroups in $ℓ^1$ which are interesting from the Banach space theory point of view are discussed.
4
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On the topological dimension of the solutions sets for some classes of operator and differential inclusions

51%
Bader R., Gel'man B., Kamenskii M., Obukhovskii V.
Discussiones Mathematicae, Differential Inclusions, Control and Optimization
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2002
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tom 22
|
nr 1
17-32
EN
In the present paper, we give the lower estimation for the topological dimension of the fixed points set of a condensing continuous multimap in a Banach space. The abstract result is applied to the fixed point set of the multioperator of the form $𝓕 = S 𝓟_F$ where $𝓟_F$ is the superposition multioperator generated by the Carathéodory type multifunction F and S is the shift of a linear injective operator. We present sufficient conditions under which this set has the infinite topological dimension. In the last section of the paper, we consider the applications of the solutions sets for Cauchy and periodic problems for semilinear differential inclusions in a Banach space.
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