Displaying similar documents to “Stretching the Oxtoby-Ulam Theorem”

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...

Tietze Extension Theorem for n-dimensional Spaces

Karol Pąk (2014)

Formalized Mathematics

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In this article we prove the Tietze extension theorem for an arbitrary convex compact subset of εn with a non-empty interior. This theorem states that, if T is a normal topological space, X is a closed subset of T, and A is a convex compact subset of εn with a non-empty interior, then a continuous function f : X → A can be extended to a continuous function g : T → εn. Additionally we show that a subset A is replaceable by an arbitrary subset of a topological space that is homeomorphic...

On a one-dimensional analogue of the Smale horseshoe

Ryszard Rudnicki (1991)

Annales Polonici Mathematici

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We construct a transformation T:[0,1] → [0,1] having the following properties: 1) (T,|·|) is completely mixing, where |·| is Lebesgue measure, 2) for every f∈ L¹ with ∫fdx = 1 and φ ∈ C[0,1] we have ∫ φ ( T n x ) f ( x ) d x → ∫ φ d μ , where μ is the cylinder measure on the standard Cantor set, 3) if φ ∈ C[0,1] then n - 1 ∑ i = 0 n - 1 φ ( T i x ) → ∫ φ d μ for Lebesgue-a.e. x.

Commuting functions and simultaneous Abel equations

W. Jarczyk, K. Łoskot, M. C. Zdun (1994)

Annales Polonici Mathematici

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The system of Abel equations α(ft(x)) = α(x) + λ(t), t ∈ T, is studied under the general assumption that f t are pairwise commuting homeomorphisms of a real interval and have no fixed points (T is an arbitrary non-empty set). A result concerning embeddability of rational iteration groups in continuous groups is proved as a simple consequence of the obtained theorems.

On extension of the group operation over the Čech-Stone compactification

Jan Jełowicki (1993)

Colloquium Mathematicae

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The convolution of ultrafilters of closed subsets of a normal topological group is considered as a substitute of the extension onto ( β ) 2 of the group operation. We find a subclass of ultrafilters for which this extension is well-defined and give some examples of pathologies. Next, for a given locally compact group and its dense subgroup , we construct subsets of β algebraically isomorphic to . Finally, we check whether the natural mapping from β onto β is a homomorphism with respect to...