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Let G be a locally compact group, and let U be its unitary representation on a Hilbert space H. Endow the space ℒ(H) of bounded linear operators on H with the weak operator topology. We prove that if U is a measurable map from G to ℒ(H) then it is continuous. This result was known before for separable H. We also prove that the following statement is consistent with ZFC: every measurable homomorphism from a locally compact group into any topological group is continuous.

On continuity structures and spaces of mappings

Miroslav Katětov (1965)

Commentationes Mathematicae Universitatis Carolinae

On continuous actions commutingwith actions of positive entropy

Mark Shereshevsky (1996)

Colloquium Mathematicae

Let F and G be finitely generated groups of polynomial growth with the degrees of polynomial growth d(F) and d(G) respectively. Let $S = {S^{f}}_{f \in F}$ be a continuous action of F on a compact metric space X with a positive topological entropy h(S). Then (i) there are no expansive continuous actions of G on X commuting with S if d(G)

On continuous extension of uniformly continuous functions and metrics

T. Banakh, N. Brodskiy, I. Stasyuk, E. D. Tymchatyn (2009)

Colloquium Mathematicae

We prove that there exists a continuous regular, positive homogeneous extension operator for the family of all uniformly continuous bounded real-valued functions whose domains are closed subsets of a bounded metric space (X,d). In particular, this operator preserves Lipschitz functions. A similar result is obtained for partial metrics and ultrametrics.

On continuous extensions.

Narici, Lawrence, Beckenstein, Edward (1996)

Georgian Mathematical Journal

On continuous images of almost Tychonoff cubes

Aleksander V. Arhangel'skii, Juraj Činčura (1984)

Commentationes Mathematicae Universitatis Carolinae

On continuous images of Eberlein compacts

Petr Simon (1976)

Commentationes Mathematicae Universitatis Carolinae

On continuous maps in closure spaces.

Boonpok, C. (2009)

General Mathematics

On continuous self-maps and homeomorphisms of the Golomb space

Taras O. Banakh, Jerzy Mioduszewski, Sławomir Turek (2018)

Commentationes Mathematicae Universitatis Carolinae

The Golomb space $ℕ_{τ}$ is the set $ℕ$ of positive integers endowed with the topology $τ$ generated by the base consisting of arithmetic progressions ${a + b n : n \geq 0}$ with coprime $a, b$ . We prove that the Golomb space $ℕ_{τ}$ has continuum many continuous self-maps, contains a countable disjoint family of infinite closed connected subsets, the set $Π$ of prime numbers is a dense metrizable subspace of $ℕ_{τ}$ , and each homeomorphism $h$ of $ℕ_{τ}$ has the following properties: $h (1) = 1$ , $h (Π) = Π$ , $Π_{h (x)} = h (Π_{x})$ , and $h (x^{ℕ}) = h {(x)}^{ℕ}$ for all $x \in ℕ$ . Here $x^{ℕ} : = {x^{n} : n \in ℕ}$ and $Π_{x}$ denotes the set of prime divisors...

On continuous surjections from Cantor set.

Félix Cabello Sánchez (2004)

Extracta Mathematicae

It is a famous result of Alexandroff and Urysohn that every compact metric space is a continuous image of a Cantor set ∆. In this short note we complement this result by showing that a certain uniqueness property holds. Namely, if (K,d) is a compact metric space and f and g are two continuous mappings from ∆ onto K, the, for every e > 0 there exists a homeomorphism phi of ∆ such that d(g(x), f(phi(x))) < e for all x∆.

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Displaying 341 – 360 of 1530

On continua having three types of open subsets

On continua which resemble simple closed curves

On continuity and monotonicity of Darboux functions

On continuity of measurable group representations and homomorphisms

On continuity structures and spaces of mappings

On continuous actions commutingwith actions of positive entropy

On continuous extension of uniformly continuous functions and metrics

On continuous extensions.

On continuous images of almost Tychonoff cubes

On continuous images of Eberlein compacts

On continuous maps in closure spaces.

On continuous self-maps and homeomorphisms of the Golomb space

On continuous surjections from Cantor set.

On ℳ-contractibility

On contractible dendroids

On contractible fans

On contraction mappings

On contractive mappings in metric spaces

On contra-precontinuous functions.

On convergence groups and convergence uniformities

Currently displaying 341 – 360 of 1530