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On a universality property of some abelian Polish groups

Su Gao, Vladimir Pestov (2003)

Fundamenta Mathematicae

We show that every abelian Polish group is the topological factor group of a closed subgroup of the full unitary group of a separable Hilbert space with the strong operator topology. It follows that all orbit equivalence relations induced by abelian Polish group actions are Borel reducible to some orbit equivalence relations induced by actions of the unitary group.

On a weak form of uniform convergence

Jaroslav Fuka, Petr Holický (2005)

Commentationes Mathematicae Universitatis Carolinae

The notion of Δ -convergence of a sequence of functions is stronger than pointwise convergence and weaker than uniform convergence. It is inspired by the investigation of ill-posed problems done by A.N. Tichonov. We answer a question posed by M. Katětov around 1970 by showing that the only analytic metric spaces X for which pointwise convergence of a sequence of continuous real valued functions to a (continuous) limit function on X implies Δ -convergence are σ -compact spaces. We show that the assumption...

On analyticity in cosmic spaces

Oleg Okunev (1993)

Commentationes Mathematicae Universitatis Carolinae

We prove that a cosmic space (= a Tychonoff space with a countable network) is analytic if it is an image of a K -analytic space under a measurable mapping. We also obtain characterizations of analyticity and σ -compactness in cosmic spaces in terms of metrizable continuous images. As an application, we show that if X is a separable metrizable space and Y is its dense subspace then the space of restricted continuous functions C p ( X Y ) is analytic iff it is a K σ δ -space iff X is σ -compact.

On clopen sets in Cartesian products

Raushan Z. Buzyakova (2001)

Commentationes Mathematicae Universitatis Carolinae

The results concern clopen sets in products of topological spaces. It is shown that a clopen subset of the product of two separable metrizable (or locally compact) spaces is not always a union of clopen boxes. It is also proved that any clopen subset of the product of two spaces, one of which is compact, can always be represented as a union of clopen boxes.

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