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On a theorem of W.W. Comfort and K.A. Ross

Aleksander V. Arhangel'skii (1999)

Commentationes Mathematicae Universitatis Carolinae

A well known theorem of W.W. Comfort and K.A. Ross, stating that every pseudocompact group is C -embedded in its Weil completion [5] (which is a compact group), is extended to some new classes of topological groups, and the proofs are very transparent, short and elementary (the key role in the proofs belongs to Lemmas 1.1 and 4.1). In particular, we introduce a new notion of canonical uniform tightness of a topological group G and prove that every G δ -dense subspace Y of a topological group G , such...

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 absolute retracts of ω*

A. Bella, A. Błaszczyk, A. Szymański (1994)

Fundamenta Mathematicae

An extremally disconnected space is called an absolute retract in the class of all extremally disconnected spaces if it is a retract of any extremally disconnected compact space in which it can be embedded. The Gleason spaces over dyadic spaces have this property. The main result of this paper says that if a space X of π-weight ω 1 is an absolute retract in the class of all extremally disconnected compact spaces and X is homogeneous with respect to π-weight (i.e. all non-empty open sets have the same...

On absolute stability

Roger C. McCann (1972)

Annales de l'institut Fourier

Absolute stability of a compact set is characterized by the cardinality of a fundamental system of positively invariant neighborhoods.

On absolutely submetrizable spaces

Raushan Z. Buzyakova (2006)

Commentationes Mathematicae Universitatis Carolinae

We introduce a notion of absolute submetrizability (= ``every Tychonoff subtopology is submetrizable'') and investigate its behavior under basic topological operations. The main result is an example of an absolutely submetrizable space that contains an uncountable set of isolated points (hence the space is neither separable nor hereditarily Lindelöf). This example is used to show that absolute submetrizability is not preserved by some topological operations, in particular, by free sums.

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