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On linear functorial operators extending pseudometrics

Taras O. Banakh, Oleg Pikhurko (1997)

Commentationes Mathematicae Universitatis Carolinae

For a functor F ⊃ I d on the category of metrizable compacta, we introduce a conception of a linear functorial operator T = { T X : P c ( X ) → P c ( F X ) } extending (for each X ) pseudometrics from X onto F X ⊃ X (briefly LFOEP for F ). The main result states that the functor S P G n of G -symmetric power admits a LFOEP if and only if the action of G on { 1 , ⋯ , n } has a one-point orbit. Since both the hyperspace functor exp and the probability measure functor P contain S P 2 as a subfunctor, this implies that both exp and P do not admit LFOEP.

On minimal Hausdorff and minimal Urysohn functions

Filippo Cammaroto, Andrei Catalioto, Jack Porter (2011)

Open Mathematics

In this article, we extend the work on minimal Hausdorff functions initiated by Cammaroto, Fedorchuk and Porter in a 1998 paper. Also, minimal Urysohn functions are introduced and developed. The properties of heredity and productivity are examined and developed for both minimal Hausdorff and minimal Urysohn functions.

On open maps of Borel sets

A. Ostrovsky (1995)

Fundamenta Mathematicae

We answer in the affirmative [Th. 3 or Corollary 1] the question of L. V. Keldysh [5, p. 648]: can every Borel set X lying in the space of irrational numbers ℙ not G δ · F σ and of the second category in itself be mapped onto an arbitrary analytic set Y ⊂ ℙ of the second category in itself by an open map? Note that under a space of the second category in itself Keldysh understood a Baire space. The answer to the question as stated is negative if X is Baire but Y is not Baire.

Openly factorizable spaces and compact extensions of topological semigroups

Taras O. Banakh, Svetlana Dimitrova (2010)

Commentationes Mathematicae Universitatis Carolinae

We prove that the semigroup operation of a topological semigroup S extends to a continuous semigroup operation on its Stone-Čech compactification β S provided S is a pseudocompact openly factorizable space, which means that each map f : S → Y to a second countable space Y can be written as the composition f = g ∘ p of an open map p : X → Z onto a second countable space Z and a map g : Z → Y . We present a spectral characterization of openly factorizable spaces and establish some properties of such spaces.

Prescribed ultrametrics

J. Higgins, D. Campbell (1993)

RAIRO - Theoretical Informatics and Applications - Informatique Théorique et Applications

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