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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 projectively quotient functors

T. F. Zhuraev (2001)

Commentationes Mathematicae Universitatis Carolinae

We introduce notions of projectively quotient, open, and closed functors. We give sufficient conditions for a functor to be projectively quotient. In particular, any finitary normal functor is projectively quotient. We prove that the sufficient conditions obtained are necessary for an arbitrary subfunctor ℱ of the functor 𝒫 of probability measures. At the same time, any “good” functor is neither projectively open nor projectively closed.

On the functor of order-preserving functionals

Taras Radul (1998)

Commentationes Mathematicae Universitatis Carolinae

We introduce a functor of order-preserving functionals which contains some known functors as subfunctors. It is shown that this functor is weakly normal and generates a monad.

On the quantification of uniform properties

Robert Lowen, Bart Windels (1997)

Commentationes Mathematicae Universitatis Carolinae

Approach spaces ([4], [5]) turned out to be a natural setting for the quantification of topological properties. Thus a measure of compactness for approach spaces generalizing the well-known Kuratowski measure of non-compactness for metric spaces was defined ([3]). This article shows that approach uniformities (introduced in [6]) have the same advantage with respect to uniform concepts: they allow a nice quantification of uniform properties, such as total boundedness and completeness.

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.

Ordered spaces and quasi-uniformities on spaces of continuous order-preserving functions.

Koena Rufus Nailana (2000)

Extracta Mathematicae

In this paper we introduce and investigate the notions of point open order topology, compact open order topology, the order topology of quasi-uniform pointwise convergence and the order topology of quasi-uniform convergence on compacta. We consider the functorial correspondence between function spaces in the categories of topological spaces, bitopological spaces and ordered topological spaces. We obtain extensions to the topological ordered case of classical topological results on function spaces....

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