On geometric properties of functors of positive-homogenous and semiadditive functionals

Abstract

515.12We investigate the functor OH of positive-homogenous functionals and the functor OS of semiadditive functionals. We prove that OH X ( ) is an absolute retract if and only if X is an open-generated compactum, and OS X ( ) is an absolute retract if and only if X is an opengenerated compactum of weight ≤ ω 1 . We investigate the softness of mappings of multiplication of monads generated by these functors. IntroductionIn [1], Fedorchuk posed the following general question: How do functors affect certain geometric properties of spaces and mappings between them? Geometric properties are understood here as the property of a space to be an absolute retract, the property of a mapping to be soft or to be a Tikhonov fibering, etc.There have been many investigations in this direction involving such functors as the hyperspace functor, probability-measure functor, superextension functor, and other functors (see, e.g., [2]).As an example, consider the probability-measure functor and the superextension functor. There is a natural structure of linear convexity on the space P X ( ) for an arbitrary compactum X. For λ, de Groot [3] constructed an abstract (not linear) convexity on an arbitrary space of the form λ( ) X , and this convexity is binary, whereas the linear convexity on P X ( ) is not binary. The functors P and λ have different geometric properties. For example, the functor P, in contrast to λ, does not give absolute retracts in weights greater than ω 1 .The notion of convexity considered in the present paper is considerably broader than the classic one because it does not depend on the presence of a linear structure on a space. Our approach is based on the notion of topological convexity introduced in [4]. In [5], Radul associated every monad F with a certain abstract convexity structure on every space FX, where F is the functorial part of the monad F. If a monad F is an L-monad that weakly preserves preimages, then this convexity generates a topology of the space FX. It was also shown in [5] that L-monads that weakly preserve preimages and for which the introduced convexity structure is binary possess good geometric properties in weights greater than ω 1 (Theorem 3.3).In the present paper, we consider functors OS and OH (introduced in [6, 7]) that generate L-monads. The monad OS does not generate binary convexities, whereas the monad OH is binary, which is one of the factors that cause the difference in the geometric properties of these monads: the properties of OS are close to those of P, and the properties of OH are closer to those of λ. Definitions and FactsIn the present paper, we consider only objects and morphisms of the category Comp, i.e., compact Hausdorff spaces (or, briefly, compacta) and continuous mappings between them.