Generalized Helly spaces, continuity of monotone functions, and metrizing maps

• Autorzy: Lech Drewnowski , Artur Michalak
• Języki publikacji: EN

Abstrakty

EN

Given an ordered metric space (in particular, a Banach lattice) E, the generalized Helly space H(E) is the set of all increasing functions from the interval [0,1] to E considered with the topology of pointwise convergence, and E is said to have property (λ) if each of these functions has only countably many points of discontinuity. The main objective of the paper is to study those ordered metric spaces C(K,E), where K is a compact space, that have property (λ). In doing so, the guiding idea comes from the fact that there is a natural one-to-one correspondence between increasing functions f: [0,1] → C(K,E) (with countably many discontinuities) and continuous maps F: K → H(E) (with metrizable ranges). It leads to the investigation of general continuous metrizing maps (those with metrizable ranges), and especially of the so called separately metrizing maps, and the results obtained are then used to derive some permanence properties of the class of spaces C(K,E) with property (λ). For instance, it is shown that if K is the product of compact spaces $K_{j}$ (j ∈ J) such that each of the spaces $C(K_{j},E)$ has property (λ), so does C(K,E); and, for any compact space K, if both C(K) and a Banach lattice E have property (λ), so does C(K,E).

Kategorie tematyczne

• 46E15: Banach spaces of continuous, differentiable or analytic functions
• 54E35: Metric spaces, metrizability
• 54D30: Compactness
• 54B10: Product spaces
• 54C05: Continuous maps
• 54F05: Linearly ordered topological spaces, generalized ordered spaces, and partially ordered spaces
• 46B42: Banach lattices

• Wydawca: Institute of Mathematics Polish Academy of Sciences
• Czasopismo: Fundamenta Mathematicae
• Rocznik: 2008
• Tom: 200
• Numer: 2
• Strony: 161-184

Daty

wydano

2008

Twórcy

autor

Lech Drewnowski

• Faculty of Mathematics and Computer Science, A. Mickiewicz University, Umultowska 87, 61-614 Poznań, Poland

autor

Artur Michalak

• Faculty of Mathematics and Computer Science, A. Mickiewicz University, Umultowska 87, 61-614 Poznań, Poland

Identyfikatory

DOI

10.4064/fm200-2-4