The countable sup property for lattices of continuous functions

JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS

卷:465

Article

Kandic, M.¹  Vavpetic, A.¹ 

[1] Univ Ljubljani, Fak Matemat Fiziko, Jadranska Ulica 19, SI-1000 Ljubljana, Slovenia

关键词: Vector lattices;    Continuous functions;    Countable sup property;    Chain conditions;    Strictly positive functionals;   

DOI  :  10.1016/j.jmaa.2018.05.028

来源: Elsevier

【 摘 要 】

In this paper we find sufficient and necessary conditions under which vector lattice C(X) and its sublattices C-b(X), C-0(X) and C-c(X) have the countable sup property. It turns out that the countable sup property is tightly connected to the countable chain condition of the underlying topological space X. We also consider the countable sup property of C(X x Y). Even when both C(X) and C(Y) have the countable sup property it is possible that C(X x Y) fails to have it. For this construction one needs to assume the continuum hypothesis. In general, we present a positive result in this direction and also address the question when C(Pi(lambda is an element of Lambda) X-lambda) has the countable sup property. Our results can be understood as vector lattice theoretical versions of results regarding products of spaces satisfying the countable chain condition. We also present new results for general vector lattices that are of an independent interest. (C) 2018 Elsevier Inc. All rights reserved.

【 授权许可】

Free