Zbigniew Lipecki
Order intervals in C(K). Compactness, coincidence of topologies, metrizability

Comment.Math.Univ.Carolin. 63,3 (2022) 295-306.

Abstract: Let K be a compact space and let C(K) be the Banach lattice of real-valued continuous functions on K. We establish eleven conditions equivalent to the strong compactness of the order interval [0,x] in C(K), including the following ones: (i) \{x>0\} consists of isolated points of K; (ii) [0,x] is pointwise compact; (iii) [0,x] is weakly compact; (iv) the strong topology and that of pointwise convergence coincide on [0,x]; (v) the strong and weak topologies coincide on [0,x]. \noindent Moreover, the weak topology and that of pointwise convergence coincide on [0,x] if and only if \{x>0\} is scattered. Finally, the weak topology on [0,x] is metrizable if and only if the topology of pointwise convergence on [0,x] is such if and only if \{x>0\} is countable.

Keywords: real linear lattice; order interval; locally solid; Banach lattice C(K); strongly compact; weakly compact; pointwise compact; coincidence of topologies; metrizable; scattered; Čech-Stone compactification

DOI: DOI 10.14712/1213-7243.2022.006
AMS Subject Classification: 46A40 46B42 46E05 54C35 54D30