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

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.