Compactness of order intervals in a locally solid linear lattice

Abstract

. Let X be a linear lattice, let x ∈ X + , and let τ be a locally solid topology on X . We present four conditions equivalent to the τ -compactness of the order interval [0 , x ] in X , including the following ones: (i) there is a set S and an affine homeomorphism of [0 , x ] onto the Tychonoff cube [0 , 1] S which preserves order; (ii) C x , the set of components of x , is τ -compact and [0 , x ] is order σ -complete. In the special case where X is a Banach lattice and τ is its norm topology, another equivalent condition is: (iii) C x is weakly compact and [0 , x ] is order σ -complete.