The interplay between weak topologies on topological semilattices

Abstract

We study the interplay between three weak topologies on a topological semilattice X: the weak∘ topology wX∘ (generated by the base consisting of open subsemilattices of X), the weak• topology wX• (generated by the subbase consisting of complements to closed subsemilattices), and the I-weak topology wX (which is the weakest topology in which all continuous homomorphisms h:X→[0,1] remain continuous). Also we study the interplay between the weak topologies w•X, wX∘, wX of a topological semilattice X and some intrinsic topologies, determined by the order structure of the semilattice.We prove that the weak• topology wX• on a Hausdorff semitopological semilattice X is compact if and only if X is chain-compact in the sense that each closed chain in X is compact. For a compact Hausdorff topological semilattice X with topology τX we prove that τX=wX iff τX=wX• iff τX=wX∘.