The interplay between weak topologies on topological semilattices

Abstract

We study the interplay between three weak topologies on a topological semilattice $X$: the weak$^\circ$ topology $\mathcal W^\circ_X$ (generated by the base consiting of open subsemilattices of $X$), the weak$^\bullet$ topology $\mathcal W^\bullet_X$ (generated by the subbase consisting of complements to closed subsemilattices), and the $\mathbb I$-weak topology $\mathcal W_X$ (which is the weakest topology in which all continuous homomorphisms $h:X\to [0,1]$ remain continuous). Also we study the interplay between the weak topologies $\mathcal W^\bullet_X$, $\mathcal W^\circ_X$, $\mathcal W_X$ of a topological semilattice $X$ and the Scott and Lawson topologies $\mathcal S_X$ and $\mathcal L_X$, which are determined by the order structure of the semilattice. We prove that the weak$^\bullet$ topology $\mathcal W^\bullet$ 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. This result implies that the Lawson topology $\mathcal L_X$ on a semilattice $X$ is compact if and only if $X$ is a continuous semilattice if and only if $X$ complete in the sense that each non-empty chain $C$ in $X$ has $\inf(C)$ and $\sup(C)$ in $X$. For a chain-compact Hausdorff topological semilattice $X$ with topology $\mathcal T_X$ we prove the inclusions $\mathcal W_X\subset\mathcal L_X\subset\mathcal W^\bullet_X\subset\mathcal T_X$. For a compact topological semilattice $X$ we prove that $\mathcal T_X=\mathcal W^\bullet_X$ if and only if $\mathcal T_X=\mathcal L_X$ if and only if $\mathcal T_X=\mathcal L_X$.