The interplay between weak topologies on topological semilattices

TL;DR

This paper investigates the relationships between various weak topologies on topological semilattices, establishing conditions for their compactness and their connections with Scott and Lawson topologies, revealing new characterizations of continuous and complete semilattices.

Contribution

It introduces and analyzes three weak topologies on topological semilattices and characterizes their compactness and relationships with order-based topologies, providing new insights into the structure of semilattices.

Findings

Weak$^ullet$ topology is compact iff the semilattice is chain-compact.

Lawson topology is compact iff the semilattice is continuous and complete.

Inclusions among weak and order topologies are established for chain-compact 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 $\mathcal T_X=\mathcal L_X$.