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 ${\mathcal{W}}_X^{\circ }$ (generated by the base consiting of open subsemilattices of $X$), the weak${}^{\bullet }$ topology ${\mathcal{W}}_X^{\bullet }$ (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 ${\mathcal{W}}_X^{\bullet }$, ${\mathcal{W}}_X^{\circ }$, ${\mathcal{W}}_X$ 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 ${\mathcal{W}}_X\subset {\mathcal{L}}_X\subset {\mathcal{W}}_X^{\bullet }\subset {\mathcal{T}}_X$. For a compact topological semilattice $X$ we prove that ${\mathcal{T}}_X={\mathcal{W}}_X^{\bullet }$ if and only if ${\mathcal{T}}_X={\mathcal{L}}_X$ if and only if ${\mathcal{T}}_X={\mathcal{L}}_X$.

18 pages; dedicated to the memory of W.W. Comfort

References in corpus (4)

• On linearly ordered $H$-closed topological semilattices
• Completeness and absolute $H$-closedness of topological semilattices
• The closedness of complete subsemilattices in functionally Hausdorff semitopological semilattices
• Complete topologized posets and semilattices

Cited by in corpus (5)

• The closedness of complete subsemilattices in functionally Hausdorff semitopological semilattices
• Complete topologized posets and semilattices
• Characterizing categorically closed commutative semigroups
• On images of complete topologized subsemilattices in sequential semitopological semilattices
• The Lawson number of a semitopological semilattice