Characterizing chain-compact and chain-finite topological semilattices

Abstract

In the paper we present various characterizations of chain-compact and chain-finite topological semilattices. A topological semilattice $X$ is called chain-compact (resp. chain-finite) if each closed chain in $X$ is compact (finite). In particular, we prove that a (Hausdorff) $T_1$-topological semilattice $X$ is chain-finite (chain-compact) if and only if for any closed subsemilattice $Z\subset X$ and any continuous homomorphism $h:X\to Y$ to a (Hausdorff) $T_1$-topological semilattice $Y$ the image $h(X)$ is closed in $Y$.

12 pages

References in corpus (3)

• On linearly ordered $H$-closed topological semilattices
• Categorically closed topological groups
• $H$-closed quasitopological groups

Cited by in corpus (6)

• Completeness and absolute $H$-closedness of topological semilattices
• The closedness of complete subsemilattices in functionally Hausdorff semitopological semilattices
• Characterizing categorically closed commutative semigroups
• On images of complete topologized subsemilattices in sequential semitopological semilattices
• The Lawson number of a semitopological semilattice
• A semigroup is finite if and only if it is chain-finite and antichain-finite