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Banakh, Taras; Bardyla, Serhii (2017)
Languages: English
Types: Preprint
Subjects: Mathematics - General Topology, 22A26, 54D30, 54D35, 54H12

Classified by OpenAIRE into

arxiv: Mathematics::General Topology, Mathematics::General Mathematics
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 $Y$.
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