Characterizing chain-compact and chain-finite topological semilattices
Banakh, Taras, 1968- ; Bardyla, Serhii
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 ▫$h:Z \to Y$▫ to a (Hausdorff) ▫$T_1$▫-topological semilattice ▫$Y$▫ the image ▫$h(Z)$▫ is closed in ▫$Y$▫.
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