Characterizing chain-compact and chain-finite topological semilattices

Preprint English OPEN
Banakh, Taras ; Bardyla, Serhii (2017)
  • Related identifiers: doi: 10.1007/s00233-018-9921-x
  • Subject: Mathematics - General Topology | 22A26, 54D30, 54D35, 54H12
    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 $Y$.
  • References (2)

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