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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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    • U ∗ V ∋ [1] T. Banakh, S. Bardyla, Completeness and absolute H-closedness of topological semilattices, preprint
    • (https://arxiv.org/abs/1702.02791). [2] S. Bardyla, O. Gutik, On H-complete topological semilattices, Mat. Stud. 38:2 (2012), 118-123. [3] S. Bardyla, O. Gutik, A. Ravsky, H-closed topological groups, Topology Appl. 217 (2017) 51-58. [4] R. Engelking, General Topology, Heldermann, Berlin, 1989. [5] O. Gutik, D. Repovˇs, On linearly ordered H-closed topological semilattices, Semigroup Forum 77:3 (2008), 474-481. [6] N. Hindman, D. Strauss, Algebra in the Stone-Cˇech compactifiation, Walter de Gruyter, Berlin, NY, 1998. [7] J.W. Stepp, Algebraic maximal semilattices. Pacific J. Math. 58:1 (1975), 243-248. [8] N.V. Velicko, H-closed topological spaces, Mat. Sb. (N.S.) 70 (112) (1966) 98-112. [9] N.V. Velicko, On the theory of H-closed topological spaces, Sibirsk. Mat. Z. 8 (1967) 754-763.
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