On images of complete topologized subsemilattices in sequential semitopological semilattices
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On images of complete topologized subsemilattices in sequential semitopological semilattices Taras Banakh1,2 · Serhii Bardyla3 Received: 5 March 2019 / Accepted: 10 September 2019 © The Author(s) 2019
Abstract A topologized semilattice X is called complete if each non-empty chain C ⊂ X ¯ We prove that for any continuous homomorphism has inf C ∈ C¯ and sup C ∈ C. h : X → Y from a complete topologized semilattice X to a sequential Hausdorff semitopological semilattice Y the image h(X ) is closed in Y . Keywords Sequential space · Complete semitopological semilattice · The tower number This paper is a continuation of the investigations [1–3] of complete topologized semilattices. A semilattice is any commutative semigroup of idempotents (an element x of a semigroup is called an idempotent if x x = x). A semilattice endowed with a topology is called a topologized semilattice. A topologized semilattice X is called a (semi)topological semilattice if the semigroup operation X × X → X , (x, y) → x y, is (separately) continuous. Each semilattice carries a natural partial order ≤ defined by x ≤ y iff x y = x = yx. Endowed with this partial order, the semilattice is a poset, i.e., partially ordered set. Many properties of a semilattice can be expressed in the language of this partial order. In particular, a subset C of a semilattice (more generally, poset) X is called a chain if
Communicated by Jimmie D. Lawson. The second author was supported by the Austrian Science Fund FWF (Grant I 3709-N35).
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Taras Banakh [email protected] Serhii Bardyla [email protected]
1
Ivan Franko National University of Lviv, Lviv, Ukraine
2
Jan Kochanowski University in Kielce, Kielce, Poland
3
Institute of Mathematics, Kurt Gödel Research Center, Vienna, Austria
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T. Banakh, S. Bardyla
any points x, y ∈ C are comparable in the sense that x ≤ y or y ≤ x. A poset X is called chain-finite if each chain in X is finite. In [13] Stepp proved that for any homomorphism h : X → Y from a chain-finite semilattice to a Hausdorff topological semilattice Y the image h(X ) is closed in Y . In [1], the authors improved this result of Stepp proving the following theorem. Theorem 1 For any homomorphism h : X → Y from a chain-finite semilattice to a Hausdorff semitopological semilattice Y , the image h(X ) is closed in Y . A topological counterpart of the notion of a chain-finite poset is the notion of a complete topologized poset. A topologized poset is a poset (X , ≤) endowed with a topology. A topologized poset X is called complete if each chain C ⊂ X has inf C and sup C that belong to the closure C¯ of the chain C in X . Complete topologized semilattices were introduced in [1] under the name kcomplete topologized semilattices. But we prefer to call such topologized semilattice complete (taking into account the fundamental role of complete topologized semilattices in the theory of absolutely closed topologized semilattices, see [1–4,7,11]). In [1] the authors proved the following closedness property of complete topologized semilattices. Theore
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