Completeness and absolute H-closedness of topological semilattices

Банах, Тарас; Bardyla, Serhii

doi:10.1016/j.topol.2019.04.001

“…In this paper we are interested in detecting topological semilattices which are C-closed for various categories C of topologized semilattices. This topics will be continued in the paper [2].…”

Section: Introductionmentioning

confidence: 93%

Characterizing chain-compact and chain-finite topological semilattices

Банах

¹

,

Bardyla

²

2018

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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) T1-topological semilattice X is chain-finite (chain-compact) if and only if for any closed subsemilattice Z ⊂ X and any continuous homomorphism h :1991 Mathematics Subject Classification. 22A26; 54D30; 54D35; 54H12.

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“…In this paper we introduce a new cardinal invariant¯ (X ) of a Hausdorff topologized semilattice X , called the Lawson number of X . This was motivated by studying the closedness properties of complete topologized semilattices, see [1][2][3][4][5][6]. It turns out that complete semitopological semilattices share many common properties with compact topological semilattices, in particular their continuous homomorphic images in Hausdorff topological semilattices are closed.…”

Section: Introductionmentioning

confidence: 99%

“…Here C stands for the closure of C in X . Chain-compact and complete topologized semilattices appeared to be very helpful in studying the closedness properties of topologized semilattices, see [1][2][3][4][5][6]11]. By Theorem 3.1 [1], a Hausdorff semitopological semilattice is chain-compact if and only if it is complete (see also Theorem 4.3 [5] for generalization of this characterization to topologized posets).…”

Section: Introductionmentioning

confidence: 99%

The Lawson number of a semitopological semilattice

Банах

¹

,

Bardyla

²

,

Гутік

³

2021

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For a Hausdorff topologized semilattice X its Lawson number$$\bar{\Lambda }(X)$$ Λ ¯ ( X ) is the smallest cardinal $$\kappa $$ κ such that for any distinct points $$x,y\in X$$ x , y ∈ X there exists a family $$\mathcal U$$ U of closed neighborhoods of x in X such that $$|\mathcal U|\le \kappa $$ | U | ≤ κ and $$\bigcap \mathcal U$$ ⋂ U is a subsemilattice of X that does not contain y. It follows that $$\bar{\Lambda }(X)\le \bar{\psi }(X)$$ Λ ¯ ( X ) ≤ ψ ¯ ( X ) , where $$\bar{\psi }(X)$$ ψ ¯ ( X ) is the smallest cardinal $$\kappa $$ κ such that for any point $$x\in X$$ x ∈ X there exists a family $$\mathcal U$$ U of closed neighborhoods of x in X such that $$|\mathcal U|\le \kappa $$ | U | ≤ κ and $$\bigcap \mathcal U=\{x\}$$ ⋂ U = { x } . We prove that a compact Hausdorff semitopological semilattice X is Lawson (i.e., has a base of the topology consisting of subsemilattices) if and only if $$\bar{\Lambda }(X)=1$$ Λ ¯ ( X ) = 1 . Each Hausdorff topological semilattice X has Lawson number $$\bar{\Lambda }(X)\le \omega $$ Λ ¯ ( X ) ≤ ω . On the other hand, for any infinite cardinal $$\lambda $$ λ we construct a Hausdorff zero-dimensional semitopological semilattice X such that $$|X|=\lambda $$ | X | = λ and $$\bar{\Lambda }(X)=\bar{\psi }(X)=\mathrm {cf}(\lambda )$$ Λ ¯ ( X ) = ψ ¯ ( X ) = cf ( λ ) . A topologized semilattice X is called (i) $$\omega $$ ω -Lawson if $$\bar{\Lambda }(X)\le \omega $$ Λ ¯ ( X ) ≤ ω ; (ii) complete if each non-empty chain $$C\subseteq X$$ C ⊆ X has $$\inf C\in {\overline{C}}$$ inf C ∈ C ¯ and $$\sup C\in {\overline{C}}$$ sup C ∈ C ¯ . We prove that for any complete subsemilattice X of an $$\omega $$ ω -Lawson semitopological semilattice Y, the partial order $$\le _X=\{(x,y)\in X\times X:xy=x\}$$ ≤ X = { ( x , y ) ∈ X × X : x y = x } of X is closed in $$Y\times Y$$ Y × Y and hence X is closed in Y. This implies that for any continuous homomorphism $$h:X\rightarrow Y$$ h : X → Y from a complete topologized semilattice X to an $$\omega $$ ω -Lawson semitopological semilattice Y the image h(X) is closed in Y.

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“…In this paper we continue to study the closedness properties of complete semitopological semilattices, which were introduced and studied by the authors in [1], [2], [3], [4], [5]. It turns out that complete semitopological semilattices share many common properties with compact topological semilattices, in particular their continuous homomorphic images in Hausdorff topological semilattices are closed.…”

mentioning

confidence: 99%

“…• complete if each non-empty chain C ⊂ X has inf C ∈C and sup C ∈C. HereC stands for the closure of C in X. Chain-compact and complete topologized semilattices appeared to be very helpful in studying the closedness properties of topologized semilattices, see [1], [2], [3], [4], [5]. By Theorem 3.1 [1], a Hausdorff semitopological semilattice is chain-compact if and only if complete (see also Theorem 4.3 [5] for generalization of this characterization to topologized posets).…”

mentioning

confidence: 99%

The closedness of complete subsemilattices in functionally Hausdorff semitopological semilattices

Банах

¹

,

Bardyla

²

,

Ravsky

³

2019

Topology and its Applications

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A topologized semilattice X is complete if each non-empty chain C ⊂ X has inf C ∈C and sup C ∈C. It is proved that for any complete subsemilattice X of a functionally Hausdorff semitopological semilattice Y the partial order P = {(x, y) ∈ X × X : xy = x} of X is closed in Y × Y and hence X is closed in Y . This implies that for any continuous homomorphism h : X → Y from a compete topologized semilattice X to a functionally Hausdorff semitopological semilattice Y the image h(X) is closed in Y . The functional Hausdorffness of Y in these two results can be replaced by the weaker separation axiom T 2δ , defined in this paper.

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Completeness and absolute H-closedness of topological semilattices

Abstract: We find (completeness type) conditions on topological semilattices X, Y guaranteeing that each continuous homomorphism h : X → Y has closed image h(X) in Y .1991 Mathematics Subject Classification. 22A26; 54D30; 54D35; 54H12.

Cited by 12 publications

References 11 publications

Characterizing chain-compact and chain-finite topological semilattices

Characterizing chain-compact and chain-finite topological semilattices

The Lawson number of a semitopological semilattice

The closedness of complete subsemilattices in functionally Hausdorff semitopological semilattices

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