The interplay between weak topologies on topological semilattices

Банах, Тарас; Bardyla, Serhii

doi:10.1016/j.topol.2019.02.028

Cited by 7 publications

(14 citation statements)

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“…Now assume that X is compact. In this case the conditions ( 1)-( 4) are equivalent by Theorem 7.1 in [2]. In fact, the equivalence of the conditions (2) and ( 4) is a classical result of Lawson [13,14].…”

Section: Propositionmentioning

confidence: 80%

“…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%

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The Lawson number of a semitopological semilattice

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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Section: Propositionmentioning

confidence: 80%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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The Lawson number of a semitopological semilattice

2021

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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

Bardyla

Ravsky

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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