The interplay between weak topologies on topological semilattices
aa r X i v : . [ m a t h . GN ] J un THE INTERPLAY BETWEEN WEAK TOPOLOGIES ON TOPOLOGICALSEMILATTICES
TARAS BANAKH, SERHII BARDYLA
Dedicated to the memory of W.W. Comfort
Abstract.
We study the interplay between three weak topologies on a topological semilattice X : theweak ◦ topology W ◦ X (generated by the base consiting of open subsemilattices of X ), the weak • topology W • X (generated by the subbase consisting of complements to closed subsemilattices), and the I -weaktopology W X (which is the weakest topology in which all continuous homomorphisms h : X → [0 , W • X , W ◦ X , W X of atopological semilattice X and the Scott and Lawson topologies S X and L X , which are determined by theorder structure of the semilattice. We prove that the weak • topology W • X on a Hausdorff semitopologicalsemilattice X is compact if and only if X is chain-compact in the sense that each closed chain in X iscompact. This result implies that the Lawson topology L X on a semilattice X is compact if and only if X is a continuous semilattice if and only if X complete in the sense that each non-empty chain C in X has inf( C ) and sup( C ) in X .For a chain-compact Hausdorff topological semilattice X with topology T X we prove the inclusions W X ⊂ L X ⊂ W • X ⊂ T X . For a compact topological semilattice X we prove that T X = W • X if and onlyif T X = W ◦ X if and only if T X = L X . Introduction A semigroup is a set X endowed with an associative binary operation · : X × X → X , · : ( x, y ) xy . Asemigroup X is a band if each element x ∈ X is an idempotent in the sense that xx = x . A commutativeband X is called a semilattice . Each band X carries the natural partial order ≤ defined by x ≤ y if xy = x = yx . A band endowed with the natural partial order is a poset (i.e., a partially ordered set ).For a point x of a poset ( X, ≤ ) let ↑ x := { y ∈ X : x ≤ y } and ↓ x := { y ∈ X : y ≤ x } be the upper and lower set of x in ( X, ≤ ). A subset C of a poset X is called a chain if x ∈ ↑ y ∪ ↓ y for any points x, y ∈ C .A topologized semigroup is a semigroup X endowed with a topology T X . A topologized semigroup X is called a • a topological semigroup if the binary operation X × X → X , ( x, y ) xy , is continuous; • a semitopological semigroup if the binary operation X × X → X , ( x, y ) xy , is separatelycontinuous; • a subtopological semigroup if for any subsemigroup S ⊂ X its closure ¯ S in X is a subsemigroupof X .It is easy to see that each topological semigroup is semitopological and any semitopological semigroup issubtopological. More information on topological semigroups can be found in the survey [9] of Comfort,Hofmann and Remus.In this paper we shall be mainly interested in (semi)topological semilattices. An important exampleof a topological semilattice is the closed interval I = [0 ,
1] endowed with the semilattice operation I × I → I , ( x, y ) min( x, y ), of taking minimum.On each topologized semigroup X we shall consider three weaker topologies:- the weak ◦ topology W ◦ X , generated by the base consisting of open subsemigroups of X , Mathematics Subject Classification.
Key words and phrases. topologized semigroup, Lawson topology, chain-compact semilattice. - the weak • topology W • X , generated by the subbase consisting of complements to closed subsemi-groups of X ,- the I -weak topology W X , generated by the subbase consisting of the preimages h − ( U ) of opensets U ⊂ I under continuous semigroup homomorphisms h : X → I .It is clear that W ◦ X ! ! ❈❈❈❈❈❈❈❈ W X < < ③③③③③③③③ " " ❉❉❉❉❉❉❉❉ T X ( ∗ ) W • X = = ④④④④④④④④ where T X stands for the original topology of X and an arrow A → B indicates that
A ⊂ B .A topologized semigroup ( X, T X ) is called • weak ◦ if W ◦ X = T X ; • weak • if W • X = T X ; • I -weak if W X = T X .In the context of topological semilattices, weak ◦ topological semilattices were introduced by Lawson [15](who called them semilattices with small subsemilattices) and are well-studied in Topological Algebra[8, Ch.2], [12, VI]; I -weak topological semigroups also appear naturally in the theory of topologicalsemilattices, see [8, Ch.2] or [12, VI-3.7]. On the other hand, the notions of the weak • topology W • X or a weak • topologized semigroup seem to be new. In this paper we shall study the interplay between(separation properties) of the weak topologies W • X , W ◦ X , W X on a topologized semilattice X .The inclusion relations ( ∗ ) between the topologies W X , W ◦ X , W • X and T X imply that each I -weaktopologized semigroup X is both weak ◦ and weak • .To describe separation properties of the topologies W ◦ X , W • X and W X let us define a topologizedsemigroup X to be • W ◦ -Hausdorff if its weak ◦ topology W ◦ X is Hausdorff; • W • -Hausdorff if its weak • topology W • X is Hausdorff; • W -Hausdorff if its I -weak topology W X is Hausdorff; • W ◦ T -separated if any distinct points x, y ∈ X have disjoint neighborhoods O x ∈ W ◦ X and O y ∈ T X ; • W • T -separated if any distinct points x, y ∈ X have disjoint neighborhoods O x ∈ W • X and O y ∈ T X ; • I -separated if for any distinct points x, y ∈ X there exists a continuous semigroup homomorphism h : X → I with h ( x ) = h ( y ).The inclusion relations ( ∗ ) between the topologies W X , W ◦ X , W • X , and T X ensure that for a Hausdorfftopologized semigroup X the above separation properties relate as follows: W ◦ -weak (cid:11) (cid:19) I -weak k s + (cid:11) (cid:19) W • -weak (cid:11) (cid:19) W ◦ -Hausdorff (cid:11) (cid:19) W -Hausdorff k s + W • -Hausdorff (cid:11) (cid:19) W ◦ T -separated I -separated (cid:11) (cid:19) (cid:11) (cid:19) K S + k s W • T -separatedsemilattice. HE INTERPLAY BETWEEN WEAK TOPOLOGIES ON TOPOLOGICAL SEMILATTICES 3
We complete this diagram by three properties of topologized semilattices, determined by the orderand topological structure. We recall that each semilattice X carries the partial order ≤ defined by x ≤ y iff xy = x .A topologized semilattice X is defined to be • a U -semilattice if for each open set V ⊂ X and point x ∈ V there exists a point v ∈ V whoseupper set ↑ v = { y ∈ X : vy = v } contains x in its interior in X ; • a W -semilattice if for each open set V ⊂ X and point x ∈ V there exists a finite subset F ⊂ V whose upper set ↑ F := S y ∈ F ↑ y contains x in its interior in X ; • a V -semilattice if for any points x y in X there exists a point v / ∈ ↓ y in X whose upper set ↑ v contains x in its interior in X .It is clear that each (Hausdorff semitopological) U -semilattice is a W -semilattice (and a V -semilattice).In Proposition 8.4 we shall observe that each semitopological V -semilattice is W • -Hausdorff and inTheorem 8.1 we shall prove that a semitopological semilattice is a W -semilattice if and only if it is a U -semilattice if and only if X is a U -semilattice in the sense of [8, p.16]. By (the proof of) Lemma 2.10in [8], each Hausdorff (semi)topological U -semilattice X is I -separated.Therefore, for a Hausdorff semitopological semilattice we obtain the following Diagram 1 describingthe interplay between various separation properties of Hausdorff semitopological semilattices.weak ◦ (cid:11) (cid:19) I -weak k s + (cid:11) (cid:19) weak • (cid:11) (cid:19) W ◦ -Hausdorff (cid:11) (cid:19) W -Hausdorff k s + K S (cid:11) (cid:19) W • -Hausdorff (cid:11) (cid:19) W ◦ T -separated I -separated k s + W • T -separated W -semilattice K S k s + U -semilattice K S + V -semilattice K S Diagram 1.
Implications between some properties of Hausdorff semitopological semilatticesOne of the main results of this paper is Theorem 9.1 saying that for a compact Hausdorff semitopo-logical semilattice all properties in Diagram 1 are equivalent.Some separation properties of Diagram 1 remain equivalent for complete Hausdorff semitopologicalsemilattices. A topologized semilattice X is called complete if each chain in X has inf C ∈ ¯ C andsup C ∈ ¯ C in X . Complete topologized semilattices play an important role in the theory of (absolutely)H-closed semilattices, see [1, 2, 4, 6, 13, 18, 19]. By [1, 3.1], a Hausdorff semitopological semilattice X is complete if and only if each closed chain in X is compact if and only if for any continuoushomomorphism h : S → Y from a closed subsemilattice S ⊂ X to a Hausdorff topological semigroup Y the image h ( S ) is closed in Y . In Theorem 7.6 we shall use this characterization to prove that eachcomplete Hausdorff topological semilattice X has W X ⊂ L X ⊂ W • X , where L X is the Lawson topology,determined by the order structure of X .In Theorem 5.6 we prove that a Hausdorff semitopological semilattice X is complete if and only if itsweak • topology W • X is compact. We use this characterization to prove in Theorem 6.5 that the Lawsontopology L X of a semilattice X is compact if and only if each non-empty chain C ⊂ X has sup C andinf C in X . This characterization improves the classical Theorem III-1.9 in [12].In Section 7 we study the interplay between the Scott and Lawson topologies S X and L X with weaktopologies W and W • on a semitopological semilattice X . The main result here is Theorem 7.6 sayingthat for a complete Hausdorff semitopological semilattice X the inclusions W X ⊂ L X ⊂ W • X hold if X TARAS BANAKH, SERHII BARDYLA is a topological semilattice or X satisfies the separation axiom ~T δ (which is weaker than the functionalHausdorffness of X ).In Theorem 10.1 we shall prove that for a complete semitopological semilattice X satisfying theseparation axiom T the following properties are equivalent: I -separated ⇔ W • T -separated ⇔ W -Hausdorff ⇔ W • -Hausdorff ⇔ V -semilattice ⇔ ( W X = W • X ).2. Categorial properties of the weak ◦ , weak • , and I -weak topologies In this section we shall establish some elementary categorial properties of the weak ◦ , weak • , and I -weak topologies of topologized semigroups.We recall that a function f : X → Y between topological spaces is • continuous if for any open set U ⊂ Y the preimage f − ( U ) is open in X ; • open if for any open set V ⊂ X the image f ( V ) is open in Y ; • closed if for any closed set A ⊂ X the image f ( A ) is closed in Y ; • perfect if f is closed and for every y ∈ Y the preimage f − ( y ) is compact; • a topological embedding if f : X → f ( X ) is a homeomorphism of X onto its image f ( X ) in Y . Proposition 2.1.
Let h : X → Y be a homomorphism between topologized semigroups. (1) If h is continuous, then h remains continuous with respect to the weak ◦ , weak • , and I -weaktopologies on X and Y . (2) If h is open, then it remains open in the weak ◦ topologies on X and Y . (3) If h is perfect, then h remains perfect in the weak • topologies on X and Y . (4) If h is a topological embedding and Y is a subtopological semigroup, then h remains a topologicalembedding in the weak • topologies on X and Y .Proof.
1. Assume that the homomorphism h is continuous. Then for any open (or closed) subsemigroup S ⊂ Y the preimage h − ( S ) is an open (or closed) subsemigroup of X . This implies that the preimageof any (sub)basic open set of the weak ◦ (weak • ) topology on Y is open in the weak ◦ (weak • ) topology of X , and hence h is continuous in the weak ◦ (or weak • ) topologies on X and Y . Also for any continuoushomomorphism ϕ : Y → I the composition ϕ ◦ h : X → I is a continuous homomorphism, which impliesthat h is continuous in the I -weak topologies on X and Y .2. If the homomorphism h is open, then for any set U ∈ W ◦ X in the weak ◦ topology on X and anypoint y ∈ h ( U ) we can choose a point x ∈ U ∩ h − ( y ) and find an open subsemigroup V ⊂ U containing x . Since the homomorphism h is open, the image h ( V ) is an open subsemigroup of Y and hence belongsto the weak ◦ topology of Y . Taking into account that y = h ( x ) ∈ h ( V ) ⊂ h ( U ), we see that y is aninterior point of h ( U ) in the weak ◦ topology of Y and hence h ( U ) ∈ W ◦ Y . So, the homomorphism h : ( X, W ◦ X ) → ( Y, W ◦ Y ) is open.3. Now assuming that the homomorphism h is perfect, we shall show that the map h : ( X, W • X ) → ( Y, W • Y ) is closed. Fix any closed set F ⊂ X in the weak • topology W • X . To show that h ( F ) is closedin the weak • topology W • Y , take any point y ∈ Y \ h ( F ). If y / ∈ h ( X ), then Y \ h ( X ) ∈ W • Y is anopen neighborhood of y , disjoint with h ( F ). So, we assume that y ∈ h ( X ). Since h is perfect and y / ∈ h ( F ), the set h − ( y ) is compact and disjoint with F . Since F is closed in ( X, W • X ), for every x ∈ h − ( y ) there is a finite family F x of closed subsemilattices of X such that the basic open set V x := X \ S F x ∈ W • X is an open neighborhood of x , disjoint with the set F . For the open cover { V x : x ∈ f − ( y ) } of the compact subset f − ( y ) of X , there exists a finite subset Φ ⊂ f − ( y ) suchthat f − ( y ) ⊂ S x ∈ Φ V x = X \ T x ∈ Φ S F x . It follows that the intersection T x ∈ Φ S F x contains F andis disjoint with h − ( y ). Since the homomorphism h is closed, for any E ∈ S x ∈ Φ F x the image h ( E ) isa closed subsemilattice of Y and hence h ( E ) is closed in the weak • topology W • Y of Y . Then the set C = T x ∈ Φ S E ∈F x h ( E ) is closed in ( Y, W • Y ), and h ( F ) ⊂ C ⊂ Y \ { y } , witnessing that the set h ( F )is closed in ( Y, W • Y ). Since for every y ∈ Y , the compact set f − ( y ) remains compact in the weak • topology of X , the closed map h : ( X, W • X ) → ( Y, W • Y ) is perfect. HE INTERPLAY BETWEEN WEAK TOPOLOGIES ON TOPOLOGICAL SEMILATTICES 5
4. Finally assume that h : X → Y is a topological embedding and Y is a subtopological semigroup.By Proposition 2.1(1), the homomorphism h : ( X, W • X ) → ( Y, W • Y ) is continuous. To prove that h is atopological embedding, it suffices to show that for every W • X -closed subset F ⊂ X and any x ∈ X \ F there exists an open set V ∈ W • Y such that h ( x ) ∈ V and V ∩ h ( F ) = ∅ . Since F is closed in ( X, W • X ),there are closed subsemigroups F , . . . , F n of X such that x ∈ X \ ( F ∪ · · · ∪ F n ) ⊂ X \ F . Since h is a homomorphic topological embedding, for every i ≤ n the image h ( F i ) is a closed subsemigroupof h ( X ). Since Y is a subtopological semigroup, the closure E i of h ( F i ) in Y is a subsemigroup of Y such that E i ∩ h ( X ) = h ( F i ). Then V = Y \ ( E ∪ · · · ∪ E n ) is an open set in ( Y, W • Y ) such that V ∩ h ( X ) = h ( X \ ( F ∪ · · · ∪ F n )) and hence h ( x ) ∈ V and V ∩ h ( F ) = ∅ . (cid:3) Proposition 2.1(4) implies
Corollary 2.2.
For any subtopological semigroup Y and any subsemigroup X ⊂ Y the weak • topology W • X coincides with the subspace topology on X inherited from the weak • topology of Y . Let us also observe the following characterization of the T i -axiom of the weak • topology on a subtopo-logical band for i ∈ { , } .We recall that a topological space X satisfies the separation axiom • T if for any distinct points x, y ∈ X there exists closed set F ⊂ X containing exactly one ofthe points x, y ; • T if each singleton { x } ⊂ X is closed in X . Proposition 2.3.
A subtopological band X satisfies the separation axiom T if and only if the weak • topology W • X satisfies the separation axiom T .Proof. Since W • X ⊂ T X , the T -separation property of the weak • topology W • X implies that property ofthe topology T X .If the topology T X satisfies the separation axiom T , then for any distinct points x, y ∈ X thereexists a closed subset F ⊂ X containing exactly one of the points x, y . If x ∈ F , then { x } ⊂ F is aclosed subsemigroup of X (because X is a subtopological band) and then E := { x } is a W • X -closed setcontaining x but not y . If x / ∈ F , then E := { y } is a W • X -closed set containing x but not y . In bothcases we have found a W • X -closed set E containing exactly one of the points x, y . This means that theweak • topology W • X satisfies the separation axiom T . (cid:3) By analogy we can prove
Proposition 2.4.
A topologized band X satisfies the separation axiom T if and only if the weak • topology W • X satisfies the separation axiom T . The definitions of the weak ◦ and weak • topologies imply the following simple characterizations of W ◦ -Hausdorff and W • -Hausdorff topologized semigroups. Proposition 2.5.
The weak ◦ topology W ◦ X of a topologized semigroup X is Hausdorff if and only ifany two distinct points x, y are contained in disjoint open subsemigroups of X . Proposition 2.6.
The weak • topology W • X of a topologized semigroup X is Hausdorff if and only if forany distinct points x, y ∈ X there exists a finite cover F of X by closed subsemigroups of X such noset F ∈ F contains both points x and y . Shift-continuity of the I -weak, weak ◦ and weak • topologies A topology τ on a semigroup X is called shift-continuous if for any a ∈ X the left shift ℓ a : X → X , ℓ a : x ax , and the right shift r a : X → X , r a : x xa , both are continuous. This happens if andonly if ( X, τ ) is a semitopological semigroup.
Proposition 3.1.
For any semitopological semigroup X , the I -weak topology W X is shift-continuousand hence ( X, W X ) is a semitopological semigroup. TARAS BANAKH, SERHII BARDYLA
Proof.
We need to prove that for every a ∈ X the left and right shifts ℓ a , r a : X → X are continuouswith respect to the I -weak topology W X . This will follow as soon as we prove that for any continuoushomomorphism h : X → I the compositions hℓ a := h ◦ ℓ a and hr a := h ◦ r a are homomorphisms of X into I . Indeed, for any x, y ∈ X we get hℓ a ( xy ) = h ( axy ) = h ( a ) h ( x ) h ( y ) = h ( a ) h ( a ) h ( x ) h ( y ) == h ( a ) h ( x ) h ( a ) h ( y ) = h ( ax ) h ( ay ) = hℓ a ( x ) · hℓ a ( y ) , which means that hℓ a : X → I is a continuous homomorphism. By analogy we can check that hr a is ahomomorphism. (cid:3) Example 3.2.
For the discrete two-element group X = { , a } with generator a the weak ◦ and weak • topologies W ◦ X = (cid:8) ∅ , { } , X (cid:9) and W • X = (cid:8) ∅ , { a } , X (cid:9) are not shift-continuous and hence ( X, W ◦ X ) and ( X, W • X ) are not semitopological semigroups. On theother hand, the I -weak topology W X = {∅ , X } is anti-discrete and hence is shift-continuous.A topological semigroup X is called shift-homomorphic if for any a ∈ X the left shift ℓ a : X → X , ℓ a : x ax , and the right shift r a : X → X , r a : x xa , both are homomorphisms of X . The followingcharacterization can be derived from the definitions. Proposition 3.3.
A semigroup X is shift-homomorphic if and only if axay = axy and xaya = xya for any x, y, a ∈ X . Proposition 3.3 implies that each semilattice is shift-homomorphic. It is easy to construct examplesof shift-homomorphic semigroups which are not semilattices.Proposition 2.1(1) implies the following useful fact.
Proposition 3.4. If X is a shift-homomorphic semitopological semigroup, then ( X, W ◦ X ) and ( X, W • X ) are semitopological semigroups. Since semilattices are shift-homomorphic semigroups, Proposition 3.4 implies
Corollary 3.5. If X is a semitopological semilattice, then ( X, W ◦ X ) and ( X, W • X ) are semitopologicalsemilattices. Problem 3.6.
Is there a semitopological band X for which the topologized bands ( X, W ◦ X ) and ( X, W • X )are not semitopological?4. Examples of weak ◦ and weak • topological semigroups In this section we present some examples of weak ◦ , weak • and I -weak topologized semigroups. Werecall that a topologized semigroup ( X, T X ) is called I -weak , (resp. weak ◦ , weak • ) if T X = W X (res. T X = W ◦ X , T X = W • X ).A semigroup X is called linear if xy ∈ { x, y } for any x, y ∈ X . It is clear that each subset of a linearsemigroup is a subsemigroup. Consequently, we have Proposition 4.1.
Each linear topologized semigroup is both weak ◦ and weak • . Example 4.2.
There exists a linear topological semilattice which is not I -weak. Proof.
Consider the set X = { } ∪ { n : n ∈ N } endowed with the semilattice operation of minimum.Endow X with the topology T X consisting of sets U ⊂ X having the property:( ⋆ ) if 0 ∈ U , then n ∈ U for all but finitely many numbers n ∈ N . HE INTERPLAY BETWEEN WEAK TOPOLOGIES ON TOPOLOGICAL SEMILATTICES 7
It is clear that X is a linear topological semilattice. So, X is both weak ◦ and weak • . We claim that X is not I -weak. In the opposite case we could find continuous homomorphisms h , . . . , h k : X → I andopen sets U , . . . , U k ⊂ I such that0 ∈ k \ i =1 h − i ( U i ) ⊂ { } ∪ { n : n ∈ N } ∈ T X . Replacing each set U i by a smaller open set, we can assume that U i is order-convex in I . Since 0 isnon-isolated in X , there is a number n ∈ N such that n ∈ T ni =1 h − i ( U i ). For every i ≤ n the inequality0 ≤ n +1 ≤ n implies h i (0) ≤ h i ( n +1 ) ≤ h i ( n ). Since h i (0) , h i ( n ) ∈ U i , the order-convexity of U i ensures that h i ( n +1 ) ∈ U i . Then n +1 ∈ T ki =1 h − i ( U i ) ⊂ { } ∪ { m : m ∈ N } , which is a desiredcontradiction showing that T X = W X .The above example X was first considered by Gutik and Repovˇs in [13]. (cid:3) Proposition 4.3.
Each subsemigroup X of an I -weak (resp. weak ◦ , weak • ) semigroup Y is I -weak(resp. weak ◦ , weak • ).Proof. By Proposition 2.1(1), the identity embedding i : X → Y is continuous in the I -weak topologieson X and Y . If the topologized semigroup Y is I -weak, then T Y = W Y and we have the chain ofcontinuous identity embeddings:( X, T X ) → ( X, W X ) → ( Y, W Y ) = ( Y, T Y ) . Taking into account that the identity map ( X, T X ) → ( Y, T Y ) is a topological embedding, we concludethat the identity map ( X, W X ) → ( X, T X ) is continuous and hence T X = W X , which means that thetopologized semigroup X is I -weak.By analogy we can prove that X is weak ◦ (or weak • ) if so is the topologized semigroup Y . (cid:3) Proposition 4.4.
The Tychonoff product X = Q α ∈ A X α of I -weak (resp. weak ◦ , weak • ) topologizedsemigroups is I -weak (weak ◦ , weak • ).Proof. Assume that the topologized semigroups X α , α ∈ A , are I -weak. By Proposition 2.1(1), forevery α ∈ A the continuity of the projection pr α : X → X α implies the continuity of pr α in the I -weaktopologies on X and X α . This implies that the identity map( X, W X ) → Y α ∈ A ( X α , W X α )is continuous. Now the chain of continuous identity maps( X, W X ) → Y α ∈ A ( X α , W X α ) = Y α ∈ A ( X α , T X α ) = ( X, T X ) → ( X, W X )ensures that T X = W X , which means that the topologized semigroup X is I -weak.By analogy we can prove that the topologized semigroup X is weak ◦ (or weak • ) if so are the topolo-gized semigroups X α , α ∈ A . (cid:3) Propositions 4.3 and 4.4 imply the following theorem.
Theorem 4.5.
Any subsemigroup of the Tychonoff product of linear topologized semigroups is bothweak ◦ and weak • . The following lemma gives a condition under which a topologized semigroup is not weak • . Lemma 4.6.
A topologized semigroup X is not weak • if there exists an open set U ⊂ X and a point x ∈ U such that for every open neighborhood O x ⊂ U of x there exists an infinite set I ⊂ X \ O x suchthat ab ∈ O x for any distinct points a, b ∈ I . TARAS BANAKH, SERHII BARDYLA
Proof.
Assuming that X is weak • , we can find closed subsemigroups F , . . . , F n ⊂ X such that x ∈ X \ ( F ∪ · · · ∪ F n ) ⊂ U . By our assumption, for the open neighborhood O x := X \ ( F ∪ · · · ∪ F n )of x there exists an infinite set I ⊂ X \ O x such that ab ∈ O x for any distinct points a, b ∈ I . Since I ⊂ X \ O x = F ∪ · · · ∪ F n , for some i ≤ n the set I ∩ F n is infinite. Choose any distinct points a, b ∈ I ∩ F i and conclude that ab ∈ F i ⊂ X \ O x , which contradicts the choice of I . (cid:3) Now we construct a weak ◦ topological semilattice which is not weak • . Example 4.7.
Let X be an infinite discrete space and 0 ∈ X be any point of X . Endow X with thesemilattice operation xy = ( x if x = y, . It is clear that the discrete topological semilattice X satisfies the condition of Lemma 4.6 and hence isnot weak • . On the other hand, X is weak ◦ , being a discrete topological semilattice.An example of a weak • non-weak ◦ topological semilattice is more complicated. Example 4.8.
There exists a countable topological semilattice X which is metrizable and weak • butnot weak ◦ . Proof.
Let X be the set of all sequences ( x n ) n ∈ ω of non-negative rational numbers such that x n = 0 forall but finitely many numbers. Endow X with the semilattice operation of coordinatewise maximum.Consider the function Σ : X → R , Σ : ( x n ) n ∈ ω P n ∈ ω x n . Let τ be the smallest topology on X suchthat for every k ∈ ω the coordinate projection pr k : X → R , pr k : ( x n ) n ∈ ω x k , is continuous.Let T X be the topology on X generated by the base { U ∩ Σ − ([0 , a )) : U ∈ τ, a ∈ R } . It can be shownthat the topology T X is regular and has a countable base. By the Urysohn-Tychonoff MetrizabilityTheorem [11, 4.2.9], the topological space ( X, T X ) is metrizable.One can check that ( X, T X ) is a topological semilattice with respect to the semilattice operation ofmaximum. Moreover, this topological semilattice is weak • but not weak ◦ . (cid:3) Problem 4.9.
Let
X, Y be Hausdorff semitopological semilattices. Are the identity maps( X × Y, W ◦ X × Y ) → ( X, W ◦ X ) × ( Y, W ◦ Y ) , ( X × Y, W • X × Y ) → ( X, W • X ) × ( Y, W • Y ) , ( X × Y, W X × Y ) → ( X, W X ) × ( Y, W Y )homeomorphisms?In Proposition 10.3 we shall give a partial affirmative answer to this problem for I -separated completesemitopological semilattices.5. W • -compact topologized semigroups In this section we study topologized semigroups whose weak • topology is compact. Such topologizedsemigroups are called W • -compact .First, we present a simple characterization of W • -compact topologized semigroups and derive fromit a more complicated (and useful) characterization of W • -compact semitopological semilattices.A non-empty family F of subsets of a set X is called centered if each non-empty finite subfamily E ⊂ F has non-empty intersection T E . The classical Alexander’s subbase Theorem [11, 3.12.2] saysthat a topological space X is compact if and only if X has a subbase B of the topology such thatany centered subfamily F ⊂ { X \ B : B ∈ B} has non-empty intersection. This Alexander’s Theoremimplies the following characterization. Theorem 5.1.
A topologized semigroup X is W • -compact if and only if each centered family of closedsubsemigroups of X has non-empty intersection. HE INTERPLAY BETWEEN WEAK TOPOLOGIES ON TOPOLOGICAL SEMILATTICES 9
Now we shall detect topologized semilattices whose weak • topology is compact. First we recall somedefinitions related to topologized posets.By a topologized poset we understand a poset endowed with a topology. A topologized poset X iscalled • complete if each non-empty chain C ⊂ X has inf C ∈ ¯ C and sup C ∈ ¯ C ; • chain-compact if each closed chain in X is compact; • ↑↓ -closed if for any point x ∈ X the sets ↑ x and ↓ x are closed in X . Lemma 5.2.
Let X be a ↑↓ -closed topologized poset. The closure ¯ C of any chain C in X is a chain.Proof. Assuming that ¯ C is not a chain, we can find two points x, y ∈ ¯ C such that x / ∈ l y := ( ↑ y ) ∪ ( ↓ y ).Since X is ↑↓ -closed, the set X \ l y is an open neighborhood of x . Since x ∈ ¯ C , there exists a point z ∈ C \ l y . Taking into account that C is a chain, we conclude that C ⊂ l z and hence y ∈ ¯ C ⊂ l z = l z ,which contradicts z / ∈ l y . (cid:3) Complete topologized posets are studied in details in [4], where it is proved that completeness oftopologized posets can be equivalently defined using up-directed and down-directed sets instead ofchains.A subset D of a poset X is called up-directed (resp. down-directed ) if for any x, y ∈ D there exists z ∈ D such that x ≤ z and y ≤ z (resp. z ≤ x and z ≤ y ).The following characterization was proved in [4]. It is a topological version of a known characterizationof complete posets due to Iwamura [14] (see also [7] and [17]). Lemma 5.3.
A topologized poset X is complete if and only if each non-empty up-directed set U ⊂ X has sup U ∈ ¯ U and each non-empty down-directed set D ⊂ X has inf D ∈ ¯ D in X . A topologized semilattice X is called complete if it is complete as a topologized poset endowed withthe natural partial order ≤ (defined by x ≤ y iff xy = x ). Lemma 5.4.
Each complete topologized semilattice is W • -compact.Proof. By Theorem 5.1, to show that X is W • -compact, it suffices to prove that each centered family F of closed subsemilattices in X has non-empty intersection.This will be proved by induction on the cardinality |F | of the family F . If F is finite, then T F 6 = ∅ as F is centered. Assume that for some infinite cardinal κ we have proved that each centered family F consisting of |F | < κ many closed subsemilattices of X has non-empty intersection.Take any centered family F = { F α } α ∈ κ of closed subsemilattices of X . By the inductive assumption,for every α < κ the closed subsemilattice F ≤ α = T β ≤ α F β is not empty. Let M α be a maximal chainin F ≤ α . By the completeness of X , the chain M α has inf M α ∈ ¯ M α ⊂ ¯ F ≤ α = F ≤ α . We claim thatthe element x α := inf M α is the smallest element of the semilattice F ≤ α . In the opposite case, we canfind an element z ∈ F ≤ α such that x α z and conclude that zx α < x α and { zx α } ∪ M α is a chainin F ≤ α that properly contains the maximal chain M α , which contradicts the maximality of M α . Thiscontradiction shows that x α is the smallest element of the semilattice F ≤ α .Observe that for any ordinals α ≤ β < κ the inclusion x β ∈ F ≤ β ⊂ F ≤ α implies x α ≤ x β . So, forevery α ∈ κ the set C α := { x β } α ≤ β<κ is a chain in X . By the completeness of X , the chain C α hassup C α ∈ ¯ C α ⊂ F ≤ α . Since the transfinite sequence ( x α ) α ∈ κ is non-decreasing, sup C α = sup C for all α ∈ κ . Then sup C ∈ T α ∈ κ F ≤ α = T α ∈ κ F α , which means that the family F = { F α } α ∈ κ has non-emptyintersection. (cid:3) Lemma 5.5.
Each W • -compact topologized semilattice X is chain-compact.Proof. We should prove that any closed chain C in X is compact. This will follow as soon as we showthat each centered family F of closed subsets of C has non-empty intersection. Since C is a chain, eachset F ∈ F is a closed subsemilattice of X . By Theorem 5.1, the W • -compactness of the topologizedsemilattice X implies that T F 6 = ∅ . (cid:3) Lemmas 5.4 and 5.5 will be used to prove the following characterization of W • -compactness. Theorem 5.6.
For an ↑↓ -closed topologized semilattice X the following conditions are equivalent: (1) X is complete; (2) X is W • -compact; (3) X is chain-compact.Proof. The implications (1) ⇒ (2) ⇒ (3) follow from Lemmas 5.4 and 5.5. To prove that (3) ⇒ (1),assume that an ↑↓ -closed topologized semilattice X is chain-compact and take any chain C ⊂ X . ByLemma 5.2, the closure ¯ C of C in X is a chain. By the chain-compactness of X , the closed chain ¯ C iscompact. By the compactness of ¯ C and the ↑↓ -closedness of X , the centered family { ¯ C ∩ ↓ x : x ∈ ¯ C } ofclosed sets in ¯ C has non-empty intersection, consisting of a unique point min ¯ C , which is the smallestelement of the chain ¯ C . It is clear that min ¯ C is a lower bound of the set C . For any other lowerbound b ∈ X of C we get C ⊂ ↑ b and hence min ¯ C ∈ ¯ C ⊂ ↑ b = ↑ b , which means that b ≤ min ¯ C andinf C = min ¯ C ∈ ¯ C . By analogy we can prove that the chain has sup C = max ¯ C ∈ ¯ C . (cid:3) The Scott and Lawson topologies on a poset
In this section we apply Theorem 5.6 to characterize posets whose Lawson topology is compact. Werecall that a poset is a set endowed with a partial order. For a subset U ⊂ X let ↑ U := S x ∈ U ↑ x .A subset U of a poset X is called Scott-open if U = ↑ U and each up-directed set D ⊂ X withsup D ∈ U intersects U . The family S X of Scott-open subset of X is a topology, called the Scotttopology of the poset X . Observe that for any x ∈ X the set X \ ↓ x is Scott-open in X .The Lawson topology L X on a poset X is generated by the subbase S X ∪ { X \ ↑ x : x ∈ X } . The Lawson topology L X contains the interval topology I X of X , which is generated by the subbase { X \ ↑ x : x ∈ X } ∪ { X \ ↓ x : x ∈ X } . Since { x } = ↑ x ∩ ↓ x , the interval and Lawson topologies on each poset satisfy the separation axiom T .More information on the Scott and Lawson topologies can be found in the monograph [12]. Nowwe prove some lemmas on properties of complete posets. A poset X is defined to be complete if eachnon-empty chain C in X has inf( C ) and sup( C ) in X . Observe that a poset X is complete if and onlyif it is complete as a topologized poset endowed with the anti-discrete topology {∅ , X } . By Lemma 5.3,a poset X is complete if and only if each up-directed set U ⊂ X has sup U ∈ X and each down-directedset L ⊂ X has inf L ∈ X . Lemma 6.1.
If a poset X is complete, then for any maximal chain M ⊂ X and any non-empty subset C ⊂ M the elements inf( C ) and sup( C ) belong to M .Proof. By the maximality of M , it suffices to prove that inf( C ) and sup( C ) are comparable with anyelement x ∈ M . First, we prove that inf( C ) is comparable with x . If x ≥ c for some c ∈ C , theninf( C ) ≤ c ≤ x and we are done. So, we assume that x c and hence x < c for every c ∈ C ⊂ M .This means that x is a lower bound for the set C and hence x ≤ inf( C ) as inf( C ) is the largest lowerbound for C in X . So, inf( C ) is comparable with any element x ∈ M and hence inf( C ) ∈ M by themaximality of the chain M .By analogy, we can prove that sup( C ) is comparable with any element x ∈ M and hence sup( C ) ∈ M . (cid:3) Lemma 6.2.
If a poset X is complete, then for any Scott-open set U ⊂ X and any maximal chain C ⊂ X the set C \ U is either empty or has the largest element x = max( C \ U ) and coincides with C ∩ ↓ x . HE INTERPLAY BETWEEN WEAK TOPOLOGIES ON TOPOLOGICAL SEMILATTICES 11
Proof.
Assume that C \ U is not empty. Since X is chain-complete, for the non-empty chain C \ U there exists x := sup( C \ U ) in X . By Lemma 6.1, x ∈ C .It follows from x = sup( C \ U ) that C \ U ⊂ C ∩ ↓ x . Assuming that C \ U = C ∩ ↓ x , we can finda point c ∈ C ∩ ↓ x such that c / ∈ C \ U and hence c ∈ U . Then x ∈ ↑ c ⊂ ↑ U = U . Being Scott-open,the set U ∋ sup( C \ U ) intersects C \ U , which is a desired contradiction showing that C \ U = C ∩ ↓ x .Then x = sup( C \ U ) ∈ C \ U , which means that x = max( C \ U ). (cid:3) Lemma 6.3.
If a poset X is complete, then for any x ∈ X and any maximal chain C ⊂ X the chain C ∩ ↓ x is either empty or has the largest element c and coincides with C ∩ ↓ c .Proof. Observe that the set U := X \ ↓ x is Scott-open and apply Lemma 6.2. (cid:3) Lemma 6.4.
If a poset X is complete, then for any x ∈ X and any maximal chain C ⊂ X the chain C ∩ ↑ x is either empty or has the smallest element c and coincides with C ∩ ↑ c .Proof. Apply Lemma 6.3 to the poset X , endowed with the opposite partial order (cid:22) = { ( x, y ) ∈ X × X : y ≤ x } . (cid:3) A semilattice X is called complete if it is complete as a poset endowed with the partial order ≤ definedby x ≤ y iff xy = x . By Lemma 5.3, this definition of completeness is equivalent to the definition of acomplete semilattice given in [12, O-2.1(iv)].The following theorem characterizes semilattices whose Lawson topology is compact, thus generalizingTheorem III-1.9 in [12] and completing Proposition O-2.2(iv) in [12]. Theorem 6.5.
For a semilattice X the following conditions are equivalent: (1) X is complete; (2) the Lawson topology L X of X is compact; (3) the interval topology I X of X is compact.Proof. To prove that (1) ⇒ (2), assume the semilattice X is complete. By definition of the Lawsontopology, the topologized semilattice ( X, L X ) is ↑↓ -closed. We claim that it is chain-compact. Givenany L X -closed chain C ⊂ X , we need to show that C is compact in the Lawson topology. Let M be anymaximal chain, containing the chain C . By Lemma 5.2, the maximal chain M is closed in the Lawsontopology L X . By the Alexander subbase Theorem [11, 3.12.2] and Lemmas 6.2, 6.4, the compactnessof M in the Lawson topology will follow as soon as we show that any non-empty centered family F ⊂ { M ∩ ↑ x : x ∈ M } ∪ { M \ ↓ x : x ∈ M } has non-empty intersection. Let U = { x ∈ M : M ∩ ↑ x ∈ F } and L = { x ∈ M : M ∩ ↓ x ∈ F } . Since the family F is centered, x ≤ y for any x ∈ U and y ∈ L . Sincethe family F is not empty, one of the sets U or L is not empty.If U is not empty, then by the completeness of X , the non-empty chain U has sup( U ) in X andsup( U ) ∈ M according to Lemma 6.1. It follows that sup( U ) ∈ M ∩ T x ∈ U ↑ x ∩ T y ∈ L ↓ y = T F .If L is not empty, then by the completeness of X , the non-empty chain L has inf( L ) in X andinf( U ) ∈ M according to Lemma 6.1. It follows that inf( L ) ∈ M ∩ T x ∈ U ↑ x ∩ T y ∈ L ↓ y = T F .In both cases, we conclude that the family F has non-empty intersection and hence the maximalchain M is compact and so is the closed subset C in M . Therefore, the topologized semilattice ( X, L X )is chain-compact. Observe that each subbasic open set U ∈ S X ∪ { X \ ↑ x } in the Lawson topology L X coincides with the complement of a closed subsemilattice of ( X, L X ), which remains closed in the weak • topology W • X of the topologized semilattice ( X, L X ). This implies that W • X = L X . By Theorem 5.6,the topology W • X = L X of the chain-compact ↑↓ -closed topologized semilattice ( X, L X ) is compact.The implication (2) ⇒ (3) trivially follows from the inclusion I X ⊂ L X .(3) ⇒ (1) Assume that the interval topology I X on X is compact. To show that X is a completesemilattice, we need to show that each non-empty chain C ⊂ X has inf C and sup C in X .First we show that the chain C has inf C . For this observe that F = {↓ c : c ∈ C } is a centered familyof I X -closed sets in X . By the compactness of the interval topology I X , the intersection K = T F is a closed non-empty set in ( X, I X ). By the compactness of K in the interval topology, the intersection T x ∈ K K ∩ ↑ x is non-empty and contains the unique element max K , which coincides with inf C by thedefinition of the greatest lower bound inf C .By analogy we can show that C has sup( C ) ∈ X . (cid:3) A semilattice X is called meet continuous if for any up-directed subset D ⊂ X having sup D ∈ X andany a ∈ X the up-directed set aD has sup( aD ) = a · sup D . For continuous semilattices the followingtheorem was proved in Theorem III-2.8 [12]. Theorem 6.6.
For a semilattice X the following conditions are equivalent: (1) X is meet continuous; (2) ( X, S X ) is a semitopological semilattice; (3) ( X, L X ) is a semitopological semilattice.If X is complete and the Lawson topology L X is Hausdorff, then the conditions (1)–(3) are equivalentto (4) ( X, L X ) is a compact topological semilattice; (5) ( X, S X ) is a compact topological semilattice.Proof. (1) ⇒ (2). Assume that the semilattice X is meet-continuous. To show that ( X, S X ) is asemitopological semilattice, we need to show that for any a ∈ X the shift s a : X → X , s a : x ax , iscontinuous in the Scott topology S X . This will follow as as soon as we show that for any Scott-openset U ⊂ X the set s − a ( U ) = { x ∈ X : ax ∈ U } is Scott-open.To show that s − a ( U ) is Scott-open, take any up-directed set D ⊂ X with sup D ∈ s − a ( U ). Since X is meet continuous, the up-directed set aD has sup( aD ) = a · sup D ∈ a · s − a ( U ) ⊂ U . Since U isScott-open, there exists x ∈ D with ax ∈ U and hence x ∈ D ∩ s − a ( U ), which means that s − a ( U ) isScott-open and the shift s a is continuous.(2) ⇒ (3) Assume that ( X, S X ) is a semitopological semilattice. To show that ( X, L X ) is a semitopo-logical semilattice, it suffices to check that for any a ∈ X and subbasic open set U ∈ S X ∪{ X \↑ z : z ∈ X } the preimage s − a ( U ) belongs to the Lawson topology L X .If U ∈ S X , then s − a ( U ) ∈ S X ⊂ L X as ( X, S X ) is a semitopological semilattice.So, we assume that U = X \ ↑ z for some z ∈ X . If a / ∈ ↑ z , then s − a ( U ) = X ∈ L X . If a ∈ ↑ z , then s − a ( U ) = U ∈ L X .(3) ⇒ (1) Assume that ( X, L X ) is a semitopological semilattice. To prove that X is meet continuous,take any up-directed subset D ⊂ X that has sup D ⊂ X . We need to show that for every a ∈ X theelement a · sup D is the least upper bound of the set aD . To derive a contradiction, assume that this isnot true and find an upper bound b of the set aD such that a · sup D b . Replacing b by ba · sup D , wecan assume that b < a · sup D and thus a · sup D / ∈ X \ ↓ b . The continuity of the shift s a in the Lawsontopology, implies that the set s − a ( X \ ↓ b ) is an open neighborhood of sup D in the Lawson topology L X . Then sup D ∈ U \ ↑ F ⊂ s − a ( X \ ↓ b ) for some Scott-open set U ⊂ X and some finite set F ⊂ X .Since sup D ∈ U , there exists a point d ∈ D ∩ U . It follows from d ≤ sup D / ∈ ↑ F that d / ∈ ↑ F andhence d ∈ U \ ↑ F ⊂ s − a ( X \ ↓ b ) and hence ad ∈ aD \ ↓ b which is not possible as b is an upper boundfor the set aD ∋ ad .The implications (4) ⇒ (3) and (5) ⇒ (2) are trivial.Now assuming that the semilattice X is complete and the Lawson topology L X is Hausdorff, weshall prove that (3) ⇒ (4) ⇒ (5). To prove that (3) ⇒ (4), assume that ( X, L X ) is a semitopologicalsemilattice. By Theorem 6.5, the Lawson topology L X is compact and by [16], the compact Hausdorffsemitopological semilattice ( X, L X ) is a topological semilattice.To prove that (4) ⇒ (5), assume that ( X, L X ) is a topological semilattice. To show that ( X, S X ) is atopological semilattice, take any points x, y ∈ X and a Scott-open neighborhood U ∈ S X ⊂ L X of theirproduct xy . Since ( X, L X ) is a topological semilattice, the points x, y have neighborhoods U x , U y ∈ L X HE INTERPLAY BETWEEN WEAK TOPOLOGIES ON TOPOLOGICAL SEMILATTICES 13 such that U x U y ⊂ U . By the (already proved) implication (4) ⇒ (1), the complete semilattice X ismeet continuous. By Proposition III-2.5 [12], the upper sets ↑ U x and ↑ U y are Scott-open neighborhoodsof x and y , respectively. Then ( ↑ U x ) · ( ↑ U y ) ⊂ ↑ ( U x · U y ) ⊂ ↑ U = U , witnessing that the semilatticeoperation is continuous in the Scott topology of X . (cid:3) Example 6.7.
Consider the semilattice X = (cid:0) { , } × ( ω ∪ { ω, ω + 1 } ) (cid:1) \ { (1 , ω ) } endowed with the operation of coordinatewise minimum. It is easy to see that X is complete and theinterval topology I X is Hausdorff and coincides with the Lawson topology L X . By Theorem 6.5, theLawson topology L X = I X is compact. On the other hand, the semilattice X is not meet-continuousas the chain C = { } × ω has sup C = (1 , ω + 1) and for a = (0 , ω + 1) the chain aC = { } × ω has sup aC = (0 , ω ) = a · sup C . Therefore, ( X, L X ) = ( X, I X ) is a compact Hausdorff topologizedsemilattice, which is not semitopological.7. The interplay of the weak topologies W X and W • X with the Scott and Lawsontopologies S X and L X In this section we investigate the interplay between the topologies W X , W • X , and the Lawson topology L X on a chain-compact semitopological semilattice X . Lemma 7.1.
Each open subset U = ↑ U of a complete topologized semilattice X is Scott-open.Proof. Given an up-directed set D ⊂ X with sup D ∈ U , we need to prove that D ∩ U = ∅ . To derive acontradiction, assume that D ∩ U = ∅ . By Lemma 5.3, sup D ∈ ¯ D ⊂ X \ U = X \ U , which contradictsour assumption. (cid:3) Now we find conditions on a Hausdorff semitopological semilattice X guaranteeing that each Scott-open subset of X is open.Let us recall that a topological space X satisfies the separation axiom • T if for any distinct points x, y ∈ X there exists an open set U ⊂ X such that x ∈ U ⊂ X \ { y } ; • T if for any distinct points x, y ∈ X there exists an open set U ⊂ X such that x ∈ U ⊂ ¯ U ⊂ X \ { y } ; • T if X is a T -space and for any open set V ⊂ X and point x ∈ V there exists an open set U ⊂ X such that x ∈ U ⊂ ¯ U ⊂ V ; • T if X is a T -space and for any open set V ⊂ X and point x ∈ V there exists a continuousfunction f : X → [0 ,
1] such that x ∈ f − ([0 , ⊂ V ; • T δ if X is a T -space and for any open set U ⊂ X and point x ∈ U there exists a countablefamily U of closed neighborhoods of x in X such that T U ⊂ U ; • ~T i for i ∈ { , , δ, , } if X admits an injective continuous map X → Y to a T i -space Y .Topological spaces satisfying a separation axiom T i are called T i -spaces . The separation axioms T δ and ~T δ were introduced in [5].The following diagram describes the implications between the separation axioms T i and ~T i for i ∈{ , , δ, , } . T + (cid:11) (cid:19) T + (cid:11) (cid:19) T δ + (cid:11) (cid:19) T K S (cid:11) (cid:19) + T K S (cid:11) (cid:19) ~T + ~T + ~T δ + ~T + T Observe that a topological space X satisfies the separation axiom ~T if and only if it is functionallyHausdorff in the sense that for any distinct points x, y ∈ X there exists a continuous function f : X → R with f ( x ) = f ( y ). Therefore, each functionally Hausdorff space is a ~T δ -space. A topological space X is sequential if for each non-closed subset A ⊂ X there exists a sequence { a n } n ∈ ω ⊂ A that converges to a point x ∈ X \ A . Theorem 7.2.
A Scott-open subset U of a complete Hausdorff semitopological semilattice X is openin X if one of the following conditions is satisfied: (1) X is a topological semilattice; (2) X is a sequential space; (3) X is a ~T δ -space; (4) X is functionally Hausdorff.Proof. Since U = ↑ U , the complement X \ U is a subsemilattice of X . We claim that the semitopologicalsemilattice X \ U is complete. We need to prove that each non-empty chain C ⊂ X \ U has inf C ∈ ¯ C \ U and sup C ∈ ¯ C \ U . Since X is complete, there exists inf C ∈ ¯ C and sup C ∈ ¯ C . It follows from U = ↑ U ⊂ X \ C that inf C / ∈ U . Taking into account that the set C ⊂ X \ U is up-directed and U is Scott-open, we conclude that sup C / ∈ U . Therefore the topologized semilattice X \ U is complete.Now Theorem 7.3 implies that the set X \ U is closed in X and hence U is open in X . (cid:3) Theorem 7.3.
For a continuous homomorphism h : X → Y from a complete topologized semilattice X to a Hausdorff semitopological semilattice Y the image h ( X ) is closed in Y if one of the followingconditions is satisfied: (1) Y is a topological semilattice; (2) Y is a sequential space; (3) Y is a ~T δ -space; (4) Y is functionally Hausdorff. The statements (1), (2), (3) of this theorem were proved in [1], [3] and [5], respectively. The statement(4) follows from (3).
Problem 7.4.
Let X be a complete Hausdorff semitopological semilattice. Is each Scott-open set U ⊂ X open?Next, we explore the interplay between the Lawson topology L X and the weak topologies on atopologized semilattices. Lemma 7.5.
For any complete topologized semilattice X we have the inclusion W X ⊂ L X .Proof. The inclusion W X ⊂ L X will follow as soon as we prove that for any continuous homomorphism h : X → I and any a ∈ I the sets h − ( ↑ a ) and h − ( ↓ a ) are closed in the Lawson topology L X . Observethat h − ( ↑ a ) is a closed subsemilattice in X , which has the smallest element s by the completeness of X and Lemma 5.3. Now we see that h − ( ↑ a ) = ↑ s is L X -closed.To show that h − ( ↓ a ) is L X -closed, it suffices to check that its complement V := { x ∈ X : h ( x ) > a } is Scott-open. But this follows from Lemma 7.1 as V is an open upper set in X . (cid:3) Theorem 7.6.
For a complete Hausdorff semitopological semilattice X the inclusions W X ⊂ L X ⊂ W • X ⊂ T X hold, provided one of the following conditions is satisfied: (1) X is a topological semilattice; (2) X is sequential; (3) X is a ~T δ -space; (4) X is functionally Hausdorff.Proof. The inclusion W X ⊂ L X was proved in Lemma 7.5 and W • X ⊂ T X follows from the definition ofthe topology W • X . The inclusion L X ⊂ W • X follows from the definitions of the topologies L X , W • X andTheorem 7.2. (cid:3) HE INTERPLAY BETWEEN WEAK TOPOLOGIES ON TOPOLOGICAL SEMILATTICES 15 On U -semilattices, W -semilattices, and V -semilattices In this section we shall prove some characterizations involving U -semilattices, W -semilattices, and V -semilattices, which were defined in the introduction.It is clear that each U -semilattice is a W -semilattice. The converse is true for semitopologicalsemilattices. Theorem 8.1.
For a semitopological semilattice X the following conditions are equivalent: (1) X is a U -semilattice; (2) X is a W -semilattice; (3) for any open set U = ↑ U of X and any point x ∈ U there exists a point y ∈ U whose upper set ↑ y is a neighborhood of x in X .Proof. The implications (2) ⇐ (1) ⇒ (3) are trivial.(3) ⇒ (1) Assume that the semilattice X satisfies the condition (3). To show that X is a U -semilattice, fix any open set V ⊂ X and a point x ∈ V . We need to find a point v ∈ V whose upperset ↑ v contains x in its interior.The separate continuity of the semilattice operation implies that the set ↑ V = S v ∈ V ℓ − v ( V ) is open.By the condition (3), there exists a point y ∈ ↑ V whose upper set ↑ y contains the point x in its interior.For the point y ∈ ↑ V find a point v ∈ V with v ≤ y and observe that the upper set ↑ v ⊃ ↑ y contains x in its interior.(2) ⇒ (1) Assume that X is a W -semilattice. To show that X is a U -semilattice, take any open set V ⊂ X and any point x ∈ V . We need to find an element e ∈ V whose upper set ↑ e is a neighborhoodof x in X .Since X is a W -semilattice, there exists a finite set F ⊂ V whose upper set ↑ F is a neighborhood of x . By Lemma 8.2, for some e ∈ F ⊂ V the upper set ↑ e is a neighborhood of x . (cid:3) Lemma 8.2.
Let X be a semitopological semilattice and F ⊂ X be a finite subset whose upper set ↑ F contains a point x ∈ X in its interior. Then for some e ∈ F the upper set ↑ e is a neighborhood of x .Proof. Since ↑ F ∩ ↓ x is a neighborhood of x in ↓ x , it is not nowhere dense in ↓ x . Since the finite unionof nowhere dense sets is nowhere dense, for some element e ∈ F the closed set ↑ e ∩ ↓ x has non-emptyinterior W in ↓ x . We claim that x ∈ W . Indeed, for any w ∈ W we get w = wx and by the continuityof the shift ℓ w : ↓ x → ↓ x , the set O x := { z ∈ ↓ x : zw ∈ W } is an open neighborhood of x in ↓ x . It isclear that O x ⊂ ↑ W ∩ ↓ x ⊂ ↑ e ∩ ↓ x and hence O x ⊂ W .By the continuity of the shift ℓ x : X → ↓ x , ℓ x : y xy , the set ℓ − x ( W ) = { y ∈ X : xy ∈ W } is anopen neighborhood of x in X such that ℓ − x ( W ) ⊂ ↑ W ⊂ ↑ e . This means that for the point e ∈ F ⊂ U the upper set ↑ e is a neighborhood of x . (cid:3) The equivalence (1) ⇔ (3) of Theorem 8.1 means that our definition of a U -semilattice is equivalentto the classical definition of a U -semilattice given in [8, p.16]. The (proof of) Lemma 2.10 in [8] yieldsthe following important fact. Theorem 8.3 (Lawson) . Each Hausdorff semitopological U -semilattice is I -separated. Next, we establish some properties of V -semilattices. Proposition 8.4.
Each semitopological V -semilattice X satisfying the separation axiom T is W • -Hausdorff.Proof. To prove that the V -semilattice X is W • -Hausdorff, fix any distinct points x, y ∈ X . Becauseof symmetry, we can assume that x y . Since X is a V -semilattice, there exists a point v
6∈ ↓ y whose upper set ↑ v contains x in its interior V in X . By the continuity of the left shifts on X , the set ↑ V = S u ∈ V ℓ − u ( V ) is open in X . Taking into account that ↑ V ⊂ ↑ v , we conclude that ↑ V = V andthen F = X \ V is a closed subsemilattice of X . Since X is a T -space, the singleton { v } is closed and so is its preimage ↑ v = ℓ − v ( { v } ) under thecontinuous shift ℓ v . Then O x := X \ F and O y = X \ ↑ v are disjoint W • X -open neighborhoods of x and y , respectively. (cid:3) We shall say that topologized semilattice X is ↓ -chain-compact if for each x ∈ X the subsemilattice ↓ x := { y ∈ X : y ≤ x } of X is chain-compact. Theorem 8.5.
For a ↓ -chain-compact semitopological semilattice the following conditions are equiva-lent: (1) X is W • -Hausdorff; (2) X is W • T -separated; (3) X is a V -semilattice satisfying the separation axiom T .Proof. The implication (1) ⇒ (2) is trivial and (3) ⇒ (1) was proved in Proposition 8.4. To prove that(2) ⇒ (3), assume that X is W • T -separated. Then X is Hausdorff and hence satisfies the separationaxiom T . Moreover, the separate continuity of the semilattice opeartion implies that X is ↑↓ -closed.To prove that X is a V -semilattice, take any two points x y in X . We need to find a point e / ∈ ↓ y in X whose upper set ↑ e is a neighborhood of x .By our assumption, the subsemilattice ↓ y is chain-compact and by Theorem 5.6, ↓ y is W • -compact.By Corollary 2.2, the set ↓ y is compact in the topological space ( X, W • X ).Since the semilattice X is W • T -separated, for every point z ∈ ↓ y there exist disjoint open sets O z ∈ W • X and U x,z ∈ T X such that z ∈ O z and x ∈ U x,z . We can assume that O z is of basic form O z = X \ S F z for a finite family F z of closed subsemilattices of X .By the compactness of ↓ y , the W • X -open cover { O z : z ∈ ↓ y } of ↓ y has a finite subcover { O z : z ∈ E } (here E is a suitable finite subset of ↓ y ). Then ↓ y ⊂ S z ∈ E O z = S z ∈ E X \ S F z = X \ T z ∈ E S F z andthe W • X -open set X \ T z ∈ E S F z does not intersect the neighborhood U x := T z ∈ E U x,z of x .It follows that U x ⊂ T z ∈ E S F y ⊂ X \ ↓ y . Observe that T z ∈ E S F y = S K where K = { T z ∈ E F z :( F z ) z ∈ E ∈ Q z ∈ E F z } . Each non-empty set K ∈ K is a closed subsemilattice in X which has the smallestelement x K by the ↓ -chain-compactness of X . Consider the finite set F = { x K : K ∈ K \ {∅}} andobserve that U x ⊂ S K ⊂ ↑ F ⊂ X \ ↓ y . By Lemma 8.2, the set F contains a point e such that ↑ e is aneighborhood of x . It follows from e / ∈ ↓ y that e y . (cid:3) Separation properties of weak topologies on compact topological semilattices
In this section we prove that all separation properties in Diagram 1 are equivalent for compact Haus-dorff semitopological semilattices. It should be mentioned that each compact Hausdorff semitopologicalsemilattice is a topological semilattice, see [16]. By Theorem 7.6, we have the following inclusion rela-tions between various weak topologies on a chain-compact Hausdorff topological semilattice X (in thediagram an arrow A → B indicates that
A ⊂ B ). W X (cid:15) (cid:15) / / W ◦ X / / T X L X / / W • X = = ④④④④④④④④ The equivalence of the conditions (1,2,4,6,8,9,12,14) in the following theorem is a well-known resultof Lawson [15], see also [12, VI-3.4].
Theorem 9.1.
For a compact Hausdorff semitopological semilattice X with topology T X the followingconditions are equivalent: (1) W X = T X ; (2) W ◦ X = T X ; (3) W • X = T X ; (4) W X = L X ; HE INTERPLAY BETWEEN WEAK TOPOLOGIES ON TOPOLOGICAL SEMILATTICES 17 (5) L X ⊂ W ◦ X ; (6) the weak ◦ topology W ◦ X is Hausdorff; (7) the weak • topology W • X is Hausdorff; (8) the I -weak topology W X is Hausdorff; (9) the Lawson topology L X is Hausdorff; (10) X is W ◦ T -separated; (11) X is W • T -separated; (12) X is I -separated; (13) X is a W -semilattice; (14) X is a U -semilattice; (15) X is a V -semilattice.Proof. It suffices to prove the chains of implications (1) ⇒ (2) ⇒ (6) ⇒ (10) ⇒ (2) ⇒ (14) ⇒ (12) ⇒ (8) ⇒ (1), (1) ⇒ (3) ⇒ (13) ⇒ (14), (3) ⇔ (7) ⇔ (11) ⇔ (15), (1) ⇒ (4) ⇒ (12), (1) ⇒ (5) ⇒ (14)and (1) ⇒ (9) ⇒ (8).The implications (1) ⇒ (2) ⇒ (6) ⇒ (10) trivially follow from the inclusions of the topologies W X ⊂ W ◦ X ⊂ T X .(10) ⇒ (2) Assume that the compact semitopological semigroup X is W ◦ T -separated. Since W ◦ X ⊂T X , the equality W ◦ X = T X will follow as soon as we check for any point x ∈ X and neighborhood U ∈ T X of X there exists a neighborhood U x ∈ W ◦ X of x such that U x ⊂ U . Since X is W ◦ T -separated,for any y ∈ X \ U we can find disjoint neighborhoods U x,y ∈ W ◦ X and O y ∈ T X of the points x and y ,respectively. By the compactness of X the open cover { O y : y ∈ X \ U } of the closed set X \ U ⊂ X has a finite subcover { O y : y ∈ F } . Then U x := T y ∈ F U x,y is a W ◦ X -open neighborhood of x , containedin U .(2) ⇒ (14) By [16], the compact Hausdorff semitopological semilattice is a topological semilattice. If X is weak ◦ , then by Theorem 2.12 [8] and Theorem 8.1, X is a U -semilattice.(14) ⇒ (12) If X is a U -semilattice, then X is I -separated by Theorem 8.3.The implication (12) ⇒ (8) trivially follows from the definitions.(8) ⇒ (1) If X is W -Hausdorff, then the identity map ( X, T X ) → ( X, W X ) is a homeomorphism bythe compactness of X and the Hausdorff property of W X .The implications (1) ⇒ (3) ⇒ (7) trivially follows from the inclusions of the topologies W X ⊂ W • X ⊂T X .(3) ⇒ (13) Assume that W • X = T X . To prove that X is a W -semilattice, take any open set U ⊂ X and point x ∈ U . We need to find a finite subset F ⊂ U whose upper set ↑ F is a neighborhood of x .Since the space X is compact and Hausdorff, the point x has a compact neighborhood K x ⊂ U .Since W • X = T X , each point y ∈ X \ U has an open neighborhood V y ∈ W • X which is disjoint with thecompact set K x . We can assume that V y = X \ S F y for a finite family F y of closed subsemilattices of X . By the compactness of X \ U , the open cover { V y : y ∈ X \ U } has a finite subcover { V y : y ∈ E } .Here E ⊂ X \ U is a suitable finite set.It follows that X \ U ⊂ S y ∈ E V y = S y ∈ E ( X \ S F y ) = X \ T y ∈ E S F y ⊂ X \ K x and hence K x ⊂ T y ∈ E S F y ⊂ U .Observe that T y ∈ E S F y = S K where K = { T y ∈ E F y : ( F y ) y ∈ E ∈ Q y ∈ E F y } . Each set K ∈ K \ {∅} is a non-empty compact subsemilattice of X , so K ⊂ ↑ x K where x K ∈ K is the smallest element of K (which exists by the compactness of K ). It follows that F = (cid:8) x K : K ∈ K \ {∅} (cid:9) is a finite subset of S K ⊂ U such that K x ⊂ S K ⊂ ↑ F , which means that ↑ F is a neighborhood of x .The implication (13) ⇒ (14) was proved in Theorem 8.1.(3) ⇔ (7) If W • X = T X , then the weak • topology W • X = T X is Hausdorff. If W • X is Hausdorff, then theidentity map ( X, T X ) → ( X, W • X ) is a homeomorphism by the compactness of X and hence T X = W • X .The equivalences (7) ⇔ (11) ⇔ (15) follow from Theorem 8.5. The implications (1) ⇒ (4 ,
5) trivially follow from the inclusions W X ⊂ L X ⊂ W • X ⊂ T X and W X ⊂ W ◦ X ⊂ T X . The (non-trivial) inclusions W X ⊂ L X ⊂ W • X are proved in Theorem 7.6.(5) ⇒ (14). Assume that L X ⊂ W ◦ X . To prove that X is a U -semilattice, it sufficient to show thatfor every open set U = ↑ U and point x ∈ U there exists a point y ∈ U whose upper set ↑ y contains x in its interior. By Lemma 7.1, the set U is Scott-open and hence U ∈ L X ⊂ W ◦ X . By the regularity ofthe compact Hausdorff space X , the point x has a compact neighborhood K x ⊂ U . The continuity ofthe semilattice operation of X implies that the partial order of X is closed. Now the compactness of X ensures that the upper set ↑ K x is closed. Choose an open neighborhood V x ⊂ K x of x and observe thatthe upper set ↑ V x is open in X (by the separate continuity of the semilattice operation). By Lemma 7.1,the set ↑ V x is Scott-open and hence it belongs to the topologies L X ⊂ W ◦ X . By the definition of thetopology W ◦ X , there exists an open subsemilattice V ⊂ X such that x ∈ V ⊂ ↑ V x . The closure ¯ V of V in X , being a compact subsemilattice of X , contains the smallest element y ∈ ¯ V ⊂ ↑ V x ⊂ ↑ K x ⊂ U .Since x ∈ V ⊂ ↑ y , the upper set ↑ y contains x in its interior.The implications (1) ⇒ (9) ⇒ (8) follow from Theorem 7.6 and Lemma 7.5, respectively. (cid:3) Remark 9.2.
Example 4.7 and 4.8 show that the equivalence of the conditions (1) and (2) in Theo-rem 9.1 does not hold for non-compact topological semilattices.10.
Separation axioms of the topologies W X , W • X and L X on complete semitopologicalsemilattices Some statements of Theorem 9.1 remain equivalent for complete semitopological semilattices.
Theorem 10.1.
For a complete semitopological semilattice X the following conditions are equivalent: (1) the topology W • X is Hausdorff; (2) the topology W X is Hausdorff; (3) X is I -separated; (4) X is a V -semilattice satisfying the separation axiom T ; (5) X is a T -space and W X = W • X ; (6) W X = L X = W • X ; (7) X is functionally Hausdorff and the Lawson topology L X is Hausdorff.Proof. The equivalence (2) ⇔ (3) is trivial and (2) ⇒ (1) follows from the inclusion W X ⊂ W • X .(1) ⇒ (3) Assume that the topology W • X is Hausdorff. Then X is Hausdorff and by Corollaries 3.5and Theorem 5.6, ( X, W • X ) is a Hausdorff compact semitopological semilattice. Since each W • X -closedsubsemilattice in X is T X -closed, the semitopological semilattice ( X, W • X ) is weak • and by Theorem 9.1, I -separated. Since W • X ⊂ T X , the semitopological semilattice ( X, T X ) is I -separated, too.The implication (1) ⇒ (4) follows from Theorems 8.5 and 5.6. The implication (4) ⇒ (1) is provedin Proposition 8.4.(2) ⇒ (5) Assume that the topology W X is Hausdorff. Then X is Hausdorff and hence satisfies theseparation axiom T . By Theorem 5.6, the weak • topology W • X is compact, which implies that theidentity map ( X, W • X ) → ( X, W X ) to the Hausdorff space ( X, W X ) is a homeomorphism and hence W • X = W X .(5) ⇒ (3) Assume that X is a T -space and W • X = W X . By Proposition 2.3, the weak • topology W • X satisfies the separation axiom T . Assuming that X is not I -separated, we can find two distinctpoints x, y ∈ X such that h ( x ) = h ( y ) for any continuous homomorphism h : X → I . Then the points x, y cannot be separated by W X -open sets and W X does not satisfy the separation axiom T . So, W X = W • X , which is a desired contradiction.(5) ⇒ (6) Assume that X is a T -space and W • X = W X . By the (already proved) implication(5) ⇒ (1), the topology W • X is Hausdorff and so is the topology T X of X . By Corollary 3.5, ( X, W • X )is a Hausdorff semitopological semilattice. By Theorem 5.6, the topology W • X is compact. By [16], HE INTERPLAY BETWEEN WEAK TOPOLOGIES ON TOPOLOGICAL SEMILATTICES 19 the compact Hausdorff semitopological semilattice ( X, W • X ) is a topological semilattice. ApplyingTheorem 7.6, we conclude that W X ⊂ L X ⊂ W • X . Now the equality W • X = W X implies that W X = L X = W • X .The implication (6) ⇒ (5) follows from the fact that the Lawson topology L X satisfies the separationaxiom T .(3) ⇒ (7) If X is I -separated, then X is functionally Hausdorff and the topology W X is Hausdorff.By Lemma 7.5, W X ⊂ L X and hence the Hausdorff property of W X implies the Hausdorff property ofthe Lawson topology L X .The implication (7) ⇒ (1) follows from the inclusion L X ⊂ W • X , proved in Theorem 7.6 for function-ally Hausdorff complete semitopological semilattices. (cid:3) Proposition 10.2.
If a Hausdorff complete semitopological semilattice X is weak • , then X is a weak ◦ compact topological semilattice.Proof. By Theorem 5.6, the weak • topology W • X on X is compact. If X is weak • , then X is compact andby Theorem 9.1 the compact semitopological semilattice X is weak ◦ . By [16], the compact Hausdorffsemitopological semilattice X is a topological semilattice. (cid:3) The following proposition gives a partial affirmative answer to Problem 4.9.
Proposition 10.3. If X = Q α ∈ A X α is the Tychonoff product of W • T -separated complete semitopo-logical semilattices, then the identity map ( X, W • X ) = ( X, W X ) → Y α ∈ A ( X α , W X α ) = Y α ∈ A ( X α , W • X α ) is a homeomorphism.Proof. By Theorem 10.1, the topologized semilattices X α , α ∈ A , are I -separated and so is theirTychonoff product X . Also Theorem 10.1 implies that W • X = W X and W • X α = W X α for all α ∈ A . By[4], the Hausdorff semitopological semilattice X is complete and by Theorem 5.6, the weak • topology W • X is compact and so is the I -weak topology W X = W • X . By Proposition 2.1(1), the identity mapid : ( X, W X ) → Y α ∈ A ( X α , W X α )is continuous and hence is a homeomorphism (by the compactness of the topology W X and the Hausdorffproperty of the topologies W X α , α ∈ A ). (cid:3) Acknowledgement
The authors express their sincere thanks to the referee for the very fruitful suggestion to explore therelation of the topologies W X , W • X and W ◦ X on a topologized semilattice X with the Lawson topology L X , which has a different nature and depends only on the order structure of X . References [1] T. Banakh, S. Bardyla,
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E-mail address : [email protected] S.Bardyla: Ivan Franko National University of Lviv (Ukraine)
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