aa r X i v : . [ m a t h . GN ] M a y COARSER COMPACT TOPOLOGIES
VALENTIN GUTEV
Abstract.
It is introduced the concept of a quasi-king space, which is a nat-ural generalisation of a king space. In the realm of suborderable spaces, kingspaces are precisely the compact spaces, so are the quasi-king spaces. In con-trast, quasi-king spaces are more flexible in handling coarser selection topolo-gies. The main purpose of this paper is to show that a weakly orderable spaceis quasi-king if and only if all of its coarser selection topologies are compact.
1. Introduction
All spaces are Hausdorff topological spaces. For a set X , let F ( X ) = { S ⊂ X : 1 ≤ | S | ≤ } . A map σ : F ( X ) → X is a weak selection for X if σ ( S ) ∈ S for every S ∈ F ( X ).Every weak selection σ for X generates an order-like relation ≤ σ on X definedby x ≤ σ y if σ ( { x, y } ) = x [19, Definition 7.1]; and we write x < σ y if x ≤ σ y and x = y . The relation ≤ σ is similar to a linear order being both total andantisymmetric, but is not necessarily transitive. If X is a topological space, then σ is continuous if it is continuous with respect to the Vietoris topology on F ( X ).This can be expressed only in terms of ≤ σ by the property that for every x, y ∈ X with x < σ y , there are open sets U, V ⊂ X such that x ∈ U , y ∈ V and s < σ t forevery s ∈ U and t ∈ V , see [9, Theorem 3.1].In 1921, studying dominance hierarchy in chickens and other birds, ThorleifSchjelderup-Ebbe coined the term “pecking order”. Subsequently, in 1951, H.G. Landau [17] (see also [18]) used this ‘order’ to show that any finite flock ofchickens has a most dominant one, called a king . Landau’s mathematical modelwas based on Graph Theory and became known as “The King Chicken Theorem”.The pecking order is rarely linear, in fact it is equivalent to the existence ofa weak selection σ on the flock X . In this interpretation, an element q ∈ X is called a σ -king if for every x ∈ X there exists y ∈ X with x ≤ σ y ≤ σ q .Thus, Landau actually showed that each weak selection σ on a finite set X has Date : September 24, 2018.2010
Mathematics Subject Classification.
Key words and phrases.
Vietoris topology, separately continuous weak selection, coarsertopology, weakly orderable space, quasi-king space, pseudocompact space. a σ -king. Extending Landau’s result to the setting of topological spaces, Nagaoand Shakhmatov called a space X to be a king space [22] if X has a continuousweak selection, and every continuous weak selection σ for X has a σ -king. Next,they showed that every compact space with a continuous weak selection is a kingspace [22, Theorem 2.3]. In the inverse direction, Nagao and Shakhmatov showedthat each linearly ordered king space is compact ([22, Corollary 3.3]); also thateach king space which is either pseudocompact, or zero-dimensional, or locallyconnected, is compact as well ([22, Theorem 3.5]). Subsequently, answering aquestion of [22], it was shown in [8, Theorem 4.1] that each locally pseudocompactking space is also compact.On the other hand, there are simple examples of connected king spaces whichare not compact. For instance, such a space is the topological sine curve X = { (0 , } ∪ { ( t, sin 1 /t ) : 0 < t ≤ } . However, X has a coarser topology — that of the interval [0 , σ for X generates a naturaltopology T σ on X [9], called a selection topology and defined following the patternof the open interval topology, see Section 2. If X is a space and σ is continuous,then T σ is a coarser topology on X , but σ is not necessarily continuous withrespect to T σ [9] (see also [11, 13]). A weak selection σ for a space X is called properly continuous if T σ is a coarser topology on X and σ is continuous withrespect to T σ [12, Definition 4.4]. Thus, every properly continuous weak selectionis continuous, but the converse is not necessarily true. For a weak selection σ for X , we will say that a point q ∈ X is a quasi σ -king if for each x ∈ X there arefinitely many points y , . . . , y n ∈ X with x ≤ σ y ≤ σ · · · ≤ σ y n ≤ q . Finally, weshall say that X is a quasi-king space if X has a weak selection σ such that T σ isa coarser topology on X , and each such weak selection σ has a quasi σ -king. Thefollowing theorem will be proved in this paper. Theorem 1.1.
Let X be a quasi-king space with a properly continuous weak se-lection. Then each coarser selection topology on X is compact. Regarding the proper place of Theorem 1.1, let us remark that a quasi-kingspace is a relaxed version of a king space allowing dominance in several interme-diate steps. Using this, in Section 3 we give a simple direct proof that each weakselection σ on a set X , which generates a compact selection topology T σ , has aquasi σ -king (Theorem 3.4). In the same section, we also give an example of aspace X with a continuous weak selection σ which admits a quasi σ -king, buthas no σ -king (Example 3.3). On the other hand, all mentioned results for king OARSER COMPACT TOPOLOGIES 3 spaces remain valid for quasi-king spaces. Namely, in Section 4 we show that eachsuborderable quasi-king space is compact (Proposition 4.3), which is an elementin the proof of Theorem 1.1. This brings the following natural question.
Question 1.
Does there exist a quasi-king space which is not a king space?Theorem 1.1 also gives a partial solution to a problem in the theory of contin-uous weak selections. Briefly, a space is called weakly orderable if it has a coarserorderable topology, see Section 2. Ernest Michael showed that each connectedspace with a continuous weak selection is weakly orderable [19, Lemma 7.2]. Sub-sequently, Jan van Mill and Evert Wattel showed that in the realm of compactspaces, Michael’s result remains valid without connectedness, namely that eachcompact space with a continuous weak selection is (weakly) orderable [20, Theo-rem 1.1]. This led them to pose the question whether a space with a continuousweak selection is weakly orderable; the question itself became known as the weakorderability problem . In 2009, Michael Hruˇs´ak and Iv´an Mart´ınez-Ruiz gave acounterexample by constructing a separable, first countable and locally compactspace which admits a continuous weak selection but is not weakly orderable [15,Theorem 2.7]; the interested reader is also referred to [12] where the constructionwas discussed in detail. However, this counterexample is a special Isbell-Mr´owkaspace which is not normal. Thus, the weak orderability problem still remains openin the realm of normal spaces, see [12, Question 5]. Another special case of thisproblem was proposed in [12], it is based on the fact that each weakly orderablespace has a properly continuous weak selection [12, Corollary 4.5]. Namely, thefollowing question was raised in [12, Question 3], also in [6, Problem 4.31].
Question 2.
Let X be a space which has a properly continuous weak selection.Then, is it true that X is weakly orderable?An essential element in the proof of Theorem 1.1 is that in the realm of quasi-king spaces, the answer to Question 2 is in the affirmative, see Corollary 6.5.The paper is organised as follows. In the next section, we give a brief accounton various orderable-like spaces. The idea of quasi-king spaces is discussed inSection 3. In Section 4, we show a special case of Theorem 1.1 that each clopencover of a weakly orderable quasi-king space has a finite subcover (Theorem 4.1).This is used further in Section 5 to show that each coarser selection topologyon a weakly orderable quasi-king space is compact (Theorem 5.1). The proof ofTheorem 1.1 is finally accomplished in Section 6 by showing that for a quasi-kingspace, the selection topology induced by any properly continuous weak selectionis pseudocompact, hence compact as well (Theorem 6.1). VALENTIN GUTEV
2. Selection Topologies
Let σ be a weak selection for X , and ≤ σ be the order-like relation generated by σ , see the Introduction. For subsets A, B ⊂ X , we write that A ≤ σ B ( A < σ B ) if x ≤ σ y (respectively, x < σ y ) for every x ∈ A and y ∈ B . For a singleton A = { x } ,we will simply write x ≤ σ B or x < σ B instead of { x } ≤ σ B or { x } < σ B ; inthe same way, we write A ≤ σ y or A < σ y for a singleton B = { y } . Finally, wewill use the standard notation for the intervals generated by ≤ σ . For instance,( ← , x ) ≤ σ will stand for all y ∈ X with y < σ x ; ( ← , x ] ≤ σ for that of all y ∈ X with y ≤ σ x ; the ≤ σ -intervals ( x, → ) ≤ σ and [ x, → ) ≤ σ are similarly defined. However,working with such intervals should be done with caution keeping in mind that therelation ≤ σ is not necessarily transitive.Each weak selection σ for X generates a natural topology T σ on X , called a selection topology [9, 11]. It is patterned after the open interval topology by takingthe collection (cid:8) ( ← , x ) ≤ σ , ( x, → ) ≤ σ : x ∈ X (cid:9) as a subbase for T σ . Thus, T σ = T ≤ σ is the usual open interval topology, whenever ≤ σ is a linear order on X . Eachselection topology T σ is Tychonoff [16, Theorem 2.7]. On the other hand, T σ maylack several of the other strong properties of the open interval topology, see [4, 13].If σ is a continuous weak selection for a topological space ( X, T ), then T σ ⊂ T .The converse is not true, and the inclusion T σ ⊂ T does not imply continuity of σ even in the realm of compact spaces, see [1, Example 1.21], [9, Example 3.6] and[12, Example 4.3]. In particular, a continuous weak selection σ is not necessarilycontinuous with respect to T σ . Based on this, a weak selection σ for a space( X, T ) was called(i) separately continuous if T σ ⊂ T [1, 12]; and(ii) properly continuous if T σ ⊂ T and σ is continuous with respect to T σ [12].Thus, each properly continuous weak selection is continuous, and each continuousone is separately continuous, but none of these implications is reversible.In what follows, for a weak selection σ for X , we will write σ ↾ Z to denotethe restriction of σ on a subset Z ⊂ X , i.e. σ ↾ Z = σ ↾ F ( Z ). Similarly, fora topology T on X , we will use T ↾ Z for the subspace topology on Z . Thefollowing properties are evident from the definitions, and are left to the reader. Proposition 2.1.
Let σ be a weak selection for X . Then (i) T σ ↾ Z ⊂ T σ ↾ Z , whenever Z ⊂ X ;(ii) σ is separately continuous with respect to T σ ;(iii) σ is continuous with respect to T σ , whenever ≤ σ is a linear order on X . For a topology T on X , we will use the prefix “ T -” to express properties ofsubsets of X with respect to this topology. If σ is a continuous selection for aconnected space X , then ≤ σ is a linear order on X and T σ is a coarser topology OARSER COMPACT TOPOLOGIES 5 on X [19, Lemma 7.2], which gives that X is weakly orderable with respect to ≤ σ . The property remains valid for separately continuous weak selections, andwill play an important role in the paper. Proposition 2.2.
Let σ be a weak selection for X and Z ⊂ X be a T σ -connectedsubset of X . Then (i) x / ∈ Z if and only if x < σ Z or Z < σ x ;(ii) T σ ↾ Z = T σ ↾ Z is the subspace topology on Z ;(iii) ≤ σ is a linear order on Z .In particular, σ ↾ Z is a continuous weak selection for ( Z, T σ ↾ Z ) .Proof. The property in (i) is [7, Proposition 2.4], while (ii) is [7, Proposition2.5]. The property (iii) is [9, Proposition 2.2]. The second part now follows fromProposition 2.1, see also [1, Proposition 1.22]. (cid:3)
Let D be a partition of X and γ be a weak selection for D . Following theidea of lexicographical sums of linear orders, to each collection of weak selections η ∆ : F (∆) → ∆, for ∆ ∈ D , we will associate the weak selection σ for X definedby(2.1) ( σ ↾ ∆ = η ∆ , for every ∆ ∈ D ,Γ < σ ∆ , whenever Γ , ∆ ∈ D with Γ < γ ∆.We will refer to σ as the lexicographical γ -sum of η ∆ , ∆ ∈ D , or simply as the lexicographical sum , and will denote it by σ = P ( γ, ∆ ∈ D ) η ∆ . In case η ∆ = η ↾ ∆,∆ ∈ D , for some weak selection η for X , the lexicographical sum P ( γ, ∆ ∈ D ) η ∆ wasused in [7] under the name of a ( D , γ ) -clone of η . Proposition 2.3.
Let D be an open partition of a space X , γ be a weak selectionfor D , and η ∆ be a separately continuous weak selection for ∆ , for each ∆ ∈ D .Then the lexicographical γ -sum σ = P ( γ, ∆ ∈ D ) η ∆ is a separately continuous weakselection for X . Moreover, σ is continuous provided so is each η ∆ , ∆ ∈ D .Proof. Let ∆ ∈ D and x ∈ ∆. According to (2.1), we have that( ← , x ) ≤ σ = ( ← , x ) ≤ η ∆ ∪ [ Γ < γ ∆ Γ . Hence, ( ← , x ) ≤ σ is open in X because η ∆ is separately continuous and D consistsof open sets. Similarly, ( x, → ) ≤ σ is also open. Thus, σ is separately continuous.Suppose that each η ∆ , ∆ ∈ D , is continuous. To show that σ is also continuous,take p, q ∈ X with p < σ q . It now suffices to find open sets U, V ⊂ X such that p ∈ U , q ∈ V and U < σ V . To this end, let ∆ p , ∆ q ∈ D be the unique elementswith p ∈ ∆ p and q ∈ ∆ q . If ∆ p = ∆ q , then by (2.1), ∆ p < σ ∆ q and we can take U = ∆ p and V = ∆ q because D consists of open sets. If ∆ p = ∆ q = ∆, we can VALENTIN GUTEV use that σ ↾ ∆ = η ∆ is continuous to take open sets U, V ⊂ ∆ such that p ∈ U , q ∈ V and U < η ∆ V . Evidently, U < σ V . (cid:3)
3. Quasi-King Spaces
Let σ be a weak selection for X , and ≪ σ , ≪ σ ⊂ X be the binary relationsdefined for x, y ∈ X by(3.1) (cid:26) x ≪ σ y if x ≤ σ y ≤ σ y, for some y ∈ X , and x ≪ σ y if x ≤ σ y ≤ σ · · · ≤ σ y n ≤ σ y, for some y , . . . , y n ∈ X .It is evident that ≤ σ ⊂≪ σ ⊂ ≪ σ , and that ≪ σ and ≪ σ are total and reflexivebecause so is ≤ σ . Furthermore, ≪ σ is always transitive. However, in general, ≪ σ and ≪ σ are not antisymmetric, and may contain properly the relation ≤ σ .In fact, ≤ σ is equal to one of these relations precisely when ≤ σ is transitive (i.e.a linear order), which is summarised in the proposition below. Proposition 3.1.
Let σ be a weak selection for X . Then ≪ σ = ≤ σ if and only if ≪ σ = ≤ σ , which is in turn equivalent to ≤ σ being transitive.Proof. Evidently, ≪ σ = ≤ σ implies that ≪ σ = ≤ σ because ≤ σ ⊂≪ σ ⊂ ≪ σ . If ≤ σ isnot transitive, then X contains points x, y, z ∈ X with x < σ y < σ z < σ x . In thiscase, ≪ σ = ≤ σ because x < σ y ≪ σ x . (cid:3) Our interest in these binary relations is the interpretation that p ∈ X is a σ -king if x ≪ σ p for all x ∈ X ; and p is a quasi σ -king if x ≪ σ p for all x ∈ X , seethe Introduction. In other words, the σ -kings of X are the ≪ σ -maximal elementsof X , and the quasi σ -kings are the ≪ σ -maximal ones. We proceed with someexamples about the difference between σ -kings and quasi σ -ones. Example 3.2.
Let X = { a, b, c, p } consist of four points, and γ be the weakselection for X defined by a < γ b < γ c < γ a and c < γ p < γ { a, b } . Graphically, ≤ γ is represented by the diagram below, where “ < γ ”=“ ← ” and the shortest chain a ← · · · ← p of arrows illustrating the relation a ≪ γ p is emphasised. ba cp Then p is a quasi γ -king for X , but not a γ -king. On the other hand, a , b and c are γ -kings for X . (cid:3) OARSER COMPACT TOPOLOGIES 7
In case of infinite spaces, we have the following similar example where all pointsof X are quasi σ -kings for some continuous weak selection σ , but X has no σ -king. Example 3.3.
Following Example 3.2, let X = ∆ a ⊎ ∆ b ⊎ ∆ c be the topologicalsum of three copies ∆ a , ∆ b and ∆ c of the interval (0 , γ be the weakselection on open partition D = (cid:8) ∆ a , ∆ b , ∆ c (cid:9) defined by ∆ a < γ ∆ b < γ ∆ c < γ ∆ a .Take the standard selection η ( { x, y } ) = min { x, y } , x, y ∈ (0 , a , ∆ b and ∆ c . Finally, let σ be the lexicographical γ -sum ofthese selections. In other words, σ is the weak selection for X which is continuouson each of these open segments, and ∆ a < σ ∆ b < σ ∆ c < σ ∆ a . According toProposition 2.3, σ is continuous. Moreover, each element of X is a quasi σ -king,but X has no σ -king because none of the open segments contains a last elementwith respect to ≤ σ . (cid:3) Regarding the existence of quasi σ -kings, we have the following natural resultwhich is complementary to [22, Theorem 2.3]. Theorem 3.4.
Let σ be a weak selection for X such that T σ is a compact topologyon X . Then X has a quasi σ -king.Proof. For every x ∈ X , let(3.2) K x = { p ∈ X : x ≪ σ p } . Evidently, each K x is nonempty because x ∈ K x . Take x, y ∈ X with x ≤ σ y , and p ∈ K y . Then p ∈ K x because x ≤ σ y ≪ σ p implies x ≪ σ p , see (3.1). Thus,every two elements of the collection { K x : x ∈ X } are comparable by inclusion.Hence, it has the finite intersection property. Let cl T σ ( A ) = A T σ be the closureof a subset A ⊂ X in the topology T σ . Since T σ is a compact topology, we getthat T x ∈ X cl T σ ( K x ) = ∅ . Let p ∈ T x ∈ X cl T σ ( K x ). If x ≤ σ p for every x ∈ X ,then clearly p is a σ -king for X . If p < σ q for some q ∈ X , then q is a quasi σ -king for X . Indeed, for every x ∈ X there exists p x ∈ K x with p x < σ q , because p ∈ ( ← , q ) ≤ σ ∩ cl T σ ( K x ). According to (3.1) and (3.2), q ∈ K x for every x ∈ X . (cid:3) Recall that a space X is quasi-king if it has a separately continuous weakselection, and each separately continuous weak selection σ for X has a quasi σ -king. We now have the following consequence, compare with [22, Theorem 2.3]. Corollary 3.5.
Let X be a space with a separately continuous weak selection. Ifeach coarser selection topology on X is compact, then X is a quasi-king space. We conclude with some remarks.
Remark 3.6.
The proof of Theorem 3.4 does not follow from that of [22, Theorem2.3]. In fact, the author is unaware if, in the setting of Theorem 3.4, X has a σ -king. VALENTIN GUTEV
Remark 3.7.
Following the idea of Example 3.3, one can easily characterise thespaces in which each quasi σ -king is a σ -king. Namely, for a quasi-king space X ,the following are equivalent:(a) ≪ σ = ≪ σ , for each separately continuous weak selection σ for X .(b) X is the topological sum of at most three connected subsets.Here, the requirement that X is a quasi-king space is important. Indeed, thespace in Example 3.3 satisfies (b), but is not quasi-king. So, implicitly, such apartition of a quasi-king space X must be of T σ -compact sets, for each (some) sep-arately continuous weak selection σ for X , see Propositions 2.2 and 4.3. Moreover,(b) implies that each separately continuous weak selection for X is continuous (byPropositions 2.2 and 2.3), therefore such quasi-king spaces are completely identicalto kings spaces. Remark 3.8.
Let σ be a weak selection for X . Following [18], a point p ∈ X willbe called a σ -emperor if it is the ≤ σ -maximal element of X , namely if x ≤ σ p forall x ∈ X . Thus, X may have at most one σ -emperor, and each σ -emperor is a(quasi) σ -king. If X is a finite set, then X has exactly one σ -king if and only ifthat king is a σ -emperor [18, Theorem 4]. In the setting of infinite sets, this isnot necessarily true, and the property defines a special class of topological spaces.To this end, for convenience, let seℓ ( X ) be the collection of all weak selectionsfor a set X . Then for a space X with a separately continuous weak selection, thefollowing are equivalent:(a) X is T σ -compact and ≤ σ is a linear order, for each separately continuous σ ∈ seℓ ( X ).(b) Each separately continuous σ ∈ seℓ ( X ) has a σ -emperor.(c) Each separately continuous σ ∈ seℓ ( X ) has exactly one quasi σ -king.(d) Each separately continuous σ ∈ seℓ ( X ) has exactly one σ -king.(e) X is the topological sum of at most two T σ -compact sets, for each sepa-rately continuous σ ∈ seℓ ( X ).By Proposition 2.1, the first condition implies that each separately continuousweak selection for X is properly continuous.
4. Clopen Compactness
Here, we show that every weakly orderable quasi-king space is compact in thetopology generated by its clopen subsets, which furnishes an essential part in theproof of Theorem 1.1.
Theorem 4.1.
Let X be a weakly orderable quasi-king space. Then each clopencover of X has a finite subcover. OARSER COMPACT TOPOLOGIES 9
The proof of Theorem 4.1 is based on several observations about quasi-kingspaces. The next proposition shows that the following property of king spaces isalso valid for quasi-king spaces, see [22, Lemma 3.1].
Proposition 4.2. If X is a quasi-king space, then each clopen subset of X is alsoa quasi-king space.Proof. Let A ⊂ X be a clopen set, and η be a separately continuous weak selectionfor A . Since X \ A is also clopen and has a separately continuous weak selection,it follows from Proposition 2.3 that X has a separately continuous weak selection σ with σ ↾ A = η and X \ A < σ A . Then by hypothesis, X has a quasi σ -king p ∈ X . For a point x ∈ A , this means that x ≤ σ y ≤ σ · · · ≤ σ y n ≤ σ p , forsome y , . . . , y n ∈ X . However, x ∈ A and X \ A < σ A , which implies that y , . . . , y n , p ∈ A . Accordingly, p is a quasi η -king of A because σ ↾ A = η . (cid:3) Subspaces of orderable spaces are not necessarily orderable, they are called suborderable . Their topology can be also described in terms of “order”-intervals.Briefly, a subset ∆ ⊂ X of an ordered set ( X, ≤ ) is called a ≤ -interval , or a ≤ -convex set , if ( a, b ) ≤ = ( a, → ) ≤ ∩ ( ← , b ) ≤ ⊂ ∆, for every a, b ∈ ∆ with a ≤ b . Atopological space ( X, T ) is called generalised ordered if it admits a linear order ≤ , called compatible , such that the corresponding open interval topology T ≤ iscoarser than the topology T , and T has a base of ≤ -intervals. According to aresult of E. ˇCech, generalised ordered spaces are precisely the suborderable spaces,see e.g. [2, 23]. We now get with ease that each suborderable quasi-king space iscompact, see [22, Lemma 3.2 and Corollary 3.3]. Proposition 4.3.
Each suborderable quasi-king space is compact.Proof.
Let X be a quasi-king space which is suborderable with respect to a linearorder ≤ . Then η ( { x, y } ) = min ≤ { x, y } , x, y ∈ X , is a continuous weak selectionfor X with ≤ η = ≤ . Hence, X has a unique quasi η -king, which is the ≤ -maximalelement of X , see Proposition 3.1. Since X is also suborderable with respect tothe reverse linear order, it has a ≤ -minimal element as well. This implies that X is actually orderable with respect to ≤ . Indeed, let E and D be nonempty clopensubsets of X such that E < D and X = E ∪ D . By Proposition 4.2, both E and D are quasi-king spaces. Hence, by what has been shown above, E has a maximalelement and D has a minimal one. Thus, the pair ( E, D ) is a clopen jump and,consequently, X is orderable with respect to ≤ , see e.g. [5, Lemma 6.4]. This alsoimplies that X must be compact. Namely, each nonempty clopen set A ⊂ X isboth a quasi-king space (by Proposition 4.2) and suborderable with respect to ≤ .So, by the same token, it has maximal and minimal elements. Therefore, X iscompact [14], see also [5, Proposition 6.1]. (cid:3) Corollary 4.4.
Let X be a quasi-king space which is weakly orderable with respectto a linear order ≤ . Then the open interval topology T ≤ is a coarser compacttopology on X .Proof. The topology T ≤ is a coarser topology on X , and, in particular, eachseparately continuous weak selection for ( X, T ≤ ) is a separately continuous weakselection for X . Therefore, the orderable space ( X, T ≤ ) is also quasi-king. Hence,by Proposition 4.3, ( X, T ≤ ) is compact. (cid:3) We are now ready for the proof of Theorem 4.1.
Proof of Theorem 4.1.
Let X be a weakly orderable space with respect to a linearorder ≤ . According to Corollary 4.4, it suffices to show that each clopen subsetof X is open in ( X, T ≤ ). So, let A ⊂ X be clopen in X . Then A is quasi-king (byProposition 4.2) and suborderable in the subspace topology T ≤ ↾ A . In fact, A isa quasi-king space with respect T ≤ ↾ A because T ≤ ↾ A is a coarser topology on A and the weak selection min ≤ { x, y } , x, y ∈ A , is continuous with respect to thistopology (by Proposition 2.1). Thus, by Proposition 4.3, A is a compact subsetof ( X, T ≤ ). For the same reason, so is X \ A . Therefore, A = X \ ( X \ A ) is openin ( X, T ≤ ). (cid:3)
5. Coarser Compact Selection Topologies
Here, we prove the following special case of Theorem 1.1.
Theorem 5.1.
Let X be a weakly orderable quasi-king space, and σ be a separatelycontinuous weak selection for X . Then T σ is a compact coarser topology on X . The proof of Theorem 5.1 is based on properties of components relative toselection topologies. The components (called also connected components ) are themaximal connected subsets of a space X . They form a closed partition C [ X ]of X , and each point x ∈ X is contained in a unique component C [ x ] calledthe component of x in X . The quasi-component Q [ x ] of a point x ∈ X is theintersection of all clopen subsets of X containing x . The quasi-components alsoform a partition Q [ X ] of X , thus they are simply called quasi-components of X .Each component of a point is contained in the quasi-component of that point,but the converse is not necessarily true. However, if X has a continuous weakselection, then C [ x ] = Q [ x ] for every x ∈ X [10, Theorem 4.1]. The propertyremains valid for the components of selection topologies. Proposition 5.2.
Let σ be a weak selection for X . Then each quasi-componentof ( X, T σ ) is connected.Proof. By Proposition 2.1, σ is a separately continuous weak selection for ( X, T σ ).Then the property follows from [7, Corollary 2.3]. (cid:3) OARSER COMPACT TOPOLOGIES 11
Regarding Proposition 5.2, let us explicitly remark that if C ⊂ X is a componentof a space X and σ is a separately continuous weak selection for X , then C is alsoa connected subset of ( X, T σ ). However, C is not necessarily a T σ -component,namely a component of the space ( X, T σ ). Keeping this in mind, we have thefollowing construction of clopen sets associated to T σ -components. Proposition 5.3.
Let η be a weak selection for X , and Z ⊂ X be a T η -componentof X which has no ≤ η -maximal element. Then Z is contained in a T η -clopen set Y ⊂ X with Y \ Z < η Z .Proof. The set Y = S z ∈ Z ( ← , z ) ≤ η is T η -open. Moreover, Z ⊂ Y because Z hasno last element with respect to ≤ η . If y ∈ X \ Z and y ≤ η z for some z ∈ Z ,then y < η Z because Z is T η -connected, see Proposition 2.2. This implies that Y \ Z < η Z . It also implies that Y = ( ← , x ] ≤ η ∪ Z for some (any) point x ∈ Z .Since both ( ← , x ] ≤ η and Z are T η -closed, so is Y . (cid:3) We now have the following crucial property of selection topologies.
Lemma 5.4.
Let σ be a weak selection for X such that ( X, T σ ) is a quasi-kingspace. Then each T σ -component is T σ -compact.Proof. Take a non-degenerate T σ -component Z ⊂ X . Then by Proposition 2.2,( Z, T σ ↾ Z ) is orderable with respect to ≤ σ being a connected space. Hence, toshow that Z is T σ -compact, it now suffices to show that it has both ≤ σ -minimaland ≤ σ -maximal elements. To this end, we will use that σ determines a unique‘complementary’ selection σ ∗ : F ( X ) → X , defined by S = (cid:8) σ ( S ) , σ ∗ ( S ) (cid:9) , S ∈ F ( X ). The important property of σ ∗ is that T σ ∗ = T σ because ≤ σ ∗ isreverse to ≤ σ . Thus, given a weak selection η for X with T η = T σ , it suffices toshow that Z has a ≤ η -maximal element. To see this, assume the contrary that X has a weak selection η with T η = T σ , but Z has no ≤ η -maximal element.Then by Proposition 5.3, Z is contained in a T η -clopen set Y with Y \ Z < η Z .Using that T η = T σ , it follows from Proposition 4.2 that ( Y, T η ↾ Y ) is also aquasi-king space. Moreover, γ = η ↾ Y is a separately continuous weak selectionfor ( Y, T η ↾ Y ), hence Y has a quasi γ -king q ∈ Y . Since Y \ Z < γ Z and Z hasno ≤ γ -maximal element, q < γ x for some x ∈ Z . For the same reason, q < γ y ,for every y ∈ Y with x ≤ γ y , because ≤ γ is a linear order on Z . Accordingly, q cannot be a quasi γ -king for Y . A contradiction. (cid:3) Finally, we also have that each T σ -component has a base of T σ -clopen sets. Proposition 5.5.
Let σ be a weak selection for X such that ( X, T σ ) is a quasi-king space. Then each T σ -component has a base of clopen sets in ( X, T σ ) .Proof. A space is rim-finite if it has a base of open sets whose boundaries arefinite. Evidently, ( X, T σ ) is rim-finite. Take a T σ -component Z of X , and a T σ -open set V ⊂ X with Z ⊂ V . Since Z is T σ -compact (by Lemma 5.4) and ( X, T σ ) is rim-finite, there exists W ∈ T σ such that Z ⊂ W ⊂ V and the boundary of W is finite. However, by Proposition 5.2, Z is also a quasi-component of ( X, T σ ).Hence, there exists a T σ -clopen set U ⊂ X with Z ⊂ U ⊂ W ⊂ V . (cid:3) Proof of Theorem 5.1.
Let σ be a separately continuous weak selection for X .Take an open cover U ⊂ T σ of X , and let U F be the cover of X consisting ofall finite unions of elements of U . According to Lemma 5.4 and Proposition 5.5, U F has a clopen refinement V . Then by Theorem 4.1, V has a finite subcover.This implies that U has a finite subcover as well. (cid:3)
6. Coarser Pseudocompact Topologies
For simplicity, we shall say that a space ( X, T ) is selection-orderable if it has acontinuous weak selection ϕ with T = T ϕ . The main idea behind this conventionis that for a space X with a properly continuous weak selection ϕ , the space( X, T ϕ ) is selection-orderable.We now finalise the proof of Theorem 1.1 by showing the following general resultinvolving implicitly pseudocompactness. Theorem 6.1.
Each selection-orderable quasi-king space is compact.
To prepare for the proof of Theorem 6.1, we first extend the following propertyof king spaces to the case of quasi-king spaces, see [22, Lemma 3.4].
Proposition 6.2.
Let X be a space with a continuous weak selection. If X admitsan infinite open partition, then X is not a quasi-king space.Proof. Let U be an infinite open partition of X , and ≤ be a linear order on U such that U has no last ≤ -element. Take a weak selection γ for U with ≤ γ = ≤ .Also, for every U ∈ U , take a continuous weak selection η U for U . Finally, let σ bethe lexicographical γ -sum of these selections. By Proposition 2.3, σ is continuous.Moreover, σ induces the same linear order on U as that of γ , see (2.1). Thisimplies that σ has no quasi σ -king. Indeed, let q ∈ V for some V ∈ U . Next,using that U has no last ≤ σ -element, take any U ∈ U with V < σ U . If x ∈ U and x ≤ σ y , then y has the same property as x in the sense that y ∈ W for some W ∈ U with V < σ W . Hence, for any finite number of points y , . . . , y n ∈ X with x ≤ σ y ≤ σ · · · ≤ σ y n , we have that q < y k for all k ≤ n . (cid:3) Let C [ X ] = (cid:8) C [ x ] : x ∈ X (cid:9) be the decomposition space determined by thecomponents of X . Recall that a subset U ⊂ C [ X ] is open in C [ X ] if S U isopen in X . Alternatively, C [ X ] is the quotient space obtained by the equivalencerelation x ∼ y iff C [ x ] = C [ y ]. Since the elements of C [ X ] are closed sets, thedecomposition space C [ X ] is a T -space. The following property of the decompo-sition space was essentially established in [8, Corollary 3.7]. OARSER COMPACT TOPOLOGIES 13
Proposition 6.3.
Let X be a quasi-king space, and ϕ be a continuous weak se-lection for X such that T ϕ is the topology of X . Then for every x, y ∈ X with C [ x ] ∩ C [ y ] = ∅ and x < ϕ y , there are clopen sets U, V ⊂ X such that C [ x ] ⊂ U , C [ y ] ⊂ V and U < ϕ V .Proof. Since x < ϕ y , by Proposition 2.2, we get that C [ x ] < ϕ C [ y ]. Then theexistence of such clopen sets U, V ⊂ X follows by applying Lemma 5.4 and thecondition that T ϕ is the topology of X . Namely, by Proposition 5.5, it suffices toconstruct open sets U, V ⊂ X with C [ x ] ⊂ U , C [ y ] ⊂ V and U < ϕ V . Since C [ y ]is compact (by Lemma 5.4) and ϕ is continuous, for each z ∈ C [ x ] there are opensets U z , V z ⊂ X such that z ∈ U z , C [ y ] ⊂ V z and U z < ϕ V z . Finally, since C [ x ]is also compact, there exists a finite set S ⊂ C [ x ] with C [ x ] ⊂ S z ∈ S U z . Then U = S z ∈ S U z and V = T z ∈ S V z are as required. (cid:3) The crucial final step in the preparation for the proof of Theorem 6.1 is thefollowing result.
Lemma 6.4.
Let X be a selection-orderable quasi-king space. Then the decompo-sition space C [ X ] is a zero-dimensional sequentially compact space.Proof. In this proof, we first show that C [ X ] has a continuous weak selection(following [8, Theorem 3.1]), and next that it is pseudocompact (following [22,Theorem 3.5]). To this end, let ϕ be a continuous weak selection for X such that T ϕ is the topology of X . By Proposition 5.5, each element of C [ X ] has a base ofclopen sets. Hence, the decomposition space C [ X ] is zero-dimensional. Moreover,for every x, y ∈ X with C [ x ] ∩ C [ y ] = ∅ and x < ϕ y , just as in the previous proof,we have that C [ x ] < ϕ C [ y ]. Therefore, this defines a weak selection C [ ϕ ] for C [ X ] such that C [ x ] < C [ ϕ ] C [ y ], whenever x < ϕ y with C [ x ] ∩ C [ y ] = ∅ . Finally,according to Proposition 6.3, the selection C [ ϕ ] is continuous.To show that X is pseudocompact, take a discrete family { V n : n ∈ N } ofnonempty open sets V n ⊂ C [ X ]. Since C [ X ] is zero-dimensional, each V n , n ∈ N ,contains a nonempty clopen subset U n ⊂ C [ X ]. Then each U n = S U n , n ∈ N ,is a nonempty clopen subset of X , and the family { U n : n ∈ N } is discrete in X . Hence, U = X \ S n ∈ N U n is also a clopen subset of X , and { U n : n < ω } is an infinite pairwise disjoint open cover of X . According to Proposition 6.2,this is impossible because X is a quasi-king space. Thus, every discrete familyof open subsets of C [ X ] is finite. Since C [ X ] is a Tychonoff space (being zero-dimensional), this implies that it is pseudocompact.Having already established this, we can use each pseudocompact space with acontinuous weak selection is sequentially compact [1, 3, 21], see also [6, Corollary3.9]. Accordingly, C [ X ] is sequentially compact. (cid:3) We now have also the proof of Theorem 6.1.
Proof of Theorem 6.1.
Each pseudocompact space X with a continuous weak se-lection is suborderable, see [1, 3, 21]; also [6, Theorems 3.7 and 3.8]. Moreover, byProposition 4.3, each suborderable quasi-king space is compact. Hence, it sufficesto show that X is pseudocompact. In this, we follow the proof of [8, Theorem4.1]. Namely, assume to the contrary that X is not pseudocompact. Then ithas a continuous unbounded function g : X → [0 , + ∞ ). Take a point x ∈ X with g ( x ) ≥
1, and let K = C [ x ] be the component of x . Since X is aselection-orderable quasi-king space, by Lemma 5.4, K is compact, and conse-quently g ↾ K is bounded. Hence, there exists a point x ∈ X \ K with g ( x ) ≥ K = C [ x ] and extend the arguments by induction. Thus, there exists a pair-wise disjoint sequence { K n : n ∈ N } of components of X and points x n ∈ K n with g ( x n ) ≥ n , for every n ∈ N . We claim that the sequence { K n : n ∈ N } is discretein X . Indeed, suppose that y ∈ S n ≥ k K n \ S n ≥ k K n for some k ∈ N . Since C [ y ] iscompact, g ↾ C [ y ] is bounded, and so is g ↾ U for some neighbourhood U of C [ y ].By Proposition 5.5, this implies that g ↾ H is bounded for some clopen subset H ⊂ X with C [ y ] ⊂ H ⊂ U . However, y ∈ H ∩ S n ≥ k K n and, therefore, H meetsinfinitely many terms of the sequence { K n : n ≥ k } . In fact, being a clopen set, H must contain infinitely many terms of this sequence because K n ⊂ H , whenever H ∩ K n = ∅ . Hence, g ↾ H must be also unbounded because g ( x n ) ≥ n for every n ∈ N . A contradiction. Thus, { K n : n ∈ N } is discrete.We complete the proof as follows. Since { K n : n ∈ N } ⊂ C [ X ] is discrete in X ,by Proposition 5.5, it defines a discrete sequence of elements in the decompositionspace C [ X ]. However, this is impossible because, by Lemma 6.4, the decomposi-tion space C [ X ] is sequentially compact. We have duly arrived at a contradiction,showing that X must be pseudocompact. (cid:3) The proof of Theorem 1.1 now follows from Theorem 5.1 and the followingconsequence of Theorem 6.1.
Corollary 6.5.
Let X be a quasi-king space with a properly continuous weakselection. Then X is weakly orderable.Proof. Let ϕ be a properly continuous weak selection for X . Then ϕ is continuouswith respect to its selection topology T ϕ . Moreover, T ϕ is a coarser topology on X .Hence, ( X, T ϕ ) remains a quasi-king space. Thus, ( X, T ϕ ) is a selection-orderablequasi-king space and by Theorem 6.1, it is compact. Finally, by a result of vanMill and Wattel [20, Theorem 1.1], ( X, T ϕ ) is an orderable space. Therefore, X is weakly orderable. (cid:3) References [1] G. Artico, U. Marconi, J. Pelant, L. Rotter, and M. Tkachenko,
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