Selections generating new topologies

Publ. Mat. 51 (2007), 3–15 SELECTIONS GENERATING NEW TOPOLOGIES Valentin Gutev and Artur Tomita Abstract Every (continuous) selection for the non-empty 2-point subsets of a space Xnaturally defines an interval-like topology on X. In the present paper, we demonstrate that, for a second-countable zerodimensional space X, this topology may fail to be first-countable at some (or, even any) point of X. This settles some problems stated in [7]. 1. Introduction Let Xbe a topological space, and let F(X) be the set of all nonempty closed subsets of X. Also, let D⊂F(X). A map f:D→Xis aselection for Dif f(S)∈Sfor every S∈D. A selection f:D→X is continuous if it is continuous with respect to the relative Vietoris topology τVon D. Let us recall that τVis generated by all collections of the form hVi=nS∈F(X) : S⊂[Vand S∩V6=∅,whenever V∈Vo, where Vruns over the finite families of open subsets of X. In the sequel, all spaces are assumed to be at least Hausdorff and infinite. In the present paper, we are interested in continuous selections for D, when Dis the family F2(X) = {S∈F(X) : |S| ≤ 2}. In this case, a selection f:F2(X)→Xis usually called a weak selection for X. Every weak selection ffor Xdefines an order-like relation fon X (see [10]) by letting that xfyiff f({x, y}) = x. For convenience, we write that x≺fyif xfyand x6=y. We note that the relation “f” may fail to be transitive (see, for instance, [4, Proposition 2.2]). Nevertheless, to every continuous weak selection ffor Xwe may associate a 2000 Mathematics Subject Classification. 54B20, 54C65. Key words. Hyperspace topology, Vietoris topology, continuous selection. The second author would like to thank UMALCA for the support to cover a part of the air fare expenses for his visit to UNAM, Campus Morelia, Mexico in December 2004, where the research was partially conducted. 4 V. Gutev, A. Tomita topology Tfon Xgenerated by all “open f-intervals” {y∈X:y≺fx} and {y∈X:x≺fy},x∈X. According to [10, Lemma 7.2] (see, also, [4, Lemma 3.3]), these “f-intervals” are always open in the original topology of X. Hence, Tfis a coarser topology on X, and, consequently, it is the original topology on Xprovided Xis compact. In fact, by [11, Theorem 1.1], for a compact space Xthe topology Tfcoincides with the open interval topology on Xgenerated by a linear ordering on X (i.e., Tfis an order topology on X). According to [10, Lemma 7.2], Tfis also an order topology on Xprovided Xis connected. Finally, by [12, Theorem 4 and Remark 16], Tfcoincides with the original topology on Xprovided Xis connected and locally connected. Some further properties of this topology were studied in [4], [7]. For instance, by [7, Corollary 2.3], Tfis always a regular topology on X. On the other hand, by [7, Corollary 2.4], Tfis the usual Euclidean topology on the rational numbers Q, whenever fis a continuous weak selection for Q. Hence, it become quite natural to study this topology on the irrational numbers Pwhich is an uncountable, second countable, zero-dimensional space. We are now ready to state the main purpose of this paper. Namely, in this paper, we show that every uncountable, non-compact, secondcountable, zero-dimensional space Xhas a continuous weak selection f such that Tfis not first-countable at some point of X, see Theorem 4.1. In the same theorem, we also demonstrate that Tfis not first-countable at any point of Xprovided Xhas an infinite pairwise disjoint cover consisting of uncountable open sets. Thus, in particular, there exists a continuous weak selection ffor the irrational numbers Psuch that Tfis not first-countable at any point of P(Corollary 4.2), which provides a negative answer to [7, Question 2], and a positive one to [7, Question 3]. Another interesting consequence is that an uncountable metrizable space X, with a covering dimension dim(X) = 0, is compact if and only if Tfis second-countable for every continuous weak selection fon X(see Corollary 4.4). For other applications, we refer the interested reader to Sections 4 and 5 of the paper. A preparation for the proof of Theorem 4.1 is given in Sections 2 and 3, while its proof will be finally accomplished in Section 4. A part of this preparation is based on a criterion for the existence of continuous weak selections (see Theorem 5.1), which is analogous to a result of Eilenberg on orderability [1]. It has a list of interesting independent consequences (see Section 5). Selections Generating New Topologies 5 In conclusion, the second author would like to express his best gratitude to Professor Salvador Garc´ıa-Ferreira for his support and hospitality, and for discussing some questions related to this research. 2. A relation generated by weak selections Let Xbe a set, and let E⊂X×Xbe a relation on X. As usual, we write xEy to denote that (x, y)∈E. Let us recall that a relation E on Xis anti-symmetric if xEy and yEx implies x=y. Following [7], we say that an anti-symmetric relation Eon Xis a selection relation if xEy or yEx for every x, y ∈X. Let us emphasize that, in this terminology, a relation Eon Xis a linear order on Xif Eis a selection relation which is also transitive (i.e., xEy and yEz implies xEz). It should be mentioned that the set of all possible weak selections for Xcorresponds precisely to all possible selection relations on X. Namely, any selection relation Eon Xdefines a weak selection fEby letting fE({x, y}) = xiff xEy. On the other hand, if fis a weak selection for X, then the order-like relation fgenerated by fis a selection relation. In the sequel, we will refer to fas a selection relation. In the present section, we are interested in a natural extension of such relations to the subsets of X. Following [3], for a selection relation “” and (not necessarily non-empty) subsets B, C ⊂X, we shall write that BC(respectively, B≺C) if yz(respectively, y≺z) for every y∈Band z∈C. Obviously, B≺Cimplies B∩C=∅. In these terms, we have the following simple criterion for continuity in F2(X) which is, in fact, [4, Theorem 3.1]. Proposition 2.1 ([4]).Let Xbe a space, fbe a weak selection for X, and let “f” be the selection relation generated by f. Also, let x, y ∈X be such that x≺fy. Then, fis continuous at {x, y}if and only if there are open sets Uand Vsuch that x∈U,y∈V, and U≺fV. On the other hand, we have the following property of weak selections. It was implicitly used in several papers and summarized in [6, Proposition 4.1]. Proposition 2.2. Let Xbe a space, and let fbe a weak selection. Then, fis continuous on the singletons of X. Motivated by Propositions 2.1 and 2.2, we may consider only the subset [X]2={S∈F2(X) : |S|= 2}, 6 V. Gutev, A. Tomita which will play a crucial role in this paper. In fact, we will make no difference between weak selections f:F2(X)→Xand weak selections f: [X]2→X. The following simple observation about special weak selections will be also useful. Proposition 2.3. Let Xbe a space which has a continuous weak selection, and an infinite pairwise disjoint cover Vconsisting of non-empty open subsets. Then, there exists a continuous weak selection g: [X]2→ Xsuch that Vis an unbounded well-ordered set with respect to the selection relation generated by g. Proof: Let f: [X]2→Xbe a continuous weak selection. Also, let h:δ→ Vbe a one-to-one map, where δ=|V|. Then, for every x∈X, let α(x)< δ be such that x∈h(α(x)). Finally, define g: [X]2→Xby letting for distinct points x, y ∈Xthat g({x, y}) = xif α(x)< α(y), and g({x, y}) = f({x, y}) if α(x) = α(y). Clearly, gis continuous because so is f, and Vis a discrete open cover of X. On the other hand, by the definition of g, the selection relation gdefines the same order on V as that one of the infinite cardinal δ. Hence, Vis unbounded and wellordered with respect to g. We conclude this section with some properties of the topology generated by weak selections. Suppose that fis a weak selection for X, and fis the selection relation generated by f. For every x∈X, we consider the corresponding “open f-intervals” If(x, ∞) = {y∈X:x≺fy},and If(∞, x) = {y∈X:y≺fx}. Also, for convenience, we let If(X) = {If(∞, x),If(x, ∞) : x∈X}. In these terms, the topology Tfis generated by all finite intersections of members of If(X). This is the place to recall that, in general, the relation fis not transitive. Hence, we may have points x, y, z ∈X which generate an infinite “monotone” sequence · · · ≺fx≺fy≺fz≺fx≺f··· In particular, for such points, we also have that {t∈X:x≺ft≺fy} 6=∅6={t∈X:y≺ft≺fx}. Motivated by this, for every a, b ∈Xwe will associate the set If(a, b) = If(a, ∞)∩If(∞, b) = {x∈X:a≺fx≺fb}. Selections Generating New Topologies 7 However, we don’t require that a≺fb. Hence, both f-intervals If(a, b) and If(b, a) make sense, and could be non-empty. In what follows, we shall say that a point x∈Xis an f-cutting point if there are points a, b ∈X, with x∈If(a, b). Otherwise, we shall say that xis an f-extreme point of X. Clearly, Xmay have at most two f-extreme points, which could be different for different selections f. Proposition 2.4. Let Xbe a space, fbe a weak selection for X, and let A, B ⊂Xbe non-empty subsets such that If(A, B) = \{If(a, b) : (a, b)∈A×B} 6=∅. Then, A∩B=∅. In particular, if x∈Xis an f-cutting point and U∈Tf, then x∈Uif and only if there are non-empty finite disjoint subsets A, B ⊂X, with x∈If(A, B)⊂U. Proof: The first part of this statement follows from the fact that If(z,z)= ∅for every z∈X. As for the second part, by the definition of Tf,x∈U if and only if there is a finite set K⊂If(X), with x∈TK⊂U. On the other hand, x∈If(a, b) for some a, b ∈X, because xis an f-cutting point. Let A0={y∈X:If(y, ∞)∈K}and B0={z∈X:If(∞, z)∈ K}. Then, A=A0∪ {a}and B=B0∪ {b}are as required. 3. A condition for continuity of weak selections Lemma 3.1. Let Xbe a space, f: [X]2→Xbe a selection, and let fbe the selection relation generated by f. Then, fis continuous if and only if the set L=(x, y)∈X2:x≺fyis open in X2. In particular, if fis continuous, then the map h:L→[X]2, defined by h((x, y)) = {x, y},(x, y)∈L, is a homeomorphism. Proof: Take distinct points x, y ∈Xsuch that (x, y)∈L, i.e. x≺fy. Then, by Proposition 2.1, fis continuous at {x, y}if and only if there are open sets U, V ⊂Xsuch that x∈U,y∈V, and U≺fV. According to the definition of L, this implies that fis continuous at {x, y}if and only if there are disjoint open subsets U, V ⊂Xsuch that (x, y)∈U×V⊂L. In particular, if fis continuous, then the map his a continuous open bijection which completes the proof. Lemma 3.1 suggests a natural construction of continuous weak selections. To this end, for a subset Z⊂X2, let us agree to say that π:Z→Xis a projection if π((x, y)) ∈ {x, y}for every (x, y)∈Z. Then, whenever Z⊂X2, we have always two standard continuous projections πi:Z→X,i= 0,1, defined by π0((x, y)) = xand π1((x, y)) = y, 8 V. Gutev, A. Tomita (x, y)∈Z. Here is another example of continuous projections, which will play an important role in the next section. Example 3.2. Let Xbe a space, Z⊂X2,Ube a discrete open cover of Z, and let ξ:U→2 = {0,1}be an arbitrary map. Define a map π:Z→Xby letting for (x, y)∈Zthat π((x, y)) = πξ(U)((x, y)) if (x, y)∈U∈U. Then, πis a continuous projection such that π↾U= πξ(U)↾U,U∈U. Proof: Follows from the fact that Uis a discrete open cover of Z. According to Lemma 3.1, we now have the following immediate consequence. It provides a possible way to construct continuous weak selections from given ones. Corollary 3.3. Let Xbe a space, f: [X]2→Xbe a continuous selection, and let Land h:L→[X]2be as in Lemma 3.1. Also, let π:L→Xbe a continuous projection. Then, g=π◦h−1: [X]2→Xis a continuous selection. 4. A construction of continuous weak selections Theorem 4.1. Let Xbe an uncountable, non-compact, second-countable, zero-dimensional space. Then, Xhas a continuous weak selection f such that Tfis not first-countable at some point of X. If, moreover, X has an infinite pairwise disjoint open cover consisting of uncountable sets, then Xhas a continuous weak selection fsuch that Tfis not firstcountable at any point of X. Proof: First of all, let us observe that Xis regular because it has a base of clopen sets. Hence, by the Urysohn’s metrization theorem [13] (see, also, [2]), Xis metrizable. Also, dim(X) = 0 [14] (see, also, [2]) because Xis a Lindel¨of space being regular and second-countable. In case Xhas an infinite cover consisting of pairwise disjoint uncountable open sets, we let this cover to be V. Otherwise, let us observe that Xhas an infinite cover Vconsisting of non-empty pairwise disjoint open sets such that at least three members of Vare uncountable. To this end, let Zbe the set of all points x∈Xsuch that xhas a local base consisting of uncountable open sets. Then, Zmust be closed because every neighbourhood of a point z∈Zwill contain a point of Z. In this case, X\Zmust be countable. Namely, take a countable base Ofor the topology of X, and then observe that X\Z=S{O∈O:|O| ≤ ω}. Since Xis uncountable, Zmust be also uncountable and, in particular, Selections Generating New Topologies 9 infinite. Thus, using that dim(X) = 0, we can take Vto be an infinite cover of Xconsisting of non-empty pairwise disjoint open sets, with |{V∈V:V∩Z6=∅}| ≥ 3. Having already constructed the cover V, let Bbe a countable base for the topology of Xconsisting of non-empty clopen subsets such that Bis a refinement of V. In what follows, we will use D(B) to denote the set of all non-empty subsets W⊂Bwhich are finite and pairwise disjoint. Next, for every W∈D(B), let 2Wbe the set of all maps µ:W→2 = {0,1}. Also, we let M0=[2W:W∈D(B). Finally, for every µ∈M0, we let Dom(µ) to be the domain of µ, which is clearly a non-empty finite and pairwise disjoint subset of B. Note that Xhas a continuous weak selection, because it is a subset of the Cantor set [14] (see, also, [2]) being a regular space with a countable clopen base. Hence, by Proposition 2.3, Xhas a continuous weak selection gsuch that Vis an unbounded well-ordered set with respect to the selection relation “g” generated by g. Now, let V∗={V∈V:|V|> ω}, which, by construction, has the property that |V∗| ≥ 3. Next, let V∗= mingV∗, and then take x∗∈V∗and W∗∈Bto be such that x∗∈W∗⊂V∗. Finally, define (4.1) µ∗ i:{W∗} → 2, i = 0,1,by µ∗ i(W∗) = 1 −i. Thus, we get two different elements µ∗ 0, µ∗ 1∈M0, so we let α(µ∗ 0) = mingV∗\ {V∗}and α(µ∗ 1) = mingV∗\ {V∗∪α(µ∗ 0)}. For later use, let us observe that (4.2) W∗≺gα(µ∗ 0)≺gα(µ∗ 1), while both α(µ∗ 0) and α(µ∗ 1) are uncountable. Now, we are going to extend the map α:{µ∗ 0, µ∗ 1} → Vto an injective map α:M0→Vsuch that, for every µ∈M0, (4.3) W≺gα(µ),whenever W∈Dom(µ). This can be done by transfinite induction because M0is countable, while Vis infinite, hence |M0| ≤ |V|. Namely, take a well-ordering ≪on M0 as that of the first infinite ordinal ωsuch that µ∗ 0= min≪M0and µ∗ 1= min≪M0\ {µ∗ 0}. Next, suppose that α(ν) has been already defined for every ν≪µand some µ∈M0, with µ≫µ∗ 1. Then, Vµ={α(ν) : ν≪µ} ∪ {V∈V:W⊂Vfor some W∈Dom(µ)}, 10 V. Gutev, A. Tomita is a non-empty finite subset of V, while Vis unbounded. Hence the set Vµ={V∈V: maxgVµ≺gV} is also non-empty. So, we may define α(µ) = mingVµ, which completes the construction. Now, for convenience, we let ϑ:{X} → 2 = {0,1}to be the map ϑ(X) = 0, and M=M0∪ {ϑ}. Also, we let A0=SV\α(M0), α(ϑ) = A0, and A=α(M0)∪{A0}. Thus, we get a discrete partition A of Xand a one-to-one map α:M→Asuch that (4.3) holds for every µ∈M0. Keeping in mind this, we are going to construct a discrete open partition U=Ui α(µ):µ∈Mand i= 0,1of the set L=(x, y)∈ X2:x≺gy. To this end, to every subset F⊂Xand every k∈2, we associate another subset S(k, F) defined by (4.4) S(k, F) = (X\Fif k= 0 Fif k= 1. Then, for every µ∈M, we define U0 α(µ)=L∩[S(1 −µ(W), W)×α(µ) : W∈Dom(µ),(4.5) and U1 α(µ)=L∩[S(µ(W), W)×α(µ) : W∈Dom(µ).(4.6) It is easy to observe that U0 α(µ)∪U1 α(µ)=L∩(X×α(µ)), hence Uis a partition of L. Also, Uis defined only by products of clopen sets, hence it is clopen as well. Finally, we define a map ξ:U→2 by letting ξ(U) = iif U=Ui α(µ) for some µ∈M. Thus, by Example 3.2, we get a continuous projection π:L→X, with π↾Ui α(µ)=πi↾Ui α(µ),µ∈Mand i= 0,1. Hence, by Corollary 3.3, f=π◦h−1is a continuous selection for [X]2. In what follows, let “f” be the selection relation generated by f. We are going to show that fis as required. To prepare for this, take µ∈M0,W∈Dom(µ) and (x, y)∈W×α(µ), and let us observe that (4.7) f({x, y}) = (xif µ(W) = 0, yif µ(W) = 1. Indeed, by (4.3), we have W×α(µ)⊂L. If µ(W) = 0, then, by (4.4) and (4.5), S(1 −µ(W), W)×α(µ) = S(1, W)×α(µ) = W×α(µ)⊂U0 α(µ). Selections Generating New Topologies 11 So (x, y)∈U0 α(µ), which implies that f({x, y}) = π0((x, y)) = x. If µ(W) = 1, then, in the same way, by (4.4) and (4.6), S(µ(W), W)×α(µ) = S(1, W)×α(µ) = W×α(µ)⊂U1 α(µ), and therefore f({x, y}) = π1((x, y)) = y. Now, take y, z ∈Xand non-empty finite disjoint subsets A, B ⊂X. We have the following crucial property of the topology Tf. (4.8) {y, z} ∩ (A∪B) = ∅implies If(A, B)\If(y, z)6=∅. Indeed, consider the finite set K=A∪B∪ {y, z}, and then take a pairwise disjoint family W={Wx:x∈K} ⊂ Bsuch that x∈Wx for every x∈K. Next, define µ:W→2 by letting for x∈Kthat µ(Wx) = 0 if x∈Aor x=z, and µ(Wx) = 1 otherwise. Thus, we get a particular element µof M0. Take a point e∈α(µ), and let us observe that, by the definition of µand (4.7), x≺feif x∈Aor x=z, and e≺fxif x∈Bor x=y. That is, e∈If(A, B), but e /∈If(y, z). We are finally ready to show that the selection fis as required by showing that each point of Xis an f-cutting point. Take a point x∈X, and let W∈Bbe such that x∈W. Just like in (4.1), define µi:{W}→2, i=0,1, by µi(W)=1−i. Thus, we get two different elements µ0, µ1∈M0, so α(µ0)6=α(µ1). Then, (4.9) x∈If(y0, y1),whenever (y0, y1)∈α(µ0)×α(µ1). Indeed, by (4.7), µ0(W) = 1 implies f({x, y0}) = y0because (x, y0)∈ W×α(µ0), while µ1(W) = 0 implies f({x, y1}) = xbecause (x, y1)∈ W×α(µ1). This completes the verification of (4.9). In fact, it also implies that Tfis not first-countable at x∈Xif both α(µ0) and α(µ1) are uncountable. Namely, suppose if possible that Tfis first-countable at x∈X, but α(µ0) and α(µ1) are uncountable. Then, by Proposition 2.4, there is a countable set E(x)⊂Xsuch that for every Tf-neighbourhood Uof xthere are non-empty finite disjoint subsets A, B ⊂E(x), with x∈If(A, B)⊂U. On the other hand, there are points yi∈ α(µi)\E(x), i= 0,1, because both α(µ0) and α(µ1) are uncountable. However, by (4.9), this implies that x∈If(y0, y1), while, by (4.8), it implies that If(A, B)\If(y0, y1)6=∅for every two non-empty finite disjoint subsets A, B ⊂E(x). The contradiction so obtained implies that Tfis not first-countable at x. In particular, by (4.2), it now implies that Tfis not first-countable at the point x∗selected at the beginning of this proof. Finally, if each element of Vis uncountable, then Tfwill be not first-countable at any point of X, which completes the proof.