The well-ordered (F) spaces are D-spaces 2 Xu Yuming ∗ 1 0 School of Mathematics, Shandong University, Jinan, China 2 n a J Abstract: We studied the relationships between Collins-Roscoe mecha- 5 nismandD-spaces,provedthatwell-ordered (F)spacesareD-spaces. This 2 improved the previous results written by D.Soukup and Y.Xu ] Keywords: Collins-Roscoe mechanism; well-ordered (F); D-space N AMS classification: Primary 54D15; 54D20. Secondary 54E20 G . h t a 1. Introduction m [ 1 In[1], CollinsandRoscoefirst introduced their well-known structuring mechanism, which v 1 was abstracted from the standard proof of the fact that a separable metric space is second 8 countable and has been proved to be a flexible tool for studying generalized metric spaces. 1 5 Let X be a space and for each x ∈ X, W(x) = {W(n,x) : n ∈ ω} a family of subsets of X 1. containing x. We say that X satisfies condition (G) if, 0 given open set U containing x ∈ X, there exists an open set V(x,U) containing x 2 such that, y ∈ V(x,U) implies x ∈ W(m,y) ⊂ U for some m ∈ ω. 1 : If we strengthen the condition (G) by not allowing the natural number m to vary with y, v i then we say that X satisfies condition (A), that is, for each open set U and x ∈ U, there X exists an open set V(x,U) containing x and a natural number m = m(x,U) such that r a x ∈ W(m,y) ⊂ U for all y ∈ V(x,U). If each W(n,x) is open, we say that X satisfies open (G) or open (A) respectively. If W(n + 1,x) ⊂ W(n,x) for each n ∈ ω, we say that X satisfies decreasing (G) or decreasing (A). The Collins-Roscoe mechanism has been extensively studied, and a lot of significant results have been obtained. For example, Theorem 1.1([1],[2]) The followings are equivalent for a space X: (1) X is metrizable, (2) X satisfies decreasing open (A), (3) X satisfies decreasing open (G), (4) X satisfies decreasing neighborhood (A). Theorem 1.2([2],[3]) The followings are equivalent for a space X: (1) X is stratifiable, (2) X satisfies decreasing (G) and has countable pseudo-character, (3) X satisfies decreasing (A) and has countable pseudo-character. ∗ This work is supported by NSF of Shandong Province(No.ZR2010AM019) 1 In [4], E. Van. Douwen first introduced the concept of D-space and proved that the finite product of Sorgenfrey lines is a D-space. Definition 1.5 A neighborhood assignment for a topological space (X,T) is a function φ : X → T such that x ∈ φ(x). A space X is a D-space if for each neighborhood assignment φ, there is a closed discrete subset D of X such that {φ(d) : d ∈ D } covers X. φ φ A lot of interesting work on D-spaces have been done, in particularly the connections between the generalized metric spaces and D-spaces. Borges and Wehrly proved that semi- stratifiablespacesareD-spaces[5], Buzyakova showed thatevery strongΣ-space isaD-space [6], and thus all σ-spaces are D-spaces [7], Arhangel’skii and Buzyakova obtained the inter- esting result that every space with a point countable base is a D-space [8], and so on. For more detail about the work of D-spaces, the survey paper [9] written by Gruenhage is rec- ommended. In [10], Gruenhage developed a new technique based on the earlier work of Fleissner and Stanley, and proved that any space satisfying open (G) is a D-space. This established certain connection between the Collins-Roscoe mechanism and D-spaces. Recently, Soukup and Xu proved that the well-ordered (α A), linearly semi-stratifiable space and elastic space are all D-spaces. Since these spaces are all well-ordered (F) spaces, they asked that whether well-ordered (F) spaces are D-spaces[11]. Considering that well-ordered (F) spaces are monotonically normal paracompact spaces, so this question is a weak version of an old one asked by Borges and Wehrly in [5]: Whether monotonically normal paracompact spaces are D-spaces? In the present paper, we prove that well-ordered (F) spaces are D-spaces, and thus get more insight in the relationship between the Collins-Roscoe mechanism and D-spaces.. Throughout this paper, all spaces are T , and ω is the first countable ordinal. For other 1 undefined terms we refer the reader to [2], [7] and [9]. 2. Well-ordered (F) spaces are D-spaces Recall that, a space X has a family W satisfying condition (F) if W = {W(x) : x ∈ X} where each W(x) consists of subsets of X containing x and if x belongs to open U, then there exists open V = V(x,U) containing x such that x ∈ W ⊂ U for some W ∈ W(y) whenever y ∈ V. We say X satisfies (F) if X has W satisfying (F). If, in addition, each W(x) is a chain (well-ordered) under reverse inclusion, then we say that X satisfies chain (well-ordered) (F). Further more, we say X satisfies neighborhood (F) if X has W satisfying (F), and each element of W(x) is a neighborhood of x. In [12], Stares give a characterization of the decreasing (G) as a strongly monotone normality condition from which we can decide which one of ”x ∈ V” and ”y ∈ U” holds when H(x,U)∩H(y,V) 6= ∅. For the sake of completeness, we state it in the following. Theorem 2.1[12] A space satisfies decreasing (G) iff to each point x ∈ X and open set U containing x we can assign an open set H(x,U) containing x and, for each point a ∈ 2 H(x,U) we can assign a natural number n(a,x,U) such that if a ∈ H(x,U)∩H(y,V) and n(a,x,U) ≤ n(a,y,V) then y ∈ U. For the well-ordered (F) space, we have a similar characterization as the decreasing (G) spaces in above theorem. Essentially, the idea of its proof comes from Stares’ theorem. Theorem 2.2 A space satisfies well-ordered (F) iff to each point x ∈ X and open set U containing x we can assign an open set H(x,U) containing x and, for each point a ∈ H(x,U) we can assign an ordinal number n(a,x,U) such that if a ∈ H(x,U)∩H(y,V) and n(a,x,U) ≤ n(a,y,V) then y ∈ U. Proof If X satisfies well-ordered (F), we define H(x,U) = V(x,U) for each x ∈ X and open set U containing x. Let a ∈ H(x,U), then there is some ordinal number α such that x ∈ W(α,a) ⊂ U according to the condition (F). So, we can assign n(a,x,U) = α. It is easy to check that a ∈ H(x,U)∩H(y,V) and n(a,x,U) ≤ n(a,y,V) implies y ∈ U by the well-ordered property of W(a). Conversely, foreach a ∈ X we defineW(α,a) = {a}∩{y : a ∈ H(y,V) andn(a,y,V) ≥ α for some open V}, W(a) = {W(α,a) : α ≤ α } where α = sup{n(a,y,V) : a ∈ H(y,V) for a a some open V}, and W = {W(a) : a ∈ X}. Obviously, we have a ∈ W(α,a) for each α ≤ α , a and W(β,a) ⊂ W(γ,a) whenever γ < β ≤ α . a If a ∈ H(x,U), we consider the element W(n(a,x,U),a) of W(a). By the definition of W(n(a,x,U),a), it is obvious that x ∈ W(n(a,x,U),a). Further more, we shall prove that W(n(a,x,U),a) ⊂ U. For any y ∈ W(n(a,x,U),a) there are two cases: y = a, or y 6= a. If y = a, then we have y = a ∈ H(y,U) ⊂ U. Otherwise, a ∈ H(y,V) and n(a,y,V) ≥ n(a,x,U) for some open V according to the definition of W(n(a,x,U),a). This implies that y ∈ U. The above theorem make us to have the ability to decide which one of ”x ∈ V” and ”y ∈ U” holds when H(x,U) ∩H(y,V) 6= ∅ in the well-ordered (F) spaces. Next, we will use it to prove our main theorem in this paper. Theorem 2.3 Well-ordered (F) spaces are D-spaces. Proof Let X be a well-ordered (F) space, and φ : x → φ(x) a neighborhoodassignment on X. By theorem 2.2, for each x ∈ X there is an open subset H(x,φ(x)) containing x, and for each a ∈ H(x,φ(x))thereisanordinalnumbern(a,x,φ(x))suchthat: a ∈ H(x,φ(x))∩H(y,φ(y)) and n(a,x,φ(x)) ≤ n(a,y,φ(y)) implies y ∈ φ(x). Take an element x ∈ X, then φ(x ) ⊂ X. If there is some y ∈ X \ φ(x ) such that 0 0 0 x ∈ H(y,φ(y)), then there is an ordinal number n(x ,y,φ(y)) assigned. Put C(x ) = 0 0 0 {y ∈ X \ φ(x ) : x ∈ H(y,φ(y))}, and let m(x ) = min{n(x ,y,φ(y)) : y ∈ C(x )}, then 0 0 0 0 0 there is some y ∈ C(x ) such that m(x ) = n(x ,y ,φ(y )). For each y ∈ C(x ), we have 0 0 0 0 0 0 0 n(x ,y,φ(y)) ≥ m(x ) = n(x ,y ,φ(y )). Notice that x ∈ H(y,φ(y)) ∩ H(y ,φ(y )), by 0 0 0 0 0 0 0 0 theorem 2.2 we conclude that y ∈ φ(y ). Therefore, C(x ) ⊂ φ(y ) holds. Let x = y , then 0 0 0 1 0 C(x ) ⊂ φ(x ) and for each y ∈ X \(φ(x )∪φ(x )), we have x ∈/ H(y,φ(y)). 0 1 0 1 0 If for each y ∈ X \φ(x ) we have x ∈/ H(y,φ(y)), take an element of X \φ(x ) as x . 0 0 0 1 Then, x ∈/ H(y,φ(y)) also holds for each y ∈ X \(φ(x )∪φ(x )). 0 0 1 Suppose that α ≥ 1 is an ordinal, and for each β < α we have selected the element x ∈ X such that y ∈ X \(∪{φ(x ) : δ ≤ β}) implies x ∈/ H(y,φ(y)) for each γ < β. Next, β δ γ if X \(∪{φ(x ) : δ < α}) 6= ∅, we shall take some element of X \(∪{φ(x ) : δ < α}) as x δ δ α 3 such that y ∈ X \(∪{φ(x ) : δ ≤ α}) implies x ∈/ H(y,φ(y)) for each γ < α. (∗) δ γ First, if α is a limit ordinal, we take an element of X \ (∪{φ(x ) : δ < α}) as x . For δ α each γ < α, we have γ < γ +2 < α. If y ∈ X \(∪{φ(x ) : δ ≤ α}), then y ∈ X \(∪{φ(x ) : δ δ δ ≤ γ +2}). By the inductive assumption, we claim that x ∈/ H(y,φ(y)). γ Second, if α is a success ordinal, and y ∈ X \(∪{φ(x ) : δ ≤ α−1}). By the inductive δ assumption, we have that x ∈/ H(y,φ(y)) for each γ < α−1. (∗∗) γ If y ∈ X \ (∪{φ(xδ) : δ ≤ α − 1}) implies xα−1 ∈/ H(y,φ(y)), we take an element of X \(∪{φ(x ) : δ ≤ α−1}) as x . Obviously, for each y ∈ X \(∪{φ(x ) : δ ≤ α}), we have δ α δ xα−1 ∈/ H(y,φ(y)). With the fact (∗∗), we conclude that xα satisfies the condition (∗). Otherwise, there is some y ∈ X \ (∪{φ(xδ) : δ ≤ α − 1}) such that xα−1 ∈ H(y,φ(y)) for which the corresponding ordinal number is n(xα−1,y,φ(y)). Denote C(xα−1) = {y ∈ X \ (∪{φ(xδ) : δ ≤ α −1}) : xα−1 ∈ H(y,φ(y)), and let m(xα−1) = min{n(xα−1,y,φ(y)) : y ∈ C(xα−1)}, then there is some yα ∈ C(xα−1) such that xα−1 ∈ H(yα,φ(yα)) and n(xα−1,yα,φ(yα)) = m(xα−1). If y ∈ C(xα−1), then xα−1 ∈ H(y,φ(y)) and m(xα−1) ≤ n(xα−1,y,φ(y)), and thus n(xα−1,yα,φ(yα)) ≤ n(xα−1,y,φ(y)). It follows that y ∈ φ(yα). Hence, we have that C(xα−1) ⊂ φ(yα). Let xα = yα, then for y ∈ X \ (∪{φ(xδ) : δ ≤ α}) implies that xα−1 ∈/ H(y,φ(y)). Notice the fact (∗∗), we claim that xα satisfies the condition (∗). By the transfinite induction, we can get a subset D = {x : α < κ} of X such that φ α each x satisfies the condition (∗) and ∪{φ(x ) : α < κ} = X, where Γ is an indexed set of α α ordinals. Let x ∈ X, and α be the smallest ordinal such that x ∈ φ(x ). If α is a limit ordinal, α we claim that x ∈/ H(x,φ(x)) for each β < α. In fact, suppose x ∈ H(x,φ(x)) for some β β0 β < α, then we have x ∈ X \ (∪{φ(x ) : δ ≤ β }). By the selection of x , we know 0 δ 0 β0+1 that x ∈ C(x ) ⊂ φ(x ), a contradiction with β +1 < α. If α is a successor ordinal, by β0 β0+1 0 condition (∗∗) we have that x ∈/ H(x,φ(x)) for each β < α−1. β For each β > α, we know that x ∈ X \ (∪{φ(x ) : δ ≤ α}), and thus x ∈/ φ(x ). β δ β α Therefore, we find an open neighborhood H(x,φ(x))∩φ(x ) of x such that |(H(x,φ(x))∩ α φ(x ))∩D | ≤ 2. Since X is a T -space, we conclude that D is a closed discrete subset of α φ 1 φ X, and thus X is a D-space. The converse of the above theorem is not true. There is an example of D-space which fails to have well-ordered (F). Example 2.4 Let X = R ∪ (∪{Q × {1} : n ∈ N}). The topology is defined as following: n The point of X \ R is isolated. For x ∈ R, the element of its neighborhood base is {x} ∪ (∪{([a ,x)∩Q)×{1} : n ≥ m}). x,n n Lin proved that X is k-semi-stratifiable but not normal[13]. Since the k-semi-stratifiable space is obviously semi-stratifiable, X is a D-space[11]. On the other hand, both decreasing (G) space andwell-ordered (F)space aremonotonically normal[12, 14], so X doesnot satisfy decreasing (G) and well-ordered (F). 3. Further discussion 4 In [2], Collins et. provided us an example of a monotonically normal space having W satisfying chain (F) but not metacompact, that is, the ordinal space [0,ω ). Since mono- 1 tonically normal D-space is paracompact[5], we conclude that [0,ω ) is not a D-space. This 1 fact tells us that the chain (F) spaces need not to be D-spaces, although the well-ordered (F) spaces are D-spaces. When Borges and Wehrly proved that monotonically normal D-space is paracompact in [5], they asked the question ”Whether every monotonically normal paracompact space is D-space?” Until now, it is still open. In [2], Collins et. indicated that both well-ordered (F) space and chain neighborhood (F) space are paracompact (Moody et. gave them an interest- ing name ”the unified paracompactness theorem” [15]). In fact, they are both monotonically normal paracompact spaces, so the following question is of interesting. Question 3.1 Whether every neighborhood chain (F) space is a D-space? In [16], the well-known Smirnov metrization theorem is shown, i.e., a space is metrizable iff it is paracompact and locally metrizable. 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