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A note on D-property, monotone monolithicity and σ-product
Topology and its Applications 161 (2014) 17–25
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A note on D-property, monotone monolithicity and σ-product ✩ Liang-Xue Peng College of Applied Science, Beijing University of Technology, Beijing 100124, China
a r t i c l e
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Article history: Received 31 October 2012 Received in revised form 23 July 2013 Accepted 18 September 2013 MSC: 54D20 54C35 54B10 Keywords: D-space Monolithic Monotonically monolithic σ-product Cp (X)
a b s t r a c t In this note we firstly show that if X is a σ-product of Lindelöf Σ-spaces, then Cp (X) is a semi-monotonically monolithic space and hence it is monolithic. In the second partof this note we show that if {Xα : α ∈ Λ} is a family of topological spaces such that in Xαi is a D-space for each n ∈ N and αi ∈ Λ for each i n, then any σ-product of {Xα : α ∈ Λ} is a D-space. This generalizes a conclusion of A.V. Arhangel’skii. By this conclusion and a known conclusion we have that any σ-product of regular weak θ-refinable (or (sub)metacompact) C-scattered spaces is a D-space. We also show that if Y is a dense subspace of a topological space X with |X \ Y | ω such that Cp (Y ) is monotonically monolithic, then Cp (X) is monotonically monolithic. In the last part of this note we show that if X has a k-in-countable weak base for some k ∈ N then X is a D-space. © 2013 Elsevier B.V. All rights reserved.
1. Introduction The notion of a D-space was introduced by van Douwen and Pfeffer in [9]. A neighborhood assignment for a space X is a function φ from X to the topology of the space X such that x ∈ φ(x) for any x ∈ X. A space X is called a D-space if for any neighborhood assignment φ for X there exists a closed discrete subspace D of X such that X = {φ(d): d ∈ D} [9]. All compact T1 -spaces and all metric spaces are D-spaces. Many classes of spaces are known to be D-spaces [11]. For example, every space with a point-countable base is a D-space [3] and every space with a point-countable weak base is a D-space [6,15]. R.Z. Buzyakova [8] proved that a space Cp (X) is hereditarily D when X is compact. In [20], V.V. Tkachuk introduced a notion of monotone monolithicity and proved that every monotonically monolithic space is a D-space. Some properties of monotonically monolithic spaces are also discussed in [12] and [21]. In [16], it was proved that Cp (Cp (X)) is monotonically monolithic if and only if X is monotonically monolithic. It was also proved that if X is the product of Lindelöf Σ-spaces then Cp (X) is monotonically monolithic [16]. ✩
Research supported by the National Natural Science Foundation of China (Grant No. 11271036). E-mail address: [email protected].
0166-8641/$ – see front matter © 2013 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.topol.2013.09.013
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Let X = {Xα : α ∈ A} be a topological product of the family {Xα : α ∈ A}, let x∗ = (x∗α : α ∈ A) be a point in X, and κ be a cardinal number. The Σκ -product of the family {Xα : α ∈ A} based in x∗ is the subspace Σκ (X, x∗ ) = {x ∈ X: |{α ∈ A: xα = x∗α }| < κ} of X. When κ = ω, Σκ (X, x∗ ) is called σ-product and it is usually denoted by σ(X, x∗ ). The following question appears in [16, Question 4.9]. Let X be a Σκ -product of Lindelöf Σ-spaces. Is Cp (X) then monotonically monolithic? In the first part of this note, we show that if X is a σ-product of Lindelöf Σ-spaces, then Cp (X) is semi-monotonically monolithic. Recall that a space X is monolithic if, for every infinite subset A of X, the network weight of the closure of the set A does not exceed the cardinality of A. A space X is called semi-monotonically monolithic if one can assign to each A ⊂ X a collection N (A) of subsets of X such that (1) |N (A)| max{|A|, ω}; (2) if A ⊂ B ⊂ X, then for each C1 ∈ N (A) there is some C2 ∈ N (B) such that C1 ⊂ C2 ; (3) if U is open in X and x ∈ A ∩ U , then there is some C ∈ N (A) with x ∈ C ⊂ U . Every monotonically monolithic space is semi-monotonically monolithic and every semi-monotonically monolithic space is monolithic. In the second part of this note, we show that if {Xα : α ∈ Λ} is a family of topological spaces such that in Xαi is a D-space for each n ∈ N and αi ∈ Λ for each i n, then any σ-product of {Xα : α ∈ Λ} is a D-space. This generalizes some conclusions of A.V. Arhangel’skii [2, Theorems 3.9, 3.10, and 3.12]. By this conclusion and a known conclusion [14, Theorem 18] we have that any σ-product of regular weak θ-refinable (or (sub)metacompact) C-scattered spaces is a D-space. We also show that if Y is a dense subspace of a topological space X with |X \ Y | ω such that Cp (Y ) is monotonically monolithic, then Cp (X) is monotonically monolithic. In the last part of this note we show that if X has a k-in-countable weak base for some k ∈ N then X is a D-space. Recall that l(X) is defined as the smallest infinite cardinal κ such that every open cover of X has a subcollection of cardinality κ which covers X. e(X) = sup{|D|: D ⊂ X, D is closed discrete} + ω. A condensation is a bijective continuous onto map. iw(X) denotes the minimal weight of all spaces onto which X can be condensed. All the spaces in this note are assumed to be T1 -spaces. The set of all positive integers is denoted by N and ω is N ∪ {0}. In notation and terminology we will follow [7] and [10]. 2. Main results For a space X, we denote Cp (X) the set of real-valued continuous functions, having X as its domain, with its pointwise convergence topology. Let B(R) be a countable base of R with the usual topology, where R is the set of all real numbers. We know that Cp (X) has a base which contains sets of the form [x1 , x2 , . . . , xn , U1 , U2 , . . . , Un ] = {f ∈ Cp (X): f (xi ) ∈ Ui for i n}, where n ∈ N, xi ∈ X, Ui ∈ B(R) for each i n. For subsets E1 , E2 , . . . , En of a space X and subsets U1 , U2 , . . . , Un of R, denote [E1 , E2 , . . . , En , U1 , U2 , . . . , Un ] = {f ∈ Cp (X): f (Ei ) ⊂ Ui for i = 1, 2, . . . , n}, where n ∈ N. If X = α∈Λ Xα is the product of a family {Xα : α ∈ Λ} of topological spaces and B ⊂ A, then we denote πB : X → α∈B Xα the natural projection. Recall that a collection F of subsets of a space X is a network at x ∈ X if, given any open neighborhood U of x, there is some F ∈ F with x ∈ F ⊂ U . A space X is monotonically monolithic [20] if one can assign to each A ⊂ X a collection N (A) of subsets of X such that (1) |N (A)| max{|A|, ω}; (2) A ⊂ B ⊂ X ⇒ N (A) ⊂ N (B);
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(3) if {Aα : α < δ} is an increasing collection of subsets of X, and A = {Aα : α < δ}, then N (A) = {N (Aα ): α < δ}; (4) if U is open in X and x ∈ A ∩ U , then there is some N ∈ N (A) with x ∈ N ⊂ U . We call N a monotonically monolithic operator for X and N (A) is called an external network of A. A space X is called weakly monotonically monolithic [13] if it has an operator satisfying the above conditions but with conditions (4) replaced by (4 ) if A is not closed, then there is some x ∈ A \ A such that N (A) contains a network at x. Recall that a family F of subsets of a space X is a network modulo a cover C if for each C ∈ C and an open set U of X with C ⊂ U there exists F ∈ F such that C ⊂ F ⊂ U . A space X is a Lindelöf Σ-space if there exists a countable family F of subsets of X such that F is a network modulo a compact cover C of the space X. In [20] it was proved that for every Lindelöf Σ-space X, Cp (X) is monotonically monolithic and every monotonically monolithic space is a D-space. Proposition 1. ([1, Theorem II 6.23]) Let X = α<κ Xα be a topological product, T = σ(X, x∗ ) and Y is a subspace which contains T , and λ an infinite cardinal. Assume that for each finite set K ⊂ κ, l( α∈K Xα ) λ. If f : Y → Z is a continuous function and iw(Z) λ, then there is a set S ⊂ κ with |S| λ such that if x, x ∈ Y and πS (x) = πS (x ) then f (x) = f (x ). Proposition 2. ([10, Theorem 3.3.20]) Let n ∈ N. If Ai is compact subset of a topological space Xi for each i n and A = in Ai is contained in an open set U of X = in Xi , then there is an open set Vi of Xi for each i n such that A ⊂ in Vi ⊂ U . Theorem 3. Let X = α∈κ Xα be a topological product and let Xα be a Lindelöf Σ-space for each α ∈ κ. If Y = σ(X, a∗ ), then Cp (Y ) is a semi-monotonically monolithic space. Proof. Let a∗ = (a∗α : α ∈ κ) ∈ X and let Y = σ(X, a∗ ). For each A ⊂ Cp (Y ), we will define an external network N (A) of A in Cp (Y ) such that the operator N satisfies the conditions (1) and (2) which appear in the definition of a semi-monotonically monolithic space. For each α ∈ κ, the space Xα is a Lindelöf Σ-space. Let Fα be a countable family of subsets of Xα such that Fα is a network modulo a compact cover Cα of the space Xα . Since a finite product of Lindelöf Σ-spaces is Lindelöf and iw(R) = ω, we have that for each f ∈ Cp (Y ) there is a countable set Sf ⊂ κ such that if x, x ∈ Y and πSf (x) = πSf (x ) then f (x) = f (x ). By Proposition 1, for A ⊂ Cp (Y ), let S(A) = {Sf : f ∈ A}. Thus |S(A)| max{|A|, ω}. Claim 1. If g ∈ A and πS(A) (x) = πS(A) (x ) for x, x ∈ Y , then g(x) = g(y). Proof of Claim 1. Suppose g(x) = g(x ), then there are disjoint open sets U, V of R such that g(x) ∈ U and g(x ) ∈ V . If O = {g ∈ Cp (Y ): g (x) ∈ U and g (x ) ∈ V }, then the set O is an open neighborhood of g in Cp (Y ). So O ∩ A = ∅. If f ∈ O ∩ A, then f (x) ∈ U and f (x ) ∈ V . Thus f (x) = f (x ). Since f ∈ A, we have Sf ⊂ S(A). Since πS(A) (x) = πS(A) (x ), we have πSf (x) = πSf (x ). Thus f (x) = f (x ). This is a contradiction with f (x) = f (x ). We have finished the proof of Claim 1. 2 If y = (yα : α ∈ κ) ∈ Y , then |{α ∈ κ: yα = a∗α }| < ω. We denote Ky = {α ∈ κ: yα = a∗α }. Thus the set Ky is a finite subset of κ. Let A ⊂ Cp (Y ). If S ⊂ S(A) and, for each α ∈ S, Fα ∈ Fα , then we define −1 ∗ ∗ ∗ ∗ ∗ α∈S (πα ) (Fα ) = β∈κ Fβ , where Fβ = Fα if β = α and α ∈ S, otherwise Fβ = {aβ }.
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Claim 2. Let A ⊂ Cp (Y ) and let g ∈ A. If y ∈ Y such that Ky ∩ S(A) = ∅ and g(y) ∈ U for some U ∈ B(R), then there are some y ∈ Y and some n ∈ N such that πS(A) (y) = πS(A) (y ), g(y) = g(y ), y ∈ B, and g(B) ⊂ U , where B = α∈Ky ∩S(A) (πα−1 )∗ (Fα ) \ {fj−1 (Vj ): j n}, Fα ∈ Fα for each α ∈ Ky ∩ S(A) and fj ∈ A, Vj ∈ B(R) for each j n. Proof of Claim 2. Let y ∈ Y such that yα = a∗α if α ∈ Ky \ S(A), and yα = yα if α ∈ κ \ (Ky \ S(A)). Thus πS(A) (y) = πS(A) (y ). So g(y) = g(y ) by Claim 1. Thus g(y ) ∈ U . Hence y ∈ g −1 (U ). For each α ∈ Ky ∩ S(A), there is some Cα ∈ Cα such that yα ∈ Cα . Let P = α∈κ Pα , where Pα = Cα if α ∈ Ky ∩ S(A), otherwise Pα = {a∗α }. So y ∈ P ∩ g −1 (U ). If x ∈ P \ g −1 (U ), then g(x) = g(y ). Let Vx ∈ B(R) such that g(x) ∈ Vx and g(y ) ∈ / Vx . Thus [x, y , Vx , (R \ Vx ) ∩ U ] is an open neighborhood of g in Cp (Y ). Since g ∈ A, there is some fx ∈ [x, y , Vx , (R \ Vx ) ∩ U ] ∩ A. So fx (x) ∈ Vx and fx (y ) ∈ (R \ Vx ) ∩ U . Thus fx−1 (Vx ) is open in Y and y ∈ / fx−1 (Vx ). Thus −1 y ∈ P \ fx (Vx ). The set P is compact and P = ( {P ∩ fx−1 (Vx ): x ∈ P \ g −1 (U )}) ∪ (P ∩ g −1 (U )). There is some n ∈ N and xi ∈ P \ g −1 (U ) for each i n such that P = ( {P ∩ fx−1 (Vxi ): i n}) ∪ (P ∩ g −1 (U )). i Thus πS(A)∩Ky (P ) = α∈S(A)∩Ky Cα ⊂ ( {πS(A)∩Ky (fx−1 (Vxi )): i n}) ∪ (πS(A)∩Ky (g −1 (U ))). Since i −1 the map πS(A)∩Ky |Y is open, the family {πS(A)∩Ky (fxi (Vxi )): i n} ∪ {πS(A)∩Ky (g −1 (U ))} is an open family of α∈Ky ∩S(A) Xα and covers the set α∈Ky ∩S(A) Cα . The set α∈Ky ∩S(A) Cα is compact, thus by Proposition 2 there is an open set Vα of Xα for each α ∈ Ky ∩ S(A) such that α∈Ky ∩S(A) Cα ⊂ −1 −1 (U )). So there is some Fα ∈ Fα such α∈Ky ∩S(A) Vα ⊂ ( {πS(A)∩Ky (fxi (Vxi )): i n}) ∪ πS(A)∩Ky (g that Cα ⊂ Fα ⊂ Vα for each α ∈ Ky ∩ S(A). Thus P ⊂ F ⊂ ( {fx−1 (Vxi ): i n}) ∪ (g −1 (U )) and i −1 −1 ∗ y ∈ / fxi (Vxi ) for each i n, where F = α∈Ky ∩S(A) (πα ) (Fα ). So y ∈ F \ {fx−1 (Vxi ): i n} ⊂ i g −1 (U ). Thus g(y ) ∈ g(F \ {fx−1 (V ): i n}) ⊂ U . If f = f and V = V for each i n, then xi xi i i i i −1 x−1 −1 ∗ g(y ) ∈ g(F \ {fi (Vi ): i n}). Denote B = α∈Ky ∩S(A) (πα ) (Fα ) \ {fi (Vi ): i n}. Thus y ∈ B, g(y) = g(y ), and g(y) ∈ g(B) ⊂ U . We have finished the proof of Claim 2. 2 Let A ⊂ Cp (Y ) and g ∈ A. Let y ∈ Y and let U ∈ B(R) such that g(y) ∈ U . The set Ky is a finite subset of κ. For each α ∈ Ky , there is some Cα ∈ Cα such that yα ∈ Cα . Thus α∈Ky Cα is homeomorphic to α∈κ Pα , where Pα = Cα if α ∈ Ky , otherwise Pα = {a∗α }. Since g(y) ∈ U and U is open in R, the set g −1 (U ) is an open set of Y and y ∈ g −1 (U ). (1) If Ky ∩ S(A) = ∅, then yα = a∗α for each α ∈ S(A). Thus πS(A) (y) = πS(A) (a∗ ). Since g ∈ A and πS(A) (y) = πS(A) (a∗ ), we have g(y) = g(a∗ ) by Claim 1. Denote C(A) = {z: z ∈ Y such that Kz ∩ S(A) = ∅}. For each z ∈ C(A) we have πS(A) (z) = πS(A) (a∗ ). Thus g(z) = g(y) = g(a∗ ). So g ∈ [C(A), U ] = {f : f ∈ Cp (Y ) such that f (C(A)) ⊂ U }. If A1 ⊂ A2 ⊂ Cp (Y ), then S(A1 ) ⊂ S(A2 ). Thus C(A2 ) ⊂ C(A1 ). So we have [C(A1 ), U ] ⊂ [C(A2 ), U ]. (2) Now we assume that Ky ∩ S(A) = ∅. Thus by Claim 2 there are some y ∈ Y and some n ∈ N such that πS(A) (y) = πS(A) (y ), g(y) = g(y ), y ∈ B, and g(B) ⊂ U , where B = α∈Ky ∩S(A) (πα−1 )∗ (Fα ) \ {fj−1 (Vj ): j n}, Fα ∈ Fα for each α ∈ Ky ∩ S(A) and fj ∈ A, Vj ∈ B(R) for each j n. Denote FA = {B: there are some n ∈ N and m ∈ N such that B = im (πα−1 )∗ (Fαi ) \ {fj−1 (Vj ): i j n}, where αi ∈ S(A), Fαi ∈ Fαi for each i m and fj ∈ A, Vj ∈ B(R) for each j n}. The family FA has the following properties (1) |FA | max{|A|, ω}; (2) if A1 ⊂ A2 , then FA1 ⊂ FA2 , where A1 , A2 ⊂ Cp (Y ).
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Denote N (A) = W(FA ) = {[B1 , B2 , . . . , Bn , U1 , U2 , . . . , Un ] ∩ A: Bi ∈ FA ∪ {C(A)} and Ui ∈ B(R), n ∈ N}. The family N (A) has the following properties (1) |N (A)| max{|A|, ω}; (2) if A1 ⊂ A2 ⊂ Cp (Y ), then for every C1 ∈ N (A1 ) there is some C2 ∈ N (A2 ) such that C1 ⊂ C2 . In what follows, we show that the family N (A) is an external network of A for each A ⊂ Cp (Y ). Let A ⊂ Cp (Y ) and g ∈ A. Let O be any open neighborhood of g in Cp (Y ). There are some n ∈ N, yi ∈ Y , and Ui ∈ B(R) for each i n such that g ∈ [y1 , y2 , . . . , yn , U1 , U2 , . . . , Un ] ⊂ O. Define Kyi = {α ∈ κ: yi α = a∗α } for each 1 i n. Let 1 i n. If Kyi ∩ S(A) = ∅, then πS(A) (yi ) = πS(A) (a∗ ). Thus g(yi ) = g(a∗ ) by Claim 1. For each z ∈ C(A) we also have g(z) = g(a∗ ). So g(z) = g(yi ) = g(a∗ ) for each z ∈ C(A). Thus yi ∈ C(A), g ∈ [C(A), Ui ], and C(A) ∈ FA ∪ {C(A)}. So we denote Bi = C(A). Now we assume that Kyi ∩ S(A) = ∅. So g(yi ) ∈ Ui and Kyi ∩ S(A) = ∅. By Claim 2, there are yi ∈ Y and some ni ∈ N such that πS(A) (yi ) = πS(A) (yi ), g(yi ) = g(yi ), yi ∈ Bi , and g(Bi ) ⊂ Ui , where −1 Bi = α∈S(A)∩Ky (πα−1 )∗ (Fαi ) \ {fij (Vij ): j ni } ∈ FA , Fαi ∈ Fα for each α ∈ S(A) ∩ Kyi and fij ∈ A, i Vij ∈ B(R) for each j ni . So we know that there is some Bi ∈ FA ∪ {C(A)} for each i n such that g(yi ) ∈ g(Bi ) ⊂ Ui . Thus g ∈ [B1 , B2 , . . . , Bn , U1 , U2 , . . . , Un ]∩A. In what follows, we show that [B1 , B2 , . . . , Bn , U1 , U2 , . . . , Un ]∩A ⊂ [y1 , y2 , . . . , yn , U1 , U2 , . . . , Un ]. Let 1 i n. If Kyi ∩ S(A) = ∅, then Bi = C(A). Thus yi ∈ Bi . So f ∈ [yi , Ui ] if f ∈ [Bi , Ui ]. If Kyi ∩ S(A) = ∅, then there are yi ∈ Y and some ni ∈ N such that πS(A) (yi ) = πS(A) (yi ), g(yi ) = g(yi ), −1 yi ∈ Bi , and g(Bi ) ⊂ Ui , where Bi = α∈S(A)∩Ky (πα−1 )∗ (Fαi ) \ {fij (Vij ): j ni } ∈ FA , Fαi ∈ Fα i for each α ∈ S(A) ∩ Kyi and fij ∈ A, Vij ∈ B(R) for each j ni . Since πS(A) (yi ) = πS(A) (yi ), we have f (yi ) = f (yi ) for each f ∈ A by Claim 1. Thus f (yi ) = f (yi ) if f ∈ [Bi , Ui ] ∩ A. So [Bi , Ui ] ∩ A ⊂ [yi , Ui ] for each i n. So we have that [B1 , B2 , . . . , Bn , U1 , U2 , . . . , Un ] ∩ A ⊂ [y1 , y2 , . . . , yn , U1 , U2 , . . . , Un ] ⊂ O. The set [B1 , B2 , . . . , Bn , U1 , U2 , . . . , Un ] ∩ A ∈ N (A). Thus the family N (A) is an external network of the set A in Cp (Y ). Hence the space Cp (Y ) is a semi-monotonically monolithic space. 2 Corollary 4. Let X = α∈κ Xα be a topological product and let Xα be a Lindelöf Σ-space for each α ∈ κ. If Y = σ(X, a∗ ), then Cp (Y ) is a monolithic space. In [2], it was proved that any σ-product of semi-stratifiable spaces (or spaces with a point-countable base) is a D-space. In fact, by a similar method of A.V. Arhangel’skii, we can get the following Theorem 6. Proposition 5. ([2, Proposition 3.2]) Suppose that a space X is the union of a countable family η = {Xi : i ∈ ω} of its closed subspace Xi , where X0 is a D-space, and every closed in X subspace of Xi+1 \ Xi is a D-space for each i ∈ ω. Then X is a D-space. Theorem 6. If {Xα : α ∈ Λ} is a family of topological spaces such that in Xαi is a D-space for each n ∈ N and αi ∈ Λ for each i n, then any σ-product of {Xα : α ∈ Λ} is a D-space. Proof. Let X = α∈Λ Xα and let a∗ = (a∗α : α ∈ Λ) ∈ X. Let Y = σ(X, a∗ ). For n ∈ ω we denote Yn = {x = (xα : α ∈ Λ) ∈ X: |{α ∈ Λ: xα = a∗α }| n}. We know that Y = {Yn : n ∈ ω} and Yn is closed in Y for each n ∈ ω. So Y0 = {a∗ }. For each n 1 we denote Un = Yn \ Yn−1 . Let Bn = {K: K ⊂ Λ and |K| = n} for n 1. Let n 1 and let K ∈ Bn . We denote WK = {y ∈ Y : {α ∈ Λ: yα = a∗α } = K}. If y = (yα : α ∈ Λ) ∈ Y , then we let Cy = {α ∈ Λ: yα = a∗α }. For each y ∈ WK and for each α ∈ Cy there is an open set Vα of
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Xα such that yα ∈ Vα ⊂ Xα \ {a∗ }. If Uy = ( α∈Cy πα−1 (Vα )) ∩ Y , then Uy is an open neighborhood of y in Y and Uy ∩ Yn ⊂ Yn \ Yn−1 . We know that Uy ∩ Yn ⊂ WK . Thus the set WK is an open subspace of Yn \ Yn−1 . If K1 , K2 ∈ Bn and K1 = K2 , we know that WK1 ∩ WK2 = ∅. Thus Yn \ Yn−1 = ζn , where ζn = {WK : K ∈ Bn }. So WK is clopen in Yn \ Yn−1 for each K ∈ Bn . In what follows we show that if F is a closed in Y subspace of Yn \ Yn−1 then F is a D-space. The set F is closed in Y and F ⊂ Yn \ Yn−1 , we have that F ∩ WK is closed in Y for each K ∈ ζn . Let K ∈ ζn . The set WK is homeomorphic to α∈K Xα . Thus WK is a D-space. So FK = F ∩ WK is a D-space being a closed subset of WK . Moreover, the set F is equal to the free union of the family {FK : K ∈ ζn }. So F is a D-space. So Y is a D-space by Proposition 5. 2 Since every semi-stratifiable space is a D-space [5] and a finite product of semi-stratifiable spaces is semi-stratifiable, we have Corollary 7. ([2]) Any σ-product of semi-stratifiable spaces (or spaces with a point-countable base) is a D-space. A finite product of Lindelöf Σ-spaces is a Lindelöf Σ-space, hence it is a D-space, so we have Corollary 8. If Y is a σ-product of Lindelöf Σ-spaces, then Y is a D-space. Recall that a space X is C-scattered if every non-empty closed subset F of X there is some x ∈ F which has a compact neighborhood in F [19]. A space X is called weak θ-refinable [18] if for any open cover U of X there is an open refinement V = {Vi : i ∈ N} satisfying: (1) { Vi : i ∈ N} is a point-finite open cover of X; (2) for any x ∈ X there is some i ∈ N such that 1 ord(x, Vi ) < ω where ord(x, Vi ) = |{V ∈ Vi : x ∈ V }|. Since this covering property is not so well known, we point out that every subparacompact, metacompact, or submetacompact (also called θ-refinable [7]) implies weak θ-refinable. n Lemma 9. ([14, Theorem 18]) Let n ∈ N and let X = ( i=1 Xi ) × Y . If Xi is a regular weak θ-refinable C-scattered space for each i n and Y is a Lindelöf D-space, then X is a D-space. By Theorem 6 and Lemma 9, we have Theorem 10. Any σ-product of regular weak θ-refinable C-scattered spaces is a D-space. Corollary 11. Any σ-product of regular subparacompact (metacompact, or submetacompact) C-scattered spaces is a D-space. Let Y be a dense subspace of X. In what follows, we discuss a special case that Cp (X) is monotonically monolithic if Cp (Y ) is monotonically monolithic. We firstly list some conclusions on monotonically monolithic spaces. Lemma 12. ([12, Theorem 3.2]) A space X is monotonically monolithic (resp. weakly monotonically monolithic) if and only if one can assign to each finite subset F of X a countable collection N (F ) of subsets of X such that, for each A ⊂ X, F ∈[A]<ω N (F ) contains a network at each point of A (resp., at some point of A \ A, if A is not closed).
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By Lemma 12, we can get the following conclusion. Lemma 13. Let X be a topological space and let Y be a subspace of X. The subspace Y of X is monotonically monolithic (resp., weakly monotonically monolithic) if and only if one can assign to each finite subset F of Y a countable collection N (F ) of subsets of X such that for each A ⊂ Y , F ∈[A]<ω N (F ) contains a network at each point of A ∩ Y (resp., at some point of (A ∩ Y ) \ A if A is not closed in Y ). Proof. “⇒” It is obvious by Lemma 12. “⇐” For each finite subset F of Y , denote NY (F ) = {B∩Y : B ∈ N (F )}. So |NY (F )| ω. Let A ⊂ Y and let x ∈ A(Y ) . If U is an open neighborhood of x in Y , then there is an open set V of X such that V ∩ Y = U . So there is some B ∈ N (F ) such that x ∈ B ⊂ V . Thus x ∈ B ∩ Y ⊂ U . Thus F ∈[A]<ω NY (F ) contains a network at each point of A ∩ Y in Y . Thus Y is monotonically monolithic by Lemma 12. Similarly, we can prove that Y is weakly monotonically monolithic if one can assign to each finite subset F of Y a countable collection NY (F ) of subsets of X such that for each A ⊂ Y , F ∈[A]<ω N (F ) contains a network at some point of (A ∩ Y ) \ A if A is not closed in Y . 2 Let Y be a subspace of a topological space X. Denote πY : Cp (X) → Cp (Y ) be the map of restricting a function in Cp (X) to Y , i.e. πY (f ) = f |Y for all f ∈ Cp (X). In [1, Proposition 0.4.1], it was proved that if Y is a dense subset of X then πY : Cp (X) → πY (Cp (X)) is a bijective continuous map. Theorem 14. Let Y be a dense subspace of a topological space X. If Cp (Y ) is monotonically monolithic and |X \ Y | ω, then Cp (X) is monotonically monolithic. Proof. By Lemma 12, we know that one can assign to each finite subset F of Cp (Y ) a countable collection N (F ) of subsets of Cp (Y ) such that for each A ⊂ Cp (Y ), F ∈[A]<ω N (F ) contains a network at each point of A(Cp (Y )) . Let Z = x∈X Rx , where Rx = R with the usual topology for each x ∈ X. Thus Cp (X) is a subspace of Z. Denote πY : Z → x∈Y Rx the usual projection. Since Y is dense in X, the map g = πY |Cp (X) : Cp (X) → Cp (Y ) is injective [1, Proposition 0.4.1]. Let F be any finite subset of Cp (X). Then g(F ) is a finite subset of Cp (Y ). Thus N (g(F )) is a countable family of subsets of Cp (Y ). Denote N (F ) = {πY−1 (B) ∩ ( in πx−1 (Ui )): B ∈ N (g(F )), n ∈ N, xi ∈ X \ Y for each i n, and Ui ∈ B(R)}. i Since |N (g(F ))| ω and |X \ Y | ω, we have |N (F )| ω. Thus N (F ) is a countable family of subsets of the space Z. Let A ⊂ Cp (X) and f ∈ A(Cp (X)) . If f = f |Y , then f ∈ Cp (Y ) and f ∈ πY (A)(Cp (Y )) . Let O be any open neighborhood of f in Z. Thus there is some n ∈ N and xi ∈ X, Ui ∈ B(R) for each i n such that f ∈ in πx−1 (Ui ) ⊂ O. We can assume that x1 , x2 , . . . , xl ∈ Y and xl+1 , xl+2 , . . . , xn ∈ X \ Y i for some l < n. If Ml = {t: t ∈ Cp (Y ) and t(xi ) ∈ Ui for each i l}, then Ml is an open neighborhood of f in Cp (Y ). Since f ∈ πY (A)(Cp (Y )) , there is a finite set F ⊂ πY (A) and some B ∈ N (F ) such that f ∈ B ⊂ Ml . The set F ⊂ πY (A) = g(A) ⊂ πY (Cp (X)) = g(Cp (X)) ⊂ Cp (Y ), then g −1 (F ) ⊂ Cp (X). Since the map g is injective and |F | < ω, we have |g −1 (F )| < ω and g −1 (F ) ⊂ A. Denote F = g −1 (F ). Since f (xi ) ∈ Ui for each l +1 i n, we have f ∈ l+1in πx−1 (Ui ). Since πY (f ) = f ∈ B ⊂ Ml , we have i f ∈ πY−1 (B) ⊂ πY−1 (Ml ) ⊂ 1il πx−1 (Ui ). So f ∈ πY−1 (B) ∩ ( l+1in πx−1 (Ui )) ⊂ 1in πx−1 (Ui ) ⊂ O. i i i Since B ∈ N (F ) and xi ∈ X \ Y for each l + 1 i n, we have πY−1 (B) ∩ ( l+1in πx−1 (U )) ∈ N (F ). i i Thus Cp (X) is monotonically monolithic by Lemma 13. Similar with one point compactification of a locally compact Hausdorff space, we know that for every locally Lindelöf regular space X there exists a Lindelöf space Z such that X is a dense subspace of Z and |Z \ X| = 1. The space Z is called a one point Lindelöfication of X. Corollary 15. If X is a locally Lindelöf regular space and Cp (X) is monotonically monolithic, then for any one point Lindelöfication Z of X, the space Cp (Z) is monotonically monolithic.
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In what follows, we discuss some results on k-in-countable bases and k-in-countable weak bases, where k ∈ N. Given a natural number k, a family A of subsets of a space X is k-in-countable if every B ⊂ X with |B| = k is contained in at most countably many elements of A [1]. We know that every point-countable family A of subsets of a space X is 1-in-countable. In [4], it was proved that every countably compact regular space with a k-in-countable base for some k ∈ N is metrizable. In [22], it was proved that if k ∈ N and X is a star compact space with a k-in-countable base then X is metrizable. To get this conclusion, it was also proved that if X is a space with a k-in-countable base for some k ∈ N then e(X) = l(X) [22]. We know that if X is a D-space then e(X) = l(X). In what follows, we show that if a space X has a k-in-countable weak base for some k ∈ N then X is a D-space. Let us review the definition of weak bases [17]. A collection P = {Px : x ∈ X} is called a weak base for X, if for any x ∈ X the following conditions hold: (1) For each x ∈ X, Px is closed under finite intersections and x ∈ Px ; (2) A subset U of X is open if and only if for any x ∈ U , there is some B ∈ Px such that x ∈ B ⊂ U. In [13], it was proved that if a topological space X has a point-countable weak base then X is a weakly monotonically monolithic space. Let A be a set. We denote [A]<ω = {F : F ⊂ A and |F | < ω}. If |A| k, where k ∈ N, then we denote [A]k = {F : F ⊂ A and |A| = k}. Theorem 16. Let k ∈ N. If X has a k-in-countable family F such that for each non-closed subset A of X the family {F ∈ F: there is some B ⊂ A such that |B| = k and B ⊂ F } contains a network at some point of A \ A, then X is a weak monotonically monolithic space. Proof. Since X is a T1 -space, every finite subset of X is closed. Let B ⊂ X be a finite subset of X. If |B| < k, then we denote N (B) = {X}. If |B| k, then we denote N (B) = {F ∈ F: there is some C ⊂ B such that |C| = k and C ⊂ F } ∪ {X}. Since F is a k-in-countable family of subsets of X and B is a finite subset of X, we have |N (B)| ω. Let A ⊂ X. If A is not closed in X, then |A| ω. Thus {F ∈ F: there is some B ⊂ A such that |B| = k and B ⊂ F } ∪ {X} = B∈[A]k N (B) contains a network at some point of A \ A. By the definition of N (B) for each finite subset B of X, we have B∈[A]k N (B) = B∈[A]<ω N (B). So B∈[A]<ω N (B) contains a network at some point of A \ A. Thus X is a weakly monotonically monolithic space by Lemma 12. 2 Lemma 17. Let X be a topological space. Let P = {Px : x ∈ X} be a weak base of X. If A ⊂ X is not closed in X, then there is some x ∈ A \ A such that |B ∩ A| ω for each B ∈ Px . Proof. Suppose that for each x ∈ A \ A there is some Bx ∈ Px such that |Bx ∩ A| < ω. The set X \ (Bx ∩ A) is an open neighborhood of x, so there is some Bx ∈ Px such that Bx ⊂ Bx and x ∈ Bx ⊂ X \(Bx ∩A). Thus Bx ∩A = ∅. For each x ∈ X \A, there is some Bx ∈ Px such that Bx ∩A = ∅. Thus X \A = {Bx : x ∈ X \A}, where Bx ∈ Px for each x ∈ X\A. So the set X\A is an open set of X. Thus A is closed in X. A contradiction. Thus there is some x ∈ A \ A such that |B ∩ A| ω for each B ∈ Px . 2 Theorem 18. Let k ∈ N. If a topological space X has a k-in-countable weak base, then X is a weakly monotonically monolithic space. Proof. Let P = {Px : x ∈ X} be a k-in-countable weak base for X. Let A ⊂ X. If A is not closed, then there is some x ∈ A \ A such that |P ∩ A| ω for each P ∈ Px by Lemma 17. Let U be any open neighborhood of x in X. There is some P ∈ Px such that x ∈ P ⊂ U . So |P ∩ A| ω. Let B1 ⊂ P ∩ A such that |B1 | = k. Thus P ∈ {F ∈ P: there is some B ⊂ A such that |B| = k and B ⊂ F }. So the family {F ∈ P: there is some B ⊂ A such that |B| = k and B ⊂ F } contains a network at x. Thus X is a weakly monotonically monolithic space by Theorem 16. 2
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Proposition 19. ([13]) If X is a weakly monotonically monolithic space, then X is a D-space. By Theorem 18 and Proposition 19, we have Theorem 20. Let k ∈ N. If X has a k-in-countable weak base (or base), then X is a D-space. Corollary 21. If X has a k-in-countable weak base, then l(X) = e(X). Corollary 22. ([22, Lemma 2.3]) If X has a k-in-countable base, then l(X) = e(X). Acknowledgement The author would like to thank the referee for many valuable remarks, corrections, and suggestions which greatly improved the paper. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22]
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Topology and its Applications www.elsevier.com/locate/topol
A note on D-property, monotone monolithicity and σ-product ✩ Liang-Xue Peng College of Applied Science, Beijing University of Technology, Beijing 100124, China
a r t i c l e
i n f o
Article history: Received 31 October 2012 Received in revised form 23 July 2013 Accepted 18 September 2013 MSC: 54D20 54C35 54B10 Keywords: D-space Monolithic Monotonically monolithic σ-product Cp (X)
a b s t r a c t In this note we firstly show that if X is a σ-product of Lindelöf Σ-spaces, then Cp (X) is a semi-monotonically monolithic space and hence it is monolithic. In the second partof this note we show that if {Xα : α ∈ Λ} is a family of topological spaces such that in Xαi is a D-space for each n ∈ N and αi ∈ Λ for each i n, then any σ-product of {Xα : α ∈ Λ} is a D-space. This generalizes a conclusion of A.V. Arhangel’skii. By this conclusion and a known conclusion we have that any σ-product of regular weak θ-refinable (or (sub)metacompact) C-scattered spaces is a D-space. We also show that if Y is a dense subspace of a topological space X with |X \ Y | ω such that Cp (Y ) is monotonically monolithic, then Cp (X) is monotonically monolithic. In the last part of this note we show that if X has a k-in-countable weak base for some k ∈ N then X is a D-space. © 2013 Elsevier B.V. All rights reserved.
1. Introduction The notion of a D-space was introduced by van Douwen and Pfeffer in [9]. A neighborhood assignment for a space X is a function φ from X to the topology of the space X such that x ∈ φ(x) for any x ∈ X. A space X is called a D-space if for any neighborhood assignment φ for X there exists a closed discrete subspace D of X such that X = {φ(d): d ∈ D} [9]. All compact T1 -spaces and all metric spaces are D-spaces. Many classes of spaces are known to be D-spaces [11]. For example, every space with a point-countable base is a D-space [3] and every space with a point-countable weak base is a D-space [6,15]. R.Z. Buzyakova [8] proved that a space Cp (X) is hereditarily D when X is compact. In [20], V.V. Tkachuk introduced a notion of monotone monolithicity and proved that every monotonically monolithic space is a D-space. Some properties of monotonically monolithic spaces are also discussed in [12] and [21]. In [16], it was proved that Cp (Cp (X)) is monotonically monolithic if and only if X is monotonically monolithic. It was also proved that if X is the product of Lindelöf Σ-spaces then Cp (X) is monotonically monolithic [16]. ✩
Research supported by the National Natural Science Foundation of China (Grant No. 11271036). E-mail address: [email protected].
0166-8641/$ – see front matter © 2013 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.topol.2013.09.013
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Let X = {Xα : α ∈ A} be a topological product of the family {Xα : α ∈ A}, let x∗ = (x∗α : α ∈ A) be a point in X, and κ be a cardinal number. The Σκ -product of the family {Xα : α ∈ A} based in x∗ is the subspace Σκ (X, x∗ ) = {x ∈ X: |{α ∈ A: xα = x∗α }| < κ} of X. When κ = ω, Σκ (X, x∗ ) is called σ-product and it is usually denoted by σ(X, x∗ ). The following question appears in [16, Question 4.9]. Let X be a Σκ -product of Lindelöf Σ-spaces. Is Cp (X) then monotonically monolithic? In the first part of this note, we show that if X is a σ-product of Lindelöf Σ-spaces, then Cp (X) is semi-monotonically monolithic. Recall that a space X is monolithic if, for every infinite subset A of X, the network weight of the closure of the set A does not exceed the cardinality of A. A space X is called semi-monotonically monolithic if one can assign to each A ⊂ X a collection N (A) of subsets of X such that (1) |N (A)| max{|A|, ω}; (2) if A ⊂ B ⊂ X, then for each C1 ∈ N (A) there is some C2 ∈ N (B) such that C1 ⊂ C2 ; (3) if U is open in X and x ∈ A ∩ U , then there is some C ∈ N (A) with x ∈ C ⊂ U . Every monotonically monolithic space is semi-monotonically monolithic and every semi-monotonically monolithic space is monolithic. In the second part of this note, we show that if {Xα : α ∈ Λ} is a family of topological spaces such that in Xαi is a D-space for each n ∈ N and αi ∈ Λ for each i n, then any σ-product of {Xα : α ∈ Λ} is a D-space. This generalizes some conclusions of A.V. Arhangel’skii [2, Theorems 3.9, 3.10, and 3.12]. By this conclusion and a known conclusion [14, Theorem 18] we have that any σ-product of regular weak θ-refinable (or (sub)metacompact) C-scattered spaces is a D-space. We also show that if Y is a dense subspace of a topological space X with |X \ Y | ω such that Cp (Y ) is monotonically monolithic, then Cp (X) is monotonically monolithic. In the last part of this note we show that if X has a k-in-countable weak base for some k ∈ N then X is a D-space. Recall that l(X) is defined as the smallest infinite cardinal κ such that every open cover of X has a subcollection of cardinality κ which covers X. e(X) = sup{|D|: D ⊂ X, D is closed discrete} + ω. A condensation is a bijective continuous onto map. iw(X) denotes the minimal weight of all spaces onto which X can be condensed. All the spaces in this note are assumed to be T1 -spaces. The set of all positive integers is denoted by N and ω is N ∪ {0}. In notation and terminology we will follow [7] and [10]. 2. Main results For a space X, we denote Cp (X) the set of real-valued continuous functions, having X as its domain, with its pointwise convergence topology. Let B(R) be a countable base of R with the usual topology, where R is the set of all real numbers. We know that Cp (X) has a base which contains sets of the form [x1 , x2 , . . . , xn , U1 , U2 , . . . , Un ] = {f ∈ Cp (X): f (xi ) ∈ Ui for i n}, where n ∈ N, xi ∈ X, Ui ∈ B(R) for each i n. For subsets E1 , E2 , . . . , En of a space X and subsets U1 , U2 , . . . , Un of R, denote [E1 , E2 , . . . , En , U1 , U2 , . . . , Un ] = {f ∈ Cp (X): f (Ei ) ⊂ Ui for i = 1, 2, . . . , n}, where n ∈ N. If X = α∈Λ Xα is the product of a family {Xα : α ∈ Λ} of topological spaces and B ⊂ A, then we denote πB : X → α∈B Xα the natural projection. Recall that a collection F of subsets of a space X is a network at x ∈ X if, given any open neighborhood U of x, there is some F ∈ F with x ∈ F ⊂ U . A space X is monotonically monolithic [20] if one can assign to each A ⊂ X a collection N (A) of subsets of X such that (1) |N (A)| max{|A|, ω}; (2) A ⊂ B ⊂ X ⇒ N (A) ⊂ N (B);
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(3) if {Aα : α < δ} is an increasing collection of subsets of X, and A = {Aα : α < δ}, then N (A) = {N (Aα ): α < δ}; (4) if U is open in X and x ∈ A ∩ U , then there is some N ∈ N (A) with x ∈ N ⊂ U . We call N a monotonically monolithic operator for X and N (A) is called an external network of A. A space X is called weakly monotonically monolithic [13] if it has an operator satisfying the above conditions but with conditions (4) replaced by (4 ) if A is not closed, then there is some x ∈ A \ A such that N (A) contains a network at x. Recall that a family F of subsets of a space X is a network modulo a cover C if for each C ∈ C and an open set U of X with C ⊂ U there exists F ∈ F such that C ⊂ F ⊂ U . A space X is a Lindelöf Σ-space if there exists a countable family F of subsets of X such that F is a network modulo a compact cover C of the space X. In [20] it was proved that for every Lindelöf Σ-space X, Cp (X) is monotonically monolithic and every monotonically monolithic space is a D-space. Proposition 1. ([1, Theorem II 6.23]) Let X = α<κ Xα be a topological product, T = σ(X, x∗ ) and Y is a subspace which contains T , and λ an infinite cardinal. Assume that for each finite set K ⊂ κ, l( α∈K Xα ) λ. If f : Y → Z is a continuous function and iw(Z) λ, then there is a set S ⊂ κ with |S| λ such that if x, x ∈ Y and πS (x) = πS (x ) then f (x) = f (x ). Proposition 2. ([10, Theorem 3.3.20]) Let n ∈ N. If Ai is compact subset of a topological space Xi for each i n and A = in Ai is contained in an open set U of X = in Xi , then there is an open set Vi of Xi for each i n such that A ⊂ in Vi ⊂ U . Theorem 3. Let X = α∈κ Xα be a topological product and let Xα be a Lindelöf Σ-space for each α ∈ κ. If Y = σ(X, a∗ ), then Cp (Y ) is a semi-monotonically monolithic space. Proof. Let a∗ = (a∗α : α ∈ κ) ∈ X and let Y = σ(X, a∗ ). For each A ⊂ Cp (Y ), we will define an external network N (A) of A in Cp (Y ) such that the operator N satisfies the conditions (1) and (2) which appear in the definition of a semi-monotonically monolithic space. For each α ∈ κ, the space Xα is a Lindelöf Σ-space. Let Fα be a countable family of subsets of Xα such that Fα is a network modulo a compact cover Cα of the space Xα . Since a finite product of Lindelöf Σ-spaces is Lindelöf and iw(R) = ω, we have that for each f ∈ Cp (Y ) there is a countable set Sf ⊂ κ such that if x, x ∈ Y and πSf (x) = πSf (x ) then f (x) = f (x ). By Proposition 1, for A ⊂ Cp (Y ), let S(A) = {Sf : f ∈ A}. Thus |S(A)| max{|A|, ω}. Claim 1. If g ∈ A and πS(A) (x) = πS(A) (x ) for x, x ∈ Y , then g(x) = g(y). Proof of Claim 1. Suppose g(x) = g(x ), then there are disjoint open sets U, V of R such that g(x) ∈ U and g(x ) ∈ V . If O = {g ∈ Cp (Y ): g (x) ∈ U and g (x ) ∈ V }, then the set O is an open neighborhood of g in Cp (Y ). So O ∩ A = ∅. If f ∈ O ∩ A, then f (x) ∈ U and f (x ) ∈ V . Thus f (x) = f (x ). Since f ∈ A, we have Sf ⊂ S(A). Since πS(A) (x) = πS(A) (x ), we have πSf (x) = πSf (x ). Thus f (x) = f (x ). This is a contradiction with f (x) = f (x ). We have finished the proof of Claim 1. 2 If y = (yα : α ∈ κ) ∈ Y , then |{α ∈ κ: yα = a∗α }| < ω. We denote Ky = {α ∈ κ: yα = a∗α }. Thus the set Ky is a finite subset of κ. Let A ⊂ Cp (Y ). If S ⊂ S(A) and, for each α ∈ S, Fα ∈ Fα , then we define −1 ∗ ∗ ∗ ∗ ∗ α∈S (πα ) (Fα ) = β∈κ Fβ , where Fβ = Fα if β = α and α ∈ S, otherwise Fβ = {aβ }.
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Claim 2. Let A ⊂ Cp (Y ) and let g ∈ A. If y ∈ Y such that Ky ∩ S(A) = ∅ and g(y) ∈ U for some U ∈ B(R), then there are some y ∈ Y and some n ∈ N such that πS(A) (y) = πS(A) (y ), g(y) = g(y ), y ∈ B, and g(B) ⊂ U , where B = α∈Ky ∩S(A) (πα−1 )∗ (Fα ) \ {fj−1 (Vj ): j n}, Fα ∈ Fα for each α ∈ Ky ∩ S(A) and fj ∈ A, Vj ∈ B(R) for each j n. Proof of Claim 2. Let y ∈ Y such that yα = a∗α if α ∈ Ky \ S(A), and yα = yα if α ∈ κ \ (Ky \ S(A)). Thus πS(A) (y) = πS(A) (y ). So g(y) = g(y ) by Claim 1. Thus g(y ) ∈ U . Hence y ∈ g −1 (U ). For each α ∈ Ky ∩ S(A), there is some Cα ∈ Cα such that yα ∈ Cα . Let P = α∈κ Pα , where Pα = Cα if α ∈ Ky ∩ S(A), otherwise Pα = {a∗α }. So y ∈ P ∩ g −1 (U ). If x ∈ P \ g −1 (U ), then g(x) = g(y ). Let Vx ∈ B(R) such that g(x) ∈ Vx and g(y ) ∈ / Vx . Thus [x, y , Vx , (R \ Vx ) ∩ U ] is an open neighborhood of g in Cp (Y ). Since g ∈ A, there is some fx ∈ [x, y , Vx , (R \ Vx ) ∩ U ] ∩ A. So fx (x) ∈ Vx and fx (y ) ∈ (R \ Vx ) ∩ U . Thus fx−1 (Vx ) is open in Y and y ∈ / fx−1 (Vx ). Thus −1 y ∈ P \ fx (Vx ). The set P is compact and P = ( {P ∩ fx−1 (Vx ): x ∈ P \ g −1 (U )}) ∪ (P ∩ g −1 (U )). There is some n ∈ N and xi ∈ P \ g −1 (U ) for each i n such that P = ( {P ∩ fx−1 (Vxi ): i n}) ∪ (P ∩ g −1 (U )). i Thus πS(A)∩Ky (P ) = α∈S(A)∩Ky Cα ⊂ ( {πS(A)∩Ky (fx−1 (Vxi )): i n}) ∪ (πS(A)∩Ky (g −1 (U ))). Since i −1 the map πS(A)∩Ky |Y is open, the family {πS(A)∩Ky (fxi (Vxi )): i n} ∪ {πS(A)∩Ky (g −1 (U ))} is an open family of α∈Ky ∩S(A) Xα and covers the set α∈Ky ∩S(A) Cα . The set α∈Ky ∩S(A) Cα is compact, thus by Proposition 2 there is an open set Vα of Xα for each α ∈ Ky ∩ S(A) such that α∈Ky ∩S(A) Cα ⊂ −1 −1 (U )). So there is some Fα ∈ Fα such α∈Ky ∩S(A) Vα ⊂ ( {πS(A)∩Ky (fxi (Vxi )): i n}) ∪ πS(A)∩Ky (g that Cα ⊂ Fα ⊂ Vα for each α ∈ Ky ∩ S(A). Thus P ⊂ F ⊂ ( {fx−1 (Vxi ): i n}) ∪ (g −1 (U )) and i −1 −1 ∗ y ∈ / fxi (Vxi ) for each i n, where F = α∈Ky ∩S(A) (πα ) (Fα ). So y ∈ F \ {fx−1 (Vxi ): i n} ⊂ i g −1 (U ). Thus g(y ) ∈ g(F \ {fx−1 (V ): i n}) ⊂ U . If f = f and V = V for each i n, then xi xi i i i i −1 x−1 −1 ∗ g(y ) ∈ g(F \ {fi (Vi ): i n}). Denote B = α∈Ky ∩S(A) (πα ) (Fα ) \ {fi (Vi ): i n}. Thus y ∈ B, g(y) = g(y ), and g(y) ∈ g(B) ⊂ U . We have finished the proof of Claim 2. 2 Let A ⊂ Cp (Y ) and g ∈ A. Let y ∈ Y and let U ∈ B(R) such that g(y) ∈ U . The set Ky is a finite subset of κ. For each α ∈ Ky , there is some Cα ∈ Cα such that yα ∈ Cα . Thus α∈Ky Cα is homeomorphic to α∈κ Pα , where Pα = Cα if α ∈ Ky , otherwise Pα = {a∗α }. Since g(y) ∈ U and U is open in R, the set g −1 (U ) is an open set of Y and y ∈ g −1 (U ). (1) If Ky ∩ S(A) = ∅, then yα = a∗α for each α ∈ S(A). Thus πS(A) (y) = πS(A) (a∗ ). Since g ∈ A and πS(A) (y) = πS(A) (a∗ ), we have g(y) = g(a∗ ) by Claim 1. Denote C(A) = {z: z ∈ Y such that Kz ∩ S(A) = ∅}. For each z ∈ C(A) we have πS(A) (z) = πS(A) (a∗ ). Thus g(z) = g(y) = g(a∗ ). So g ∈ [C(A), U ] = {f : f ∈ Cp (Y ) such that f (C(A)) ⊂ U }. If A1 ⊂ A2 ⊂ Cp (Y ), then S(A1 ) ⊂ S(A2 ). Thus C(A2 ) ⊂ C(A1 ). So we have [C(A1 ), U ] ⊂ [C(A2 ), U ]. (2) Now we assume that Ky ∩ S(A) = ∅. Thus by Claim 2 there are some y ∈ Y and some n ∈ N such that πS(A) (y) = πS(A) (y ), g(y) = g(y ), y ∈ B, and g(B) ⊂ U , where B = α∈Ky ∩S(A) (πα−1 )∗ (Fα ) \ {fj−1 (Vj ): j n}, Fα ∈ Fα for each α ∈ Ky ∩ S(A) and fj ∈ A, Vj ∈ B(R) for each j n. Denote FA = {B: there are some n ∈ N and m ∈ N such that B = im (πα−1 )∗ (Fαi ) \ {fj−1 (Vj ): i j n}, where αi ∈ S(A), Fαi ∈ Fαi for each i m and fj ∈ A, Vj ∈ B(R) for each j n}. The family FA has the following properties (1) |FA | max{|A|, ω}; (2) if A1 ⊂ A2 , then FA1 ⊂ FA2 , where A1 , A2 ⊂ Cp (Y ).
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Denote N (A) = W(FA ) = {[B1 , B2 , . . . , Bn , U1 , U2 , . . . , Un ] ∩ A: Bi ∈ FA ∪ {C(A)} and Ui ∈ B(R), n ∈ N}. The family N (A) has the following properties (1) |N (A)| max{|A|, ω}; (2) if A1 ⊂ A2 ⊂ Cp (Y ), then for every C1 ∈ N (A1 ) there is some C2 ∈ N (A2 ) such that C1 ⊂ C2 . In what follows, we show that the family N (A) is an external network of A for each A ⊂ Cp (Y ). Let A ⊂ Cp (Y ) and g ∈ A. Let O be any open neighborhood of g in Cp (Y ). There are some n ∈ N, yi ∈ Y , and Ui ∈ B(R) for each i n such that g ∈ [y1 , y2 , . . . , yn , U1 , U2 , . . . , Un ] ⊂ O. Define Kyi = {α ∈ κ: yi α = a∗α } for each 1 i n. Let 1 i n. If Kyi ∩ S(A) = ∅, then πS(A) (yi ) = πS(A) (a∗ ). Thus g(yi ) = g(a∗ ) by Claim 1. For each z ∈ C(A) we also have g(z) = g(a∗ ). So g(z) = g(yi ) = g(a∗ ) for each z ∈ C(A). Thus yi ∈ C(A), g ∈ [C(A), Ui ], and C(A) ∈ FA ∪ {C(A)}. So we denote Bi = C(A). Now we assume that Kyi ∩ S(A) = ∅. So g(yi ) ∈ Ui and Kyi ∩ S(A) = ∅. By Claim 2, there are yi ∈ Y and some ni ∈ N such that πS(A) (yi ) = πS(A) (yi ), g(yi ) = g(yi ), yi ∈ Bi , and g(Bi ) ⊂ Ui , where −1 Bi = α∈S(A)∩Ky (πα−1 )∗ (Fαi ) \ {fij (Vij ): j ni } ∈ FA , Fαi ∈ Fα for each α ∈ S(A) ∩ Kyi and fij ∈ A, i Vij ∈ B(R) for each j ni . So we know that there is some Bi ∈ FA ∪ {C(A)} for each i n such that g(yi ) ∈ g(Bi ) ⊂ Ui . Thus g ∈ [B1 , B2 , . . . , Bn , U1 , U2 , . . . , Un ]∩A. In what follows, we show that [B1 , B2 , . . . , Bn , U1 , U2 , . . . , Un ]∩A ⊂ [y1 , y2 , . . . , yn , U1 , U2 , . . . , Un ]. Let 1 i n. If Kyi ∩ S(A) = ∅, then Bi = C(A). Thus yi ∈ Bi . So f ∈ [yi , Ui ] if f ∈ [Bi , Ui ]. If Kyi ∩ S(A) = ∅, then there are yi ∈ Y and some ni ∈ N such that πS(A) (yi ) = πS(A) (yi ), g(yi ) = g(yi ), −1 yi ∈ Bi , and g(Bi ) ⊂ Ui , where Bi = α∈S(A)∩Ky (πα−1 )∗ (Fαi ) \ {fij (Vij ): j ni } ∈ FA , Fαi ∈ Fα i for each α ∈ S(A) ∩ Kyi and fij ∈ A, Vij ∈ B(R) for each j ni . Since πS(A) (yi ) = πS(A) (yi ), we have f (yi ) = f (yi ) for each f ∈ A by Claim 1. Thus f (yi ) = f (yi ) if f ∈ [Bi , Ui ] ∩ A. So [Bi , Ui ] ∩ A ⊂ [yi , Ui ] for each i n. So we have that [B1 , B2 , . . . , Bn , U1 , U2 , . . . , Un ] ∩ A ⊂ [y1 , y2 , . . . , yn , U1 , U2 , . . . , Un ] ⊂ O. The set [B1 , B2 , . . . , Bn , U1 , U2 , . . . , Un ] ∩ A ∈ N (A). Thus the family N (A) is an external network of the set A in Cp (Y ). Hence the space Cp (Y ) is a semi-monotonically monolithic space. 2 Corollary 4. Let X = α∈κ Xα be a topological product and let Xα be a Lindelöf Σ-space for each α ∈ κ. If Y = σ(X, a∗ ), then Cp (Y ) is a monolithic space. In [2], it was proved that any σ-product of semi-stratifiable spaces (or spaces with a point-countable base) is a D-space. In fact, by a similar method of A.V. Arhangel’skii, we can get the following Theorem 6. Proposition 5. ([2, Proposition 3.2]) Suppose that a space X is the union of a countable family η = {Xi : i ∈ ω} of its closed subspace Xi , where X0 is a D-space, and every closed in X subspace of Xi+1 \ Xi is a D-space for each i ∈ ω. Then X is a D-space. Theorem 6. If {Xα : α ∈ Λ} is a family of topological spaces such that in Xαi is a D-space for each n ∈ N and αi ∈ Λ for each i n, then any σ-product of {Xα : α ∈ Λ} is a D-space. Proof. Let X = α∈Λ Xα and let a∗ = (a∗α : α ∈ Λ) ∈ X. Let Y = σ(X, a∗ ). For n ∈ ω we denote Yn = {x = (xα : α ∈ Λ) ∈ X: |{α ∈ Λ: xα = a∗α }| n}. We know that Y = {Yn : n ∈ ω} and Yn is closed in Y for each n ∈ ω. So Y0 = {a∗ }. For each n 1 we denote Un = Yn \ Yn−1 . Let Bn = {K: K ⊂ Λ and |K| = n} for n 1. Let n 1 and let K ∈ Bn . We denote WK = {y ∈ Y : {α ∈ Λ: yα = a∗α } = K}. If y = (yα : α ∈ Λ) ∈ Y , then we let Cy = {α ∈ Λ: yα = a∗α }. For each y ∈ WK and for each α ∈ Cy there is an open set Vα of
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Xα such that yα ∈ Vα ⊂ Xα \ {a∗ }. If Uy = ( α∈Cy πα−1 (Vα )) ∩ Y , then Uy is an open neighborhood of y in Y and Uy ∩ Yn ⊂ Yn \ Yn−1 . We know that Uy ∩ Yn ⊂ WK . Thus the set WK is an open subspace of Yn \ Yn−1 . If K1 , K2 ∈ Bn and K1 = K2 , we know that WK1 ∩ WK2 = ∅. Thus Yn \ Yn−1 = ζn , where ζn = {WK : K ∈ Bn }. So WK is clopen in Yn \ Yn−1 for each K ∈ Bn . In what follows we show that if F is a closed in Y subspace of Yn \ Yn−1 then F is a D-space. The set F is closed in Y and F ⊂ Yn \ Yn−1 , we have that F ∩ WK is closed in Y for each K ∈ ζn . Let K ∈ ζn . The set WK is homeomorphic to α∈K Xα . Thus WK is a D-space. So FK = F ∩ WK is a D-space being a closed subset of WK . Moreover, the set F is equal to the free union of the family {FK : K ∈ ζn }. So F is a D-space. So Y is a D-space by Proposition 5. 2 Since every semi-stratifiable space is a D-space [5] and a finite product of semi-stratifiable spaces is semi-stratifiable, we have Corollary 7. ([2]) Any σ-product of semi-stratifiable spaces (or spaces with a point-countable base) is a D-space. A finite product of Lindelöf Σ-spaces is a Lindelöf Σ-space, hence it is a D-space, so we have Corollary 8. If Y is a σ-product of Lindelöf Σ-spaces, then Y is a D-space. Recall that a space X is C-scattered if every non-empty closed subset F of X there is some x ∈ F which has a compact neighborhood in F [19]. A space X is called weak θ-refinable [18] if for any open cover U of X there is an open refinement V = {Vi : i ∈ N} satisfying: (1) { Vi : i ∈ N} is a point-finite open cover of X; (2) for any x ∈ X there is some i ∈ N such that 1 ord(x, Vi ) < ω where ord(x, Vi ) = |{V ∈ Vi : x ∈ V }|. Since this covering property is not so well known, we point out that every subparacompact, metacompact, or submetacompact (also called θ-refinable [7]) implies weak θ-refinable. n Lemma 9. ([14, Theorem 18]) Let n ∈ N and let X = ( i=1 Xi ) × Y . If Xi is a regular weak θ-refinable C-scattered space for each i n and Y is a Lindelöf D-space, then X is a D-space. By Theorem 6 and Lemma 9, we have Theorem 10. Any σ-product of regular weak θ-refinable C-scattered spaces is a D-space. Corollary 11. Any σ-product of regular subparacompact (metacompact, or submetacompact) C-scattered spaces is a D-space. Let Y be a dense subspace of X. In what follows, we discuss a special case that Cp (X) is monotonically monolithic if Cp (Y ) is monotonically monolithic. We firstly list some conclusions on monotonically monolithic spaces. Lemma 12. ([12, Theorem 3.2]) A space X is monotonically monolithic (resp. weakly monotonically monolithic) if and only if one can assign to each finite subset F of X a countable collection N (F ) of subsets of X such that, for each A ⊂ X, F ∈[A]<ω N (F ) contains a network at each point of A (resp., at some point of A \ A, if A is not closed).
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By Lemma 12, we can get the following conclusion. Lemma 13. Let X be a topological space and let Y be a subspace of X. The subspace Y of X is monotonically monolithic (resp., weakly monotonically monolithic) if and only if one can assign to each finite subset F of Y a countable collection N (F ) of subsets of X such that for each A ⊂ Y , F ∈[A]<ω N (F ) contains a network at each point of A ∩ Y (resp., at some point of (A ∩ Y ) \ A if A is not closed in Y ). Proof. “⇒” It is obvious by Lemma 12. “⇐” For each finite subset F of Y , denote NY (F ) = {B∩Y : B ∈ N (F )}. So |NY (F )| ω. Let A ⊂ Y and let x ∈ A(Y ) . If U is an open neighborhood of x in Y , then there is an open set V of X such that V ∩ Y = U . So there is some B ∈ N (F ) such that x ∈ B ⊂ V . Thus x ∈ B ∩ Y ⊂ U . Thus F ∈[A]<ω NY (F ) contains a network at each point of A ∩ Y in Y . Thus Y is monotonically monolithic by Lemma 12. Similarly, we can prove that Y is weakly monotonically monolithic if one can assign to each finite subset F of Y a countable collection NY (F ) of subsets of X such that for each A ⊂ Y , F ∈[A]<ω N (F ) contains a network at some point of (A ∩ Y ) \ A if A is not closed in Y . 2 Let Y be a subspace of a topological space X. Denote πY : Cp (X) → Cp (Y ) be the map of restricting a function in Cp (X) to Y , i.e. πY (f ) = f |Y for all f ∈ Cp (X). In [1, Proposition 0.4.1], it was proved that if Y is a dense subset of X then πY : Cp (X) → πY (Cp (X)) is a bijective continuous map. Theorem 14. Let Y be a dense subspace of a topological space X. If Cp (Y ) is monotonically monolithic and |X \ Y | ω, then Cp (X) is monotonically monolithic. Proof. By Lemma 12, we know that one can assign to each finite subset F of Cp (Y ) a countable collection N (F ) of subsets of Cp (Y ) such that for each A ⊂ Cp (Y ), F ∈[A]<ω N (F ) contains a network at each point of A(Cp (Y )) . Let Z = x∈X Rx , where Rx = R with the usual topology for each x ∈ X. Thus Cp (X) is a subspace of Z. Denote πY : Z → x∈Y Rx the usual projection. Since Y is dense in X, the map g = πY |Cp (X) : Cp (X) → Cp (Y ) is injective [1, Proposition 0.4.1]. Let F be any finite subset of Cp (X). Then g(F ) is a finite subset of Cp (Y ). Thus N (g(F )) is a countable family of subsets of Cp (Y ). Denote N (F ) = {πY−1 (B) ∩ ( in πx−1 (Ui )): B ∈ N (g(F )), n ∈ N, xi ∈ X \ Y for each i n, and Ui ∈ B(R)}. i Since |N (g(F ))| ω and |X \ Y | ω, we have |N (F )| ω. Thus N (F ) is a countable family of subsets of the space Z. Let A ⊂ Cp (X) and f ∈ A(Cp (X)) . If f = f |Y , then f ∈ Cp (Y ) and f ∈ πY (A)(Cp (Y )) . Let O be any open neighborhood of f in Z. Thus there is some n ∈ N and xi ∈ X, Ui ∈ B(R) for each i n such that f ∈ in πx−1 (Ui ) ⊂ O. We can assume that x1 , x2 , . . . , xl ∈ Y and xl+1 , xl+2 , . . . , xn ∈ X \ Y i for some l < n. If Ml = {t: t ∈ Cp (Y ) and t(xi ) ∈ Ui for each i l}, then Ml is an open neighborhood of f in Cp (Y ). Since f ∈ πY (A)(Cp (Y )) , there is a finite set F ⊂ πY (A) and some B ∈ N (F ) such that f ∈ B ⊂ Ml . The set F ⊂ πY (A) = g(A) ⊂ πY (Cp (X)) = g(Cp (X)) ⊂ Cp (Y ), then g −1 (F ) ⊂ Cp (X). Since the map g is injective and |F | < ω, we have |g −1 (F )| < ω and g −1 (F ) ⊂ A. Denote F = g −1 (F ). Since f (xi ) ∈ Ui for each l +1 i n, we have f ∈ l+1in πx−1 (Ui ). Since πY (f ) = f ∈ B ⊂ Ml , we have i f ∈ πY−1 (B) ⊂ πY−1 (Ml ) ⊂ 1il πx−1 (Ui ). So f ∈ πY−1 (B) ∩ ( l+1in πx−1 (Ui )) ⊂ 1in πx−1 (Ui ) ⊂ O. i i i Since B ∈ N (F ) and xi ∈ X \ Y for each l + 1 i n, we have πY−1 (B) ∩ ( l+1in πx−1 (U )) ∈ N (F ). i i Thus Cp (X) is monotonically monolithic by Lemma 13. Similar with one point compactification of a locally compact Hausdorff space, we know that for every locally Lindelöf regular space X there exists a Lindelöf space Z such that X is a dense subspace of Z and |Z \ X| = 1. The space Z is called a one point Lindelöfication of X. Corollary 15. If X is a locally Lindelöf regular space and Cp (X) is monotonically monolithic, then for any one point Lindelöfication Z of X, the space Cp (Z) is monotonically monolithic.
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In what follows, we discuss some results on k-in-countable bases and k-in-countable weak bases, where k ∈ N. Given a natural number k, a family A of subsets of a space X is k-in-countable if every B ⊂ X with |B| = k is contained in at most countably many elements of A [1]. We know that every point-countable family A of subsets of a space X is 1-in-countable. In [4], it was proved that every countably compact regular space with a k-in-countable base for some k ∈ N is metrizable. In [22], it was proved that if k ∈ N and X is a star compact space with a k-in-countable base then X is metrizable. To get this conclusion, it was also proved that if X is a space with a k-in-countable base for some k ∈ N then e(X) = l(X) [22]. We know that if X is a D-space then e(X) = l(X). In what follows, we show that if a space X has a k-in-countable weak base for some k ∈ N then X is a D-space. Let us review the definition of weak bases [17]. A collection P = {Px : x ∈ X} is called a weak base for X, if for any x ∈ X the following conditions hold: (1) For each x ∈ X, Px is closed under finite intersections and x ∈ Px ; (2) A subset U of X is open if and only if for any x ∈ U , there is some B ∈ Px such that x ∈ B ⊂ U. In [13], it was proved that if a topological space X has a point-countable weak base then X is a weakly monotonically monolithic space. Let A be a set. We denote [A]<ω = {F : F ⊂ A and |F | < ω}. If |A| k, where k ∈ N, then we denote [A]k = {F : F ⊂ A and |A| = k}. Theorem 16. Let k ∈ N. If X has a k-in-countable family F such that for each non-closed subset A of X the family {F ∈ F: there is some B ⊂ A such that |B| = k and B ⊂ F } contains a network at some point of A \ A, then X is a weak monotonically monolithic space. Proof. Since X is a T1 -space, every finite subset of X is closed. Let B ⊂ X be a finite subset of X. If |B| < k, then we denote N (B) = {X}. If |B| k, then we denote N (B) = {F ∈ F: there is some C ⊂ B such that |C| = k and C ⊂ F } ∪ {X}. Since F is a k-in-countable family of subsets of X and B is a finite subset of X, we have |N (B)| ω. Let A ⊂ X. If A is not closed in X, then |A| ω. Thus {F ∈ F: there is some B ⊂ A such that |B| = k and B ⊂ F } ∪ {X} = B∈[A]k N (B) contains a network at some point of A \ A. By the definition of N (B) for each finite subset B of X, we have B∈[A]k N (B) = B∈[A]<ω N (B). So B∈[A]<ω N (B) contains a network at some point of A \ A. Thus X is a weakly monotonically monolithic space by Lemma 12. 2 Lemma 17. Let X be a topological space. Let P = {Px : x ∈ X} be a weak base of X. If A ⊂ X is not closed in X, then there is some x ∈ A \ A such that |B ∩ A| ω for each B ∈ Px . Proof. Suppose that for each x ∈ A \ A there is some Bx ∈ Px such that |Bx ∩ A| < ω. The set X \ (Bx ∩ A) is an open neighborhood of x, so there is some Bx ∈ Px such that Bx ⊂ Bx and x ∈ Bx ⊂ X \(Bx ∩A). Thus Bx ∩A = ∅. For each x ∈ X \A, there is some Bx ∈ Px such that Bx ∩A = ∅. Thus X \A = {Bx : x ∈ X \A}, where Bx ∈ Px for each x ∈ X\A. So the set X\A is an open set of X. Thus A is closed in X. A contradiction. Thus there is some x ∈ A \ A such that |B ∩ A| ω for each B ∈ Px . 2 Theorem 18. Let k ∈ N. If a topological space X has a k-in-countable weak base, then X is a weakly monotonically monolithic space. Proof. Let P = {Px : x ∈ X} be a k-in-countable weak base for X. Let A ⊂ X. If A is not closed, then there is some x ∈ A \ A such that |P ∩ A| ω for each P ∈ Px by Lemma 17. Let U be any open neighborhood of x in X. There is some P ∈ Px such that x ∈ P ⊂ U . So |P ∩ A| ω. Let B1 ⊂ P ∩ A such that |B1 | = k. Thus P ∈ {F ∈ P: there is some B ⊂ A such that |B| = k and B ⊂ F }. So the family {F ∈ P: there is some B ⊂ A such that |B| = k and B ⊂ F } contains a network at x. Thus X is a weakly monotonically monolithic space by Theorem 16. 2
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Proposition 19. ([13]) If X is a weakly monotonically monolithic space, then X is a D-space. By Theorem 18 and Proposition 19, we have Theorem 20. Let k ∈ N. If X has a k-in-countable weak base (or base), then X is a D-space. Corollary 21. If X has a k-in-countable weak base, then l(X) = e(X). Corollary 22. ([22, Lemma 2.3]) If X has a k-in-countable base, then l(X) = e(X). Acknowledgement The author would like to thank the referee for many valuable remarks, corrections, and suggestions which greatly improved the paper. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22]
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