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The projective Menger property and an embedding of Sω into function spaces
Topology and its Applications 220 (2017) 118–130
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Virtual Special Issue – International Conference on Topology, Messina (ICTM2015)
The projective Menger property and an embedding of Sω into function spaces Masami Sakai 1 Department of Mathematics, Kanagawa University, Hiratsuka 259-1293, Japan
a r t i c l e
i n f o
Article history: Received 8 December 2015 Received in revised form 10 January 2016 Accepted 15 September 2016 Available online 10 February 2017 MSC: 54C25 54C35 54D20
a b s t r a c t For a Tychonoff space X, we denote by Cp (X) the space of all real-valued continuous functions on X with the topology of pointwise convergence. In this paper, we show that (1) Cp (X, I) is projectively Menger if and only if X is b-discrete (i.e., every countable subset of X is closed and C ∗ -embedded in X), (2) there is a Menger space L such that the sequential fan Sω can be embedded into Cp (L). The first (1) enables us to give a direct proof of Arhangel’skii’s theorem [2, Theorem 6]: If Cp (X) is Menger, then X is finite. The second (2) is an affirmative answer to Arhangel’skii’s problem [5, Problem II.2.7] under CH (the continuum hypothesis). © 2017 Elsevier B.V. All rights reserved.
Keywords: Menger Projectively Menger Function space The sequential fan Whyburn
1. Introduction Throughout this paper, all spaces are assumed to be Tychonoff. The set of positive integers is denoted by N. Let R be the real line, we put D = {0, 1}, and I = [0, 1] ⊂ R. For a space X, we denote by Cp (X) the space of all real-valued continuous functions on X with the topology of pointwise convergence. The symbol 0 stands for the constant function to 0. A basic open neighborhood of 0 is of the form [F, (−ε, ε)] = {f ∈ Cp (X) : f (F ) ⊂ (−ε, ε)}, where F is a finite subset of X and ε > 0. For a subset Y ⊂ R, we put Cp (X, Y ) = {f ∈ Cp (X) : f (X) ⊂ Y }. Let Sω = {∞} ∪ {(n, m) : n, m ∈ N} be the sequential fan, where each (n, m) is isolated in Sω and a basic open neighborhood of ∞ is of the form N (ϕ) = {∞} ∪ {(n, m) : n ∈ N, m ≥ ϕ(n)} for a function ϕ ∈ NN . In other words, Sω is the quotient space obtained by identifying the limits of countably many convergent sequences.
1
E-mail address: [email protected]. The author was supported by JSPS KAKENHI Grant Number 25400213.
http://dx.doi.org/10.1016/j.topol.2017.02.017 0166-8641/© 2017 Elsevier B.V. All rights reserved.
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A space X is said to be Menger [12] (or, [18] as a survey) if for every sequence {Un : n ∈ N} of open covers of X, there are finite subfamilies Vn ⊂ Un such that {Vn : n ∈ N} is a cover of X. In the literature, a Menger space is sometimes called a Hurewicz space, but nowadays the term “Hurewicz space” stands for a stronger property. Every σ-compact space is Menger, and a Menger space is Lindelöf. The Menger property is closed hereditary, and it is preserved by continuous maps. It is well known that the Baire space NN (hence, Rω ) is not Menger. The first motivation of this paper is Arhangel’skii’s theorem below. Theorem 1.1 ([2, Theorem 6], [3]). For a space X, Cp (X) is Menger if and only if X is finite. To prove this theorem, Arhangel’skii used the notion of a R-quotient map and some properties of such a map, the fact that every analytic Menger space is σ-compact and N.V. Velichko’s theorem: Cp (X) is σ-compact if and only if X is finite. Thus, the proof needs several extra arguments about the Menger property, and the Menger property is not applied to Cp (X) directly. In this paper, applying the Menger property to Cp (X) directly, we give a direct proof of Arhangel’skii’s theorem. To this end, it is useful to investigate projectively Menger Cp (X, I) and Cp (X). Definition 1.2 ([8,5]). A space X is projectively Menger (resp., projectively σ-compact) if every second countable continuous image of X is Menger (resp., σ-compact). Every Menger space is projectively Menger. It is known [8, Proposition 8 (1)] that a space is Menger if and only if it is Lindelöf and projectively Menger. A subset A of a space X is said to be bounded in X if for every continuous function f : X → R, f A : A → R is a bounded function. We can easily observe that a subset A ⊂ X is bounded in X if and only if for every discrete family D consisting of nonempty open subsets of X, {D ∈ D : D ∩ A = ∅} is finite. Hence, if A is a bounded subset in a second countable space, then A is compact in the second countable space. Therefore, every σ-bounded space (i.e., a space which is the union of countably many bounded subsets) is projectively σ-compact (hence, projectively Menger) [5, Proposition 1.1]. Among several characterizations of a projectively Menger space, the following one is useful. Theorem 1.3 ([8, Theorem 6]). A space X is projectively Menger if and only if for every sequence {Un : n ∈ N} of countable covers of X consisting of cozero-sets in X, there are finite subfamilies Vn ⊂ Un such that {Vn : n ∈ N} is a cover of X. A space X is said to have countable fan-tightness [2] if whenever An ⊂ X and x ∈ An (n ∈ ω), there are finite sets Fn ⊂ An such that x ∈ {Fn : n ∈ ω}. Obviously Sω does not have countable fan-tightness. Arhangel’skii [2, Theorem 4] proved that every finite power of X is Menger if and only if Cp (X) has countable fan-tightness. Hence, if every finite power of X is Menger, Sω cannot be embedded into Cp (X). The second motivation of this paper is Arhangel’skii’s problem below. Problem 1.4 ([5, Problem II.2.7]). Can Sω be embedded into Cp (X) for some Menger space X? In the third section, we solve this problem in the affirmative under CH. Also we will discuss some connections among the projective Menger property, an embedding of Sω into Cp (X) and the Whyburn property.
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2. The projective Menger property of function spaces A subset S of a space X is said to be C ∗ -embedded in X [11] if every continuous real-valued bounded function on S can be extended to a continuous function on X. Urysohn’s extension theorem [11, 1.17] says that a subset S of a space X is C ∗ -embedded in X if and only if for any disjoint zero-sets A and B in S, there are disjoint zero-sets ZA and ZB in X such that A ⊂ ZA and B ⊂ ZB . According to Tkačuk [22], a space X is said to be b-discrete if every countable subset of X is closed (equivalently, closed and discrete) and C ∗ -embedded in X. The following lemma can be proved easily by Urysohn’s extension theorem, but for completeness we give a proof. Lemma 2.1. The following are equivalent for a space X: (1) X is b-discrete; (2) For any disjoint countable subsets A and B in X, there are disjoint zero-sets ZA and ZB in X such that A ⊂ ZA and B ⊂ ZB ; (3) For any disjoint countable subsets A and B in X such that A is closed in X, there are disjoint zero-sets ZA and ZB in X such that A ⊂ ZA and B ⊂ ZB . Proof. (1)→(2): Let A and B be disjoint countable subsets in X. Then, A ∪ B is a closed and discrete, C ∗ -embedded subset in X. Since both A and B are zero-sets in A ∪ B, by Urysohn’s extension theorem, we can take disjoint zero-sets ZA and ZB in X such that A ⊂ ZA and B ⊂ ZB . (2)→(3) is trivial. (3)→(1): Let B be any countable subset in X, and take any point a ∈ X \ B. Applying the condition (3) to A = {a} and B, we can see that B is closed in X. Thus every countable subset of X is closed and discrete. Let S be a countable subset of X, and let A and B be disjoint subsets in S. Since A is closed in X, we can take disjoint zero-sets ZA and ZB in X such that A ⊂ ZA and B ⊂ ZB . By Urysohn’s extension theorem, S is C ∗ -embedded in X. 2 Tkačuk proved the following two theorems, where the statement (4) in Theorem 2.3 is due to Arhangel’skii [5, Theorem 3.2]. Theorem 2.2 ([22]). For a space X, the following are equivalent: (1) (2) (3) (4)
X is b-discrete; Cp (X, I) is pseudocompact; Cp (X, I) is σ-pseudocompact; Cp (X, I) is σ-bounded.
Theorem 2.3 ([22]). For a space X, the following are equivalent: (1) (2) (3) (4)
X is pseudocompact and b-discrete; Cp (X) is σ-pseudocompact; Cp (X) is σ-bounded; Cp (X) is projectively σ-compact (A.V. Arhangel’skii).
Therefore, if X is b-discrete (resp., pseudocompact and b-discrete), then Cp (X, I) (resp., Cp (X)) is projectively Menger. We show that the converses hold. Theorem 2.4. A space X is b-discrete if and only if Cp (X, I) is projectively Menger.
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Proof. If X is b-discrete, Cp (X, I) is projectively Menger by Theorem 2.2. Conversely, assume that Cp (X, I) is projectively Menger. We show the statement (3) in Lemma 2.1. Let A and B be disjoint countable subsets in X such that A is closed in X. Let B = {bn : n ∈ N}, and let Bn = {b1 , . . . , bn }. For each n, m ∈ N, we put Zn,m = {f ∈ Cp (X, I) : f (A) = {0} and f (Bm ) ⊂ [1/2n , 1]}. Since A and Bm are countable, each Zn,m is a zero-set in Cp (X, I). Assume that {Zn,m : m ∈ N} = ∅ for all n ∈ N. Using the projective Menger property of Cp (X, I), by Theorem 1.3, we can take some ϕ ∈ NN such that {Zn,ϕ(n) : n ∈ N} = ∅. For each n ∈ N, take any gn ∈ Cp (X, I) satisfying gn (A) = {0} and ∞ gn (Bϕ(n) ) = {1}. Let g = j=1 2−j gj . Then, g ∈ Cp (X, I) and g(A) = {0}. Fix any n ∈ N and 1 ≤ k ≤ ϕ(n). Then we have g(bk ) =
∞
2−j gj (bk ) ≥ 2−n gn (bk ) = 2−n .
j=1
Hence, g ∈ {Zn,ϕ(n) : n ∈ N}. This is a contradiction. Thus, there is some n ∈ N such that {Zn,m : m ∈ N} = ∅. Let h ∈ {Zn,m : m ∈ N}, then A ⊂ ZA = h−1 (0) and B ⊂ ZB = h−1 ([1/2n , 1]). 2 Asanov [6] proved that if Cp (X) is Lindelöf, then every finite power of X has countable tightness. The following is a special case of Asanov’s theorem. Lemma 2.5. If Cp (X, I) is Lindelöf, then X has countable tightness. Proof. Assume x ∈ A \ A for a subset A ⊂ X. Let Z = {f ∈ Cp (X, I) : f (x) = 0}. Since Z is closed in Cp (X, I), it is Lindelöf. For each a ∈ A, let Ua = {f ∈ Cp (X, I) : f (a) < 1}. Since {Ua : a ∈ A} is an open family in Cp (X, I) covering Z, there is a countable subset B ⊂ A such that Z ⊂ {Ua : a ∈ B}. Then we have x ∈ B. 2 Corollary 2.6. A space X is discrete if and only if Cp (X, I) is Menger. Proof. If X is discrete, then Cp (X, I) = IX is compact. Conversely, assume that Cp (X, I) is Menger. By Theorem 2.4 and Lemma 2.5, X is a b-discrete space of countable tightness. Then X must be discrete. 2 Now we can show Arhangel’skii’s theorem directly without using extra arguments on R-quotient maps, analytic spaces, or Velichko’s theorem. Corollary 2.7 (A.V. Arhangel’skii). A space X is finite if and only if Cp (X) is Menger. Proof. If X is finite, Cp (X) = RX is σ-compact. Conversely, assume that Cp (X) is Menger. By Corollary 2.6, we have Cp (X) = RX , so X must be finite. 2 Moreover, we note that the projective Menger property of Cp (X) can be characterized by a pseudocompact b-discrete space X. Theorem 2.8. A space X is pseudocompact and b-discrete if and only if Cp (X) is projectively Menger. Proof. If X is pseudocompact and b-discrete, by Theorem 2.3, Cp (X) is projectively Menger. Conversely, assume that Cp (X) is projectively Menger. Since Cp (X, I) is a retract of Cp (X), it is also projectively
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Menger. By Theorem 2.4, X is b-discrete. If X is not pseudocompact, since there is a countably infinite discrete family of nonempty open subsets in X, we can take a countably infinite C-embedded subset A in X. Hence, RA is a second countable continuous image of Cp (X), but it is not Menger. This is a contradiction. 2 An example of an infinite pseudocompact b-discrete space is constructed in Shakhmatov [21]. The space of all weak P -points in βω \ ω is also an infinite pseudocompact b-discrete space: see [4, Example 6.4]. The topological sum of b-discrete spaces is b-discrete, and Cp ( α<κ Xα , I) (resp., Cp ( α<κ Xα )) is homeomorphic to α<κ Cp (Xα , I) (resp., α<κ Cp (Xα )). Combining these facts and Tkačuk’s theorems above, we have: Proposition 2.9. The following assertions hold. (1) If Cp (X, I) is projectively Menger, then any power of Cp (X, I) is pseudocompact, (2) If Cp (X) is projectively Menger, then any finite power of Cp (X) is σ-pseudocompact. We conclude this section with the projective Menger property of the realcompactification υX of a space X. Proposition 2.10. A space X is projectively Menger if and only if the realcompactification υX of X is projectively Menger. Proof. Assume that X is projectively Menger, and let Un (n ∈ N) be a sequence of countable covers of υX consisting of cozero-sets in υX. By the projective Menger property of X, there are finite subfamilies Vn ⊂ Un such that X ⊂ { Vn : n ∈ N}. Since { Vn : n ∈ N} is a cozero-set in υX, we have { Vn : n ∈ N} = υX. Because, every zero-set in υX is the closure of a zero-set in X [11, 8D. 1]. Conversely, assume that υX is projectively Menger, and let Un (n ∈ N) be a sequence of countable covers of X consisting of cozero-sets in X. For each cozero-set U in X, fix a cozero-set U in υX with U = U ∩ X. Then each Un = {U : U ∈ Un } is a cover of υX, because of [11, 8D. 1]. Apply the projective Menger property of υX to these Un . 2 If a space Y satisfies X ⊂ Y ⊂ υX, then obviously υY = υX holds. Corollary 2.11. Let X be a projectively Menger space. Then every space Y with X ⊂ Y ⊂ υX is also projectively Menger. 3. An embedding of the sequential fan Sω into Cp (X) In this section, we clarify the question of when Sω can be embedded into Cp (X). An infinite cover {An : n ∈ N} of a set X is said to be a γ-cover if every point of X is contained in An for all but finitely many n ∈ N. A cover U of a set X is said to be an ω-cover [10] if every finite subset of X is contained in some member of U. A γ-cover is an ω-cover. Note that if U is an ω-cover of a set X and X∈ / U, then each finite subset of X is contained in infinitely many members of U. Lemma 3.1 ([8, Lemma 80]). Let X = {x} ∪ {xn,m : n, m ∈ N} be a T2 -space such that xn,m → x (m → ∞) for each n ∈ N, and for any ϕ ∈ NN , x ∈ / {xn,m : n ∈ N, m ≤ ϕ(n)}. Then Sω can be embedded into X. The following theorem may be regarded as a reformulation of [8, Theorem 76] which was proved under a strong additional condition, which was called Sakai’s condition. Theorem 3.2. For a space X, the following are equivalent:
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(1) Sω can be embedded into Cp (X); (2) There are sequences Un = {Un,m : m ∈ N} (n ∈ N) of covers of X consisting of cozero-sets in X and Zn = {Zn,m : m ∈ N} (n ∈ N) of covers of X consisting of zero-sets in X such that each Zn is a γ-cover of X, Zn,m ⊂ Un,m and for any ϕ ∈ NN , {Un,m : n ∈ N, m ≤ ϕ(n)} is not an ω-cover of X. Proof. (1)→(2). Let Sω = {0} ∪ {fn,m : n, m ∈ N} ⊂ Cp (X), where fn,m → 0 (m → ∞). For each n, m ∈ N, we put Un,m = {x ∈ X : |fn,m (x)| <
1 n },
Zn,m = {x ∈ X : |fn,m (x)| ≤
1 n+1 }.
Each Un,m (resp., Zn,m ) is a cozero-set (resp., zero-set) in X with Zn,m ⊂ Un,m . Let Un = {Un,m : m ∈ N} and Zn = {Zn,m : m ∈ N}. If I = {n ∈ N : X ∈ Un } is infinite, some sequence {fn,mn : n ∈ I} converges to 0 uniformly. This is a contradiction, so without loss of generality, we may assume Un,m = X for each n, m ∈ N. We can easily check that the condition fn,m → 0 (m → ∞) implies that Zn is a γ-cover of X. Let ϕ ∈ NN . Assume that Uϕ = {Un,m : n ∈ N, m ≤ ϕ(n)} is an ω-cover of X. By the condition 0∈ / {fn,m : n ∈ N, m ≤ ϕ(n)}, there are a finite set F ⊂ X and a k ∈ N such that [F, (− k1 , k1 )] ∩ {fn,m : n ∈ N, m ≤ ϕ(n)} = ∅. Since F is contained in infinitely many members of Uϕ , we have F ⊂ Un,m for some n > k and m ≤ ϕ(n). This implies |fn,m (x)| < n1 < k1 for all x ∈ F , in other words, fn,m ∈ [F, (− k1 , k1 )]. This is a contradiction. (2)→(1). For each n, m ∈ N, we take a continuous function fn,m : X → I such that fn,m (x) = 0 for all x ∈ Zn,m and fn,m (x) = 1 for all x ∈ X \ Un,m . Then fn,m → 0 (m → ∞). Let ϕ ∈ NN . Since Uϕ = {Un,m : n ∈ N, m ≤ ϕ(n)} is not an ω-cover of X, there is a finite subset F ⊂ X such that F is not contained in any member of Uϕ . Then we can easily check [Fϕ , (− 12 , 12 )] ∩ {fn,m : n ∈ N, m ≤ ϕ(n)} = ∅. By Lemma 3.1, Sω can be embedded into {0} ∪ {fn,m : n, m ∈ N} ⊂ Cp (X). 2 A space X is said to be strongly zero-dimensional if any disjoint zero sets in X can be separated by a clopen set in X. Using Theorem 3.2 and its proof, we have the following corollary immediately, where strong zero-dimensionality is used only for (1) → (2), the equivalence of (2) and (3) is true for any space X, and it is due to Bonanzinga, Cammaroto and Matveev [8, Theorem 78].2 Corollary 3.3. For a strongly zero-dimensional space X, the following are equivalent: (1) Sω can be embedded into Cp (X); (2) There is a sequence Un = {Un,m : m ∈ N} (n ∈ N) of γ-covers of X consisting of clopen sets in X such that for any ϕ ∈ NN , {Un,m : n ∈ N, m ≤ ϕ(n)} is not an ω-cover of X; (3) Sω can be embedded into Cp (X, D). Let X = Cp (Sω ). Then obviously Sω can be embedded into Cp (X). But Sω cannot be embedded into Cp (X, D), because X is connected. The author does not know if this corollary is true for a zero-dimensional space X. In general, if Sω can be embedded into j∈ω Xj , then it can be embedded into j
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Proof. We show only the case of Cp (X). Assume that Sω can be embedded into Cp (X)ω . Then Sω can be embedded into Cp (X)k for some k ∈ N. Since Cp (X)k is homeomorphic to Cp (Y ), where Y = X1 · · · Xk and each Xj is a homeomorphic copy of X, Y has sequences Un = {Un,m : m ∈ N} and Zn = {Zn,m : m ∈ N} (n ∈ N) in Theorem 3.2 (2). For each 1 ≤ j ≤ k, let idj : X → Xj be the identity map. For each n, m ∈ N, we put Un,m =
−1 {idj (Un,m ∩ Xj ) : 1 ≤ j ≤ k}, Zn,m = {id−1 j (Zn,m ∩ Xj ) : 1 ≤ j ≤ k}.
Then each Un,m (resp., Zn,m ) is a cozero-set (resp., zero-set) in X and Zn,m ⊂ Un,m is satisfied. Each Zn = {Zn,m : m ∈ N} is a γ-cover of X, because for each x ∈ X {idj (x) : 1 ≤ j ≤ k} is contained in Zn,m for all but finitely many m ∈ N. Assume that there is a ϕ ∈ NN such that {Un,m : n ∈ N, m ≤ ϕ(n)} is an ω-cover of X. Then for each finite subset F ⊂ X, there are n, m ∈ N such that F ⊂ Un,m . This implies {idj (F ) : 1 ≤ j ≤ k} ⊂ Un,m , hence {Un,m : n ∈ N, m ≤ ϕ(n)} is an ω-cover of Y . This is a contradiction. By Theorem 3.2, Sω can be embedded into Cp (X). 2
Finally, we solve Problem 1.4 in the first section in the affirmative under CH. Remark 3.5. Let (Z, +) be the usual discrete group of integers. In [13, Lemma 2.6], under CH the authors constructed a Lusin set L ⊂ Zω (i.e., L is uncountable and its intersection with every first category set in Zω is countable) such that L + L = Zω . Though L is a Lusin set in Zω , by a homeomorphism between Zω and the space of irrationals of R, it is a Lusin set in R. A space X is said to be Rothberger if for every sequence {Un : n ∈ ω} of open covers of X, there are some members Un ∈ Un such that {Un : n ∈ ω} is a cover of X. Every Rothberger space is trivially Menger. Since any Lusin set is Rothberger [13, Theorem 2.7], L is Menger. However, L × L is not Menger, because L + L = Zω implies that Zω is the continuous image of L × L. In [13, Theorem 2.8], using the condition L + L = Zω , the authors proved that this Lusin set L has a sequence Un = {Un,m : m ∈ ω} (n ∈ ω) of increasing covers of L consisting of clopen sets in L such that for each ϕ ∈ ω ω , {Un,ϕ(n) : n ∈ ω} is not an ω-cover of L. Applying the equivalence of (2) and (3) in Corollary 2.6 due to Bonanzinga, Cammaroto and Matveev, Sω can be embedded into Cp (L, D) ⊂ Cp (L). In general, if a space of countable tightness has a homeomorphic copy of Sω , then Sω can be embedded into X as a closed set [15, Corollary 3.9]. Therefore, since Cp (L) has countable tightness, Sω can be embedded into Cp (L, D) as a closed set. On the other hand, in [13, Theorem 2.13], under CH the authors constructed a Lusin set H ⊂ R such that every finite power of H is Rothberger. For this H, Sω cannot be embedded into Cp (H), because Cp (H) has countable fan-tightness. In contrast with a Lusin set, for any Sierpiński set S (i.e., S is an uncountable subset of R and its intersection with every Lebesgue measure zero set in R is countable), Sω cannot be embedded into Cp (S) by [19, Corollary 11]. 4. An embedding of Sω into Cp (X) and related properties In this section, we examine some connections among the projective Menger property, an embedding of Sω into Cp (X) and the Whyburn property. Proposition 4.1. If Sω cannot be embedded into Cp (X), then X is projectively Menger. Proof. Let Un = {Un,m : m ∈ N} (n ∈ N) be a sequence of covers of X consisting of cozero-sets in X. For each n, m ∈ N, we put Vn,m = Un,1 ∪ · · · ∪ Un,m . Since Vn = {Vn,m : m ∈ N} is an increasing cover of X consisting of cozero-sets in X, we can find zero-sets Zn,m in X such that Zn,m ⊂ Vn,m and
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Zn = {Zn,m : m ∈ N} is an increasing cover of X. Applying Theorem 3.2 to Vn and Zn , we can take some ϕ ∈ NN such that {Vn,ϕ(n) : n ∈ N} is an ω-cover of X. This implies that {Un,m : n ∈ N, m ≤ ϕ(n)} is a cover of X. 2 The converse of this proposition is not true. Recall the Lusin set L mentioned in Remark 3.5, L is Menger, but Sω can be embedded into Cp (L). In [8, Theorem 18], the authors noted that if every finite power of X is projectively Menger, then every countable subspace of Cp (X) has countable fan-tightness. We make clear when every countable subspace of Cp (X) has countable fan-tightness. It is known [13, Theorem 3.9] that every finite power of X is Menger if and only if for any sequence Un (n ∈ N) of open ω-covers of X, there are finite subfamilies Vn ⊂ Un such that {Vn : n ∈ N} is an ω-cover of X. This latter covering property is denoted by the symbol Sf in (Ω, Ω) in [13]. Definition 4.2. A space X is projectively Sf in (Ω, Ω) if every second countable continuous image of X satisfies Sf in (Ω, Ω). Theorem 4.3. For a space X, the following are equivalent: (1) Every countable subspace of Cp (X) has countable fan-tightness; (2) For any sequence Un = {Un,m : m ∈ N} (n ∈ N) of ω-covers of X consisting of cozero-sets in X, there is some ϕ ∈ NN such that {Un,m : n ∈ N, m ≤ ϕ(n)} is an ω-cover of X; (3) X is projectively Sf in (Ω, Ω). Proof. (1)→(2). Let Un = {Un,m : m ∈ N} (n ∈ N) be a sequence of ω-covers of X consisting of cozero-sets in X. We can put Un,m = {Zn,m,l : l ∈ N}, where each Zn,m,l is a zero-set in X and Zn,m,l ⊂ Zn,m,l+1 . −1 −1 Take a continuous function fn,m,l : X → I satisfying fn,m,l (0) = Zn,m,l and fn,m,l (1) = X \ Un,m . For each n ∈ N, let An = {fn,m,l : m, l ∈ N}. By the facts that each Un is an ω-cover of X and {Zn,m,l : l ∈ N} is increasing, we can see 0 ∈ {An : n ∈ N}. By our assumption (1), there is some ϕ ∈ NN such that 0 ∈ {fn,m,l : n ∈ N, m, l ≤ ϕ(n)}. Then {Un,m : n ∈ N, m ≤ ϕ(n)} is an ω-cover of X. Indeed, let F be a finite subset of X, and consider the basic open neighborhood [F, (− 21 , 12 )] of 0. Take n ∈ N and m, l ≤ ϕ(n) such that fn,m,l ∈ [F, (− 12 , 12 )]. This implies F ⊂ Un,m . (2)→(3). Let f : X → Y be a continuous map onto a second countable space Y . Let Un (n ∈ N) be a sequence of open ω-covers of Y . Every member of Un is a cozero-set in Y , and let us recall [10] that every finite power of a space is Lindelöf if and only if every open ω-cover of the space has a countable ω-subcover. Hence, we may assume that each Un is a countable ω-cover of Y consisting of cozero-sets in Y . Considering the ω-covers f −1 (Un ) = {f −1 (U ) : U ∈ Un } of X, we have (3). (3)→(1). Take any countable subset A = {fn : n ∈ N} ⊂ Cp (X), and consider the continuous map f : X → RN defined by f (x) = (fn (x))n∈N . By our assumption (3), every finite power of f (X) is Menger, hence Cp (f (X)) has countable fan-tightness by Arhangel’skii’s theorem mentioned in the first section. Consider the embedding f ∗ of Cp (f (X)) into Cp (X) induced by f . Let pn : f (X) → R be the projection of f (X) to the n-th coordinate. Then obviously f ∗ ({pn : n ∈ N}) = A. Thus, A has countable fan-tightness. 2 Thus, if X is projectively Sf in (Ω, Ω), then Sω cannot be embedded into Cp (X). But the converse is not true. Let S be a Sierpiński set in R such that S + S in R is the set of irrationals constructed in [13, p.250] under CH. As noted in Remark 3.5, Sω cannot be embedded into Cp (S), but the fact that S + S is the set of irrationals implies that S × S is not Menger. Hence, S is not projectively Sf in (Ω, Ω). The following proposition coincides with [8, Proposition 24]. Because the definition of Sf in (Ωcz , Ωcz ) in [8, Proposition 24] is just the condition (2) in Theorem 4.3.
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Proposition 4.4. If every finite power of X is projectively Menger, then X is projectively Sf in (Ω, Ω). Proof. Let Y be a second countable continuous image of X. Then each finite power Y n is a second countable continuous image of X n . Hence, Y n is Menger. 2 Remark 4.5. Bonanzinga et al. asked in [8, Question 25] whether Proposition 4.4 (i.e., Proposition 24 in [8]) can be reversed. The answer is trivially in the negative. Because, as mentioned in [8, p. 880], there is a space X which is the topological sum of two countably compact spaces such that every countable subspace of Cp (X) has countable fan-tightness, but X 2 is not projectively Menger. On the other hand, by Theorem 4.3, X is projectively Sf in (Ω, Ω). A space X is said to be Whyburn if whenever A ⊂ X and x ∈ A \ A, there is a subset B ⊂ A such that B = {x} ∪ B. This notion was considered in [24]. Every Fréchet–Urysohn T2 -space is obviously Whyburn. A sufficient condition for a space X to be Whyburn is: Lemma 4.6 ([7, Corollary 3.4]). If a space X has countable fan-tightness and each point of X is Gδ , then X is Whyburn. In particular, every countable space of countable fan-tightness is Whyburn. Definition 4.7 ([17]). A space X is ω-Whyburn if whenever A is a countable subset in X and x ∈ A \ A, there is a subset B ⊂ A such that B = {x} ∪ B. An ω-Whyburn space was called an APω -space in [8]. If a space X is ω-Whyburn, then every countable subspace of X is obviously Whyburn. But the converse does not hold. This fact was first observed by Angelo Bella. Example 4.8 (A. Bella). Let D be an infinite maximal almost-disjoint family consisting of infinite subsets of N. Let Ψ = N ∪ D be the Mrówka–Isbell space [11, 5I], and let Ψ∗ = {∞} ∪ Ψ be the one-point compactification of Ψ. Since every compact space of countable tightness has countable fan-tightness [2, Theorem 3], Ψ∗ has countable fan-tightness. By Lemma 4.6, every countable subspace of Ψ∗ is Whyburn. On the other hand, the point ∞ is in the closure of N, but every infinite subset of N has a limit point in D, thus Ψ∗ is not ω-Whyburn. In [8, Theorem 84 (2)], the authors showed that if Cp (X) is ω-Whyburn, then X is projectively Menger. We can show the following stronger assertion. Proposition 4.9. If every countable subspace of Cp (X) is Whyburn, then Sω cannot be embedded into Cp (X). Proof. Assume that Sω can be embedded into Cp (X). Then, as discussed in [17, Proposition 2.6], Sω ×(ω+1) can be embedded into Cp (X). However, the countable space Sω × (ω + 1) is not Whyburn: for example see [17, Proposition 1.1]. 2 Proposition 4.10. If X is projectively Sf in (Ω, Ω), then every countable subspace of Cp (X) is Whyburn. Proof. By Theorem 4.3, if X is projectively Sf in (Ω, Ω), then every countable subspace of Cp (X) has countable fan-tightness. Our conclusion follows from Lemma 4.6. 2 A family N of subsets of a space X is said to be a network for X if whenever x ∈ X and U is a neighborhood of x, there is some N ∈ N such that x ∈ N ⊂ U . A space X is said to be ℵ0 -stable [1] if
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whenever a space Y is a continuous image of X and Y has a continuous one-to-one map onto a second countable space, Y has a countable network. A σ-pseudocompact space is ℵ0 -stable [1, Propositions II.6.2 and II.6.6]. A space X is said to be ℵ0 -monolithic [1] if for every countable subset A ⊂ X, A has a countable network. Arhangel’skii [1] proved that a space X is ℵ0 -stable if and only if Cp (X) is ℵ0 -monolithic. Therefore, if a space X is σ-pseudocompact, the closure of each countable subset of Cp (X) has a countable network. Proposition 4.11. If X is σ-pseudocompact, then Cp (X) is ω-Whyburn. Proof. Let A be a countable subset of Cp (X) and assume that 0 ∈ A\A. Then, E = {0} ∪ A has a countable network, in particular each point of E is Gδ in E. Since a σ-pseudocompact space is projectively Sf in (Ω, Ω), by Theorem 4.3, {0} ∪A has countable fan-tightness. We use Bella and Yaschenko’s idea in [7, Corollary 3.4] (i.e., Lemma 4.6 in this paper). We put {x} = {Gn : n ∈ ω}, where each Gn is an open neighborhood of x in E, and Gn+1 ⊂ Gn is satisfied. Using countable fan-tightness of the space {0} ∪ A, we can take finite subsets Fn ⊂ Gn ∩A such that x ∈ {Fn : n ∈ ω}. Then we have {Fn : n ∈ ω} = {x} ∪ {Fn : n ∈ ω}. 2 We summarize implications observed in this section. X is σ-pseudocompact X n is projectively Menger for all n ∈ N ↓ ↓ X is projectively Sf in (Ω, Ω) Cp (X) is ω-Whyburn ↓ ↓ Every countable subset of Cp (X) is Whyburn ↓ Sω cannot be embedded into Cp (X) ↓ X is projectively Menger By this diagram, immediately we have: Proposition 4.12. Every finite power of X is projectively Menger if and only if for any n ∈ N, Sω cannot be embedded into Cp (X n ). Let us recall Proposition 2.9 (2): If Cp (X) is projectively Menger, then any finite power of Cp (X) is σ-pseudocompact. Hence, we can see that all conditions obtained by replacing X by Cp (X) in the diagram are equivalent. Corollary 4.13. For a space X, the following are equivalent: (1) (2) (3) (4)
Cp Cp (X) is ω-Whyburn; Every countable subset of Cp Cp (X) is Whyburn; Sω cannot be embedded into Cp Cp (X); X is pseudocompact and b-discrete.
Murtinová [14, Theorem 3] proved that Cp Cp (X) is Whyburn if and only if X is finite. 5. Remarks on selection principles Sf in (Γ, Ω) and Uf in (O, Ω) In this section, we remark that an embedding of Sω into Cp (X) is related with open problems on selection principles Sf in (Γ, Ω) and Uf in (O, Ω).
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A space X is said to satisfy Sf in (Γ, Ω) [13] if for every sequence {Un : n ∈ N} of open γ-covers of X, there are finite subfamilies Vn ⊂ Un such that {Vn : n ∈ N} is an ω-cover of X. A space X is said to satisfy Uf in (O, Ω) [13] if for every sequence {Un : n ∈ N} of countable open covers of X, there are finite subfamilies Vn ⊂ Un such that { Vn : n ∈ N} is an ω-cover of X. By a simple observation, we can see that Sf in (Γ, Ω) implies Uf in (O, Ω), and that every Lindelöf space with Uf in (O, Ω) is Menger. In addition, a space X is said to satisfy Uf in (O, Γ) [13] if for every sequence {Un : n ∈ N} of countable open covers of X, there are finite subfamilies Vn ⊂ Un such that every point of X is contained in Vn for all but finitely many n ∈ N. A Lindelöf space with Uf in (O, Γ) is called a Hurewicz space. We summarize the implications of these covering properties. Uf in (O, Γ) → Uf in (O, Ω) ↑ Sf in (Γ, Ω)
if Lindelöf
→
Menger
So far as the author knows, the following problems posed in [13, Problems 1 and 2] are still open, where the symbol Uf in (Γ, Ω) in [13, Problem 1] coincides with Uf in (O, Ω), and the symbol Uf in (Γ, Γ) in [13, Problem 2] coincides with Uf in (O, Γ). Problem 5.1 ([13, Problems 1 and 2], [23, Problem 2.1]). For a space of reals: (1) Is Uf in (O, Ω) = Sf in (Γ, Ω)? (2) And if not, does Uf in (O, Γ) imply Sf in (Γ, Ω)? Concerning these problems, we consider a special embedding of Sω into Cp (X). Definition 5.2. Sω can be embedded into Cp (X) (resp., Cp (X, D)) monotonically if there is an embedding e : Sω → Cp (X) (resp., e : Sω → Cp (X, D)) such that e(∞) = 0 and e((n, m)) ≥ e((n, m + 1)) for all n, m ∈ N. Proposition 5.3. For a space X, the following are equivalent: (1) Sω can be embedded into Cp (X) monotonically; (2) There is a sequence Un = {Un,m : m ∈ N} (n ∈ N) of increasing covers of X consisting of cozero-sets in X such that for any ϕ ∈ NN , {Un,ϕ(n) : n ∈ N} is not an ω-cover of X; (3) There is a sequence Un = {Un,m : m ∈ N} (n ∈ N) of covers of X consisting of cozero-sets in X such that for any ϕ ∈ NN , { m≤ϕ(n) Un,m : n ∈ N} is not an ω-cover of X. Proof. (1)→(2). Let e : Sω → Cp (X) be a monotonic embedding, and let e(Sω ) = {0} ∪ {fn,m : n, m ∈ N}. Note that each fn,m is non-negative. For each n, m ∈ N, we put Un,m = {x ∈ X : fn,m (x) < 1/2n }. The condition fn,m ≥ fn,m+1 implies Un,m ⊂ Un,m+1 , and the condition fn,m → 0 (m → ∞) implies X = {Un,m : m ∈ N}. Thus Un = {Un,m : m ∈ N} is an increasing cover of X consisting of cozero-sets in X. The rest of this proof can be done by the same arguments as in the proof of Theorem 3.2 (1) → (2). The equivalence of (2) and (3) is obvious. (2)→(1). Let Un = {Un,m : m ∈ N} (n ∈ N) be a sequence in the assertion (2). We take a zero-set Zn,m in X such that Zn,m ⊂ Un,m , Zn,m ⊂ Zn,m+1 and X = {Zn,m : m ∈ N} for n ∈ N. Let fn,m : X → I be a −1 −1 continuous function such that fn,m (0) = Zn,m and fn,m (1) = X \ Un,m . We put gn,m = fn,1 · fn,2 · · · fn,m . −1 Then obviously gn,m ≥ gn,m+1 , gn,m → 0 (m → ∞) and gn,m (1) = X \ Un,m . Let ϕ ∈ NN , and assume 0 ∈ {gn,m : n ∈ N, m ≤ ϕ(n)}. For any finite subset F ⊂ X, there is some gn,m ∈ [F, (−1/2, 1/2)] ∩ {gn,m :
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n ∈ N, m ≤ ϕ(n)}. This means F ⊂ Un,m ⊂ Un,ϕ(n) . Since {Un,ϕ(n) : n ∈ N} is not an ω-cover of X, this is a contradiction. By Lemma 3.1, Sω can be embedded into Cp (X) monotonically. 2 Thus, if X satisfies Uf in (O, Ω), Sω cannot be embedded into Cp (X) monotonically. The converse is true for a normal countably paracompact space X (in particular, for a space X of reals). Because, if {Un : n ∈ N} is an open cover of a normal countably paracompact space X, we can find a cover {Vn : n ∈ N} of X consisting of cozero-sets such that Vn ⊂ Un for all n ∈ N: for example see [9, Theorem 5.2.3]. Corollary 5.4. Let X be a normal countably paracompact space. Then, X satisfies Uf in (O, Ω) if and only if Sω cannot be embedded into Cp (X) monotonically. Corollary 5.5. If a normal countably paracompact space X satisfies Uf in (O, Γ), then Sω cannot be embedded into Cp (X) monotonically. Quite similarly to Proposition 5.3, we have: Proposition 5.6. For a space X, the following are equivalent: (1) Sω can be embedded into Cp (X, D) monotonically; (2) There is a sequence Un = {Un,m : m ∈ N} (n ∈ N) of increasing covers of X consisting of clopen sets in X such that for any ϕ ∈ NN , {Un,ϕ(n) : n ∈ N} is not an ω-cover of X; (3) There is a sequence Un = {Un,m : m ∈ N} (n ∈ N) of covers of X consisting of clopen sets in X such that for any ϕ ∈ NN , { m≤ϕ(n) Un,m : n ∈ N} is not an ω-cover of X. The Lusin set L in Remark 3.5 satisfies the condition (2) in Proposition 5.6, so Sω can be embedded into Cp (L, D) monotonically. By Propositions 5.3 and 5.6, we have: Corollary 5.7. For a strongly zero-dimensional space X, Sω can be embedded into Cp (X) monotonically if and only if it can be embedded into Cp (X, D) monotonically. In view of Theorem 3.2, we can see that if a space X satisfies Sf in (Γ, Ω), then Sω cannot be embedded into Cp (X). Therefore, we have the implications below for normal countably paracompact spaces X. Sω cannot be embedded into Cp (X) monotonically if Lindelöf Menger Uf in (O, Γ) → Uf in (O, Ω) → ↑ Sω cannot be embedded into Cp (X) ↑ Sf in (Γ, Ω) Concerning Problem 5.1, the following questions look interesting. Question 5.8. For a space X of reals: (1) If Sω can be embedded into Cp (X) (resp., Cp (X, D)), then can it be embedded into Cp (X) (resp., Cp (X, D)) monotonically? (2) Does Uf in (O, Γ) imply that Sω cannot be embedded into Cp (X)? (3) If Sω cannot be embedded into Cp (X), then does X satisfy Sf in (Γ, Ω)?
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If a space X of reals is a σ-set (i.e., every Gδ -subset of X is Fσ ), then every open γ-cover {Un : n ∈ N} of X has a closed γ-cover {Cn : n ∈ N} of X such that Cn ⊂ Un for all n ∈ N [16]. Hence, for a σ-set X of reals, Question 5.8 (3) is in the affirmative. Acknowledgements The author expresses gratitude to Angelo Bella and Boaz Tsaban. Angelo Bella kindly informed the author of Example 4.8. Boaz Tsaban pointed out that to solve Problem 1.4, the assumption cov(M) = cof (M) is enough instead of CH, in view of [20, Theorem 32]. References [1] A.V. Arhangel’skii, Factorization theorems and function spaces: stability and monolithicity, Sov. Math. Dokl. 26 (1982) 177–181. [2] A.V. Arhangel’skii, Hurewicz spaces, analytic sets and fan tightness of function spaces, Sov. Math. Dokl. 33 (1986) 396–399. [3] A.V. Arhangel’skii, Topological Function Spaces, Kluwer, 1992. [4] A.V. Arhangel’skii, Cp -theory, in: K. Kunen, J.E. Vaughan (Eds.), Handbook of Set-Theoretic Topology, Elsevier Sci. Pub., 1984, pp. 3–56. [5] A.V. Arhangel’skii, Projective σ-compactness, ω1 -caliber, and Cp -spaces, Topol. Appl. 104 (2000) 13–26. [6] M.O. Asanov, On cardinal invariants of spaces of continuous functions, Sov. Topol. Teor. Mnozhestv. Izhevsk 2 (1979) 8–12, in Russian. [7] A. Bella, I.V. Yaschenko, On AP and WAP spaces, Comment. Math. Univ. Carol. 40 (1999) 531–536. [8] M. Bonanzinga, F. Cammaroto, M. Matveev, Projective versions of selection principles, Topol. Appl. 157 (2010) 874–893. [9] R. Engelking, General Topology, revised and completed edition, Helderman Verlag, Berlin, 1989. [10] J. Gerlits, Zs. Nagy, Some properties of C(X). I, Topol. Appl. 14 (1982) 151–161. [11] L. Gillman, M. Jerison, Rings of Continuous Functions, reprint of the 1960 edition, Graduate Texts in Mathematics, vol. 43, Springer-Verlag, New York–Heidelberg, 1976. [12] W. Hurewicz, Über eine verallgemeinerung des Borelschen Theorems, Math. Z. 24 (1925) 401–421. [13] W. Just, A.W. Miller, M. Scheepers, P.J. Szeptycki, The combinatorics of open covers II, Topol. Appl. 73 (1996) 241–266. [14] E. Murtinová, On (weakly) Whyburn spaces, Topol. Appl. 155 (2008) 2211–2215. [15] T. Nogura, Y. Tanaka, Spaces which contains a copy of Sω or S2 and their applications, Topol. Appl. 30 (1988) 51–62. [16] M. Sakai, The sequence selection properties of Cp (X), Topol. Appl. 154 (2007) 552–560. [17] M. Sakai, Notes on strongly Whyburn spaces, Comment. Math. Univ. Carol. 57 (1) (2016) 123–129. [18] M. Sakai, M. Scheepers, The combinatorics of open covers, in: K.P. Hart, J. van Mill, P. Simon (Eds.), Recent Progress in General Topology III, Atlantic Press, 2014, pp. 751–799. [19] M. Scheepers, Cp (X) and Arhangel’skii’s αi -spaces, Topol. Appl. 89 (1998) 265–275. [20] M. Scheepers, B. Tsaban, The combinatorics of Borel covers, Topol. Appl. 121 (2002) 357–382. [21] D.B. Shakhmatov, A pseudocompact Tychonoff space all countable subsets of which are closed and C ∗ -embedded, Topol. Appl. 22 (1986) 139–144. [22] V.V. Tkačuk, The spaces Cp (X): decomposition into a countable union of bounded subspaces and completeness properties, Topol. Appl. 22 (1986) 241–253. [23] B. Tsaban, Selection principles and special sets of reals, in: E. Pearl (Ed.), Open Problems in Topology II, Elsevier B.V., 2007, pp. 91–108. [24] G.T. Whyburn, Mappings on inverse sets, Duke Math. J. 23 (1956) 237–240.
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Virtual Special Issue – International Conference on Topology, Messina (ICTM2015)
The projective Menger property and an embedding of Sω into function spaces Masami Sakai 1 Department of Mathematics, Kanagawa University, Hiratsuka 259-1293, Japan
a r t i c l e
i n f o
Article history: Received 8 December 2015 Received in revised form 10 January 2016 Accepted 15 September 2016 Available online 10 February 2017 MSC: 54C25 54C35 54D20
a b s t r a c t For a Tychonoff space X, we denote by Cp (X) the space of all real-valued continuous functions on X with the topology of pointwise convergence. In this paper, we show that (1) Cp (X, I) is projectively Menger if and only if X is b-discrete (i.e., every countable subset of X is closed and C ∗ -embedded in X), (2) there is a Menger space L such that the sequential fan Sω can be embedded into Cp (L). The first (1) enables us to give a direct proof of Arhangel’skii’s theorem [2, Theorem 6]: If Cp (X) is Menger, then X is finite. The second (2) is an affirmative answer to Arhangel’skii’s problem [5, Problem II.2.7] under CH (the continuum hypothesis). © 2017 Elsevier B.V. All rights reserved.
Keywords: Menger Projectively Menger Function space The sequential fan Whyburn
1. Introduction Throughout this paper, all spaces are assumed to be Tychonoff. The set of positive integers is denoted by N. Let R be the real line, we put D = {0, 1}, and I = [0, 1] ⊂ R. For a space X, we denote by Cp (X) the space of all real-valued continuous functions on X with the topology of pointwise convergence. The symbol 0 stands for the constant function to 0. A basic open neighborhood of 0 is of the form [F, (−ε, ε)] = {f ∈ Cp (X) : f (F ) ⊂ (−ε, ε)}, where F is a finite subset of X and ε > 0. For a subset Y ⊂ R, we put Cp (X, Y ) = {f ∈ Cp (X) : f (X) ⊂ Y }. Let Sω = {∞} ∪ {(n, m) : n, m ∈ N} be the sequential fan, where each (n, m) is isolated in Sω and a basic open neighborhood of ∞ is of the form N (ϕ) = {∞} ∪ {(n, m) : n ∈ N, m ≥ ϕ(n)} for a function ϕ ∈ NN . In other words, Sω is the quotient space obtained by identifying the limits of countably many convergent sequences.
1
E-mail address: [email protected]. The author was supported by JSPS KAKENHI Grant Number 25400213.
http://dx.doi.org/10.1016/j.topol.2017.02.017 0166-8641/© 2017 Elsevier B.V. All rights reserved.
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A space X is said to be Menger [12] (or, [18] as a survey) if for every sequence {Un : n ∈ N} of open covers of X, there are finite subfamilies Vn ⊂ Un such that {Vn : n ∈ N} is a cover of X. In the literature, a Menger space is sometimes called a Hurewicz space, but nowadays the term “Hurewicz space” stands for a stronger property. Every σ-compact space is Menger, and a Menger space is Lindelöf. The Menger property is closed hereditary, and it is preserved by continuous maps. It is well known that the Baire space NN (hence, Rω ) is not Menger. The first motivation of this paper is Arhangel’skii’s theorem below. Theorem 1.1 ([2, Theorem 6], [3]). For a space X, Cp (X) is Menger if and only if X is finite. To prove this theorem, Arhangel’skii used the notion of a R-quotient map and some properties of such a map, the fact that every analytic Menger space is σ-compact and N.V. Velichko’s theorem: Cp (X) is σ-compact if and only if X is finite. Thus, the proof needs several extra arguments about the Menger property, and the Menger property is not applied to Cp (X) directly. In this paper, applying the Menger property to Cp (X) directly, we give a direct proof of Arhangel’skii’s theorem. To this end, it is useful to investigate projectively Menger Cp (X, I) and Cp (X). Definition 1.2 ([8,5]). A space X is projectively Menger (resp., projectively σ-compact) if every second countable continuous image of X is Menger (resp., σ-compact). Every Menger space is projectively Menger. It is known [8, Proposition 8 (1)] that a space is Menger if and only if it is Lindelöf and projectively Menger. A subset A of a space X is said to be bounded in X if for every continuous function f : X → R, f A : A → R is a bounded function. We can easily observe that a subset A ⊂ X is bounded in X if and only if for every discrete family D consisting of nonempty open subsets of X, {D ∈ D : D ∩ A = ∅} is finite. Hence, if A is a bounded subset in a second countable space, then A is compact in the second countable space. Therefore, every σ-bounded space (i.e., a space which is the union of countably many bounded subsets) is projectively σ-compact (hence, projectively Menger) [5, Proposition 1.1]. Among several characterizations of a projectively Menger space, the following one is useful. Theorem 1.3 ([8, Theorem 6]). A space X is projectively Menger if and only if for every sequence {Un : n ∈ N} of countable covers of X consisting of cozero-sets in X, there are finite subfamilies Vn ⊂ Un such that {Vn : n ∈ N} is a cover of X. A space X is said to have countable fan-tightness [2] if whenever An ⊂ X and x ∈ An (n ∈ ω), there are finite sets Fn ⊂ An such that x ∈ {Fn : n ∈ ω}. Obviously Sω does not have countable fan-tightness. Arhangel’skii [2, Theorem 4] proved that every finite power of X is Menger if and only if Cp (X) has countable fan-tightness. Hence, if every finite power of X is Menger, Sω cannot be embedded into Cp (X). The second motivation of this paper is Arhangel’skii’s problem below. Problem 1.4 ([5, Problem II.2.7]). Can Sω be embedded into Cp (X) for some Menger space X? In the third section, we solve this problem in the affirmative under CH. Also we will discuss some connections among the projective Menger property, an embedding of Sω into Cp (X) and the Whyburn property.
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2. The projective Menger property of function spaces A subset S of a space X is said to be C ∗ -embedded in X [11] if every continuous real-valued bounded function on S can be extended to a continuous function on X. Urysohn’s extension theorem [11, 1.17] says that a subset S of a space X is C ∗ -embedded in X if and only if for any disjoint zero-sets A and B in S, there are disjoint zero-sets ZA and ZB in X such that A ⊂ ZA and B ⊂ ZB . According to Tkačuk [22], a space X is said to be b-discrete if every countable subset of X is closed (equivalently, closed and discrete) and C ∗ -embedded in X. The following lemma can be proved easily by Urysohn’s extension theorem, but for completeness we give a proof. Lemma 2.1. The following are equivalent for a space X: (1) X is b-discrete; (2) For any disjoint countable subsets A and B in X, there are disjoint zero-sets ZA and ZB in X such that A ⊂ ZA and B ⊂ ZB ; (3) For any disjoint countable subsets A and B in X such that A is closed in X, there are disjoint zero-sets ZA and ZB in X such that A ⊂ ZA and B ⊂ ZB . Proof. (1)→(2): Let A and B be disjoint countable subsets in X. Then, A ∪ B is a closed and discrete, C ∗ -embedded subset in X. Since both A and B are zero-sets in A ∪ B, by Urysohn’s extension theorem, we can take disjoint zero-sets ZA and ZB in X such that A ⊂ ZA and B ⊂ ZB . (2)→(3) is trivial. (3)→(1): Let B be any countable subset in X, and take any point a ∈ X \ B. Applying the condition (3) to A = {a} and B, we can see that B is closed in X. Thus every countable subset of X is closed and discrete. Let S be a countable subset of X, and let A and B be disjoint subsets in S. Since A is closed in X, we can take disjoint zero-sets ZA and ZB in X such that A ⊂ ZA and B ⊂ ZB . By Urysohn’s extension theorem, S is C ∗ -embedded in X. 2 Tkačuk proved the following two theorems, where the statement (4) in Theorem 2.3 is due to Arhangel’skii [5, Theorem 3.2]. Theorem 2.2 ([22]). For a space X, the following are equivalent: (1) (2) (3) (4)
X is b-discrete; Cp (X, I) is pseudocompact; Cp (X, I) is σ-pseudocompact; Cp (X, I) is σ-bounded.
Theorem 2.3 ([22]). For a space X, the following are equivalent: (1) (2) (3) (4)
X is pseudocompact and b-discrete; Cp (X) is σ-pseudocompact; Cp (X) is σ-bounded; Cp (X) is projectively σ-compact (A.V. Arhangel’skii).
Therefore, if X is b-discrete (resp., pseudocompact and b-discrete), then Cp (X, I) (resp., Cp (X)) is projectively Menger. We show that the converses hold. Theorem 2.4. A space X is b-discrete if and only if Cp (X, I) is projectively Menger.
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Proof. If X is b-discrete, Cp (X, I) is projectively Menger by Theorem 2.2. Conversely, assume that Cp (X, I) is projectively Menger. We show the statement (3) in Lemma 2.1. Let A and B be disjoint countable subsets in X such that A is closed in X. Let B = {bn : n ∈ N}, and let Bn = {b1 , . . . , bn }. For each n, m ∈ N, we put Zn,m = {f ∈ Cp (X, I) : f (A) = {0} and f (Bm ) ⊂ [1/2n , 1]}. Since A and Bm are countable, each Zn,m is a zero-set in Cp (X, I). Assume that {Zn,m : m ∈ N} = ∅ for all n ∈ N. Using the projective Menger property of Cp (X, I), by Theorem 1.3, we can take some ϕ ∈ NN such that {Zn,ϕ(n) : n ∈ N} = ∅. For each n ∈ N, take any gn ∈ Cp (X, I) satisfying gn (A) = {0} and ∞ gn (Bϕ(n) ) = {1}. Let g = j=1 2−j gj . Then, g ∈ Cp (X, I) and g(A) = {0}. Fix any n ∈ N and 1 ≤ k ≤ ϕ(n). Then we have g(bk ) =
∞
2−j gj (bk ) ≥ 2−n gn (bk ) = 2−n .
j=1
Hence, g ∈ {Zn,ϕ(n) : n ∈ N}. This is a contradiction. Thus, there is some n ∈ N such that {Zn,m : m ∈ N} = ∅. Let h ∈ {Zn,m : m ∈ N}, then A ⊂ ZA = h−1 (0) and B ⊂ ZB = h−1 ([1/2n , 1]). 2 Asanov [6] proved that if Cp (X) is Lindelöf, then every finite power of X has countable tightness. The following is a special case of Asanov’s theorem. Lemma 2.5. If Cp (X, I) is Lindelöf, then X has countable tightness. Proof. Assume x ∈ A \ A for a subset A ⊂ X. Let Z = {f ∈ Cp (X, I) : f (x) = 0}. Since Z is closed in Cp (X, I), it is Lindelöf. For each a ∈ A, let Ua = {f ∈ Cp (X, I) : f (a) < 1}. Since {Ua : a ∈ A} is an open family in Cp (X, I) covering Z, there is a countable subset B ⊂ A such that Z ⊂ {Ua : a ∈ B}. Then we have x ∈ B. 2 Corollary 2.6. A space X is discrete if and only if Cp (X, I) is Menger. Proof. If X is discrete, then Cp (X, I) = IX is compact. Conversely, assume that Cp (X, I) is Menger. By Theorem 2.4 and Lemma 2.5, X is a b-discrete space of countable tightness. Then X must be discrete. 2 Now we can show Arhangel’skii’s theorem directly without using extra arguments on R-quotient maps, analytic spaces, or Velichko’s theorem. Corollary 2.7 (A.V. Arhangel’skii). A space X is finite if and only if Cp (X) is Menger. Proof. If X is finite, Cp (X) = RX is σ-compact. Conversely, assume that Cp (X) is Menger. By Corollary 2.6, we have Cp (X) = RX , so X must be finite. 2 Moreover, we note that the projective Menger property of Cp (X) can be characterized by a pseudocompact b-discrete space X. Theorem 2.8. A space X is pseudocompact and b-discrete if and only if Cp (X) is projectively Menger. Proof. If X is pseudocompact and b-discrete, by Theorem 2.3, Cp (X) is projectively Menger. Conversely, assume that Cp (X) is projectively Menger. Since Cp (X, I) is a retract of Cp (X), it is also projectively
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Menger. By Theorem 2.4, X is b-discrete. If X is not pseudocompact, since there is a countably infinite discrete family of nonempty open subsets in X, we can take a countably infinite C-embedded subset A in X. Hence, RA is a second countable continuous image of Cp (X), but it is not Menger. This is a contradiction. 2 An example of an infinite pseudocompact b-discrete space is constructed in Shakhmatov [21]. The space of all weak P -points in βω \ ω is also an infinite pseudocompact b-discrete space: see [4, Example 6.4]. The topological sum of b-discrete spaces is b-discrete, and Cp ( α<κ Xα , I) (resp., Cp ( α<κ Xα )) is homeomorphic to α<κ Cp (Xα , I) (resp., α<κ Cp (Xα )). Combining these facts and Tkačuk’s theorems above, we have: Proposition 2.9. The following assertions hold. (1) If Cp (X, I) is projectively Menger, then any power of Cp (X, I) is pseudocompact, (2) If Cp (X) is projectively Menger, then any finite power of Cp (X) is σ-pseudocompact. We conclude this section with the projective Menger property of the realcompactification υX of a space X. Proposition 2.10. A space X is projectively Menger if and only if the realcompactification υX of X is projectively Menger. Proof. Assume that X is projectively Menger, and let Un (n ∈ N) be a sequence of countable covers of υX consisting of cozero-sets in υX. By the projective Menger property of X, there are finite subfamilies Vn ⊂ Un such that X ⊂ { Vn : n ∈ N}. Since { Vn : n ∈ N} is a cozero-set in υX, we have { Vn : n ∈ N} = υX. Because, every zero-set in υX is the closure of a zero-set in X [11, 8D. 1]. Conversely, assume that υX is projectively Menger, and let Un (n ∈ N) be a sequence of countable covers of X consisting of cozero-sets in X. For each cozero-set U in X, fix a cozero-set U in υX with U = U ∩ X. Then each Un = {U : U ∈ Un } is a cover of υX, because of [11, 8D. 1]. Apply the projective Menger property of υX to these Un . 2 If a space Y satisfies X ⊂ Y ⊂ υX, then obviously υY = υX holds. Corollary 2.11. Let X be a projectively Menger space. Then every space Y with X ⊂ Y ⊂ υX is also projectively Menger. 3. An embedding of the sequential fan Sω into Cp (X) In this section, we clarify the question of when Sω can be embedded into Cp (X). An infinite cover {An : n ∈ N} of a set X is said to be a γ-cover if every point of X is contained in An for all but finitely many n ∈ N. A cover U of a set X is said to be an ω-cover [10] if every finite subset of X is contained in some member of U. A γ-cover is an ω-cover. Note that if U is an ω-cover of a set X and X∈ / U, then each finite subset of X is contained in infinitely many members of U. Lemma 3.1 ([8, Lemma 80]). Let X = {x} ∪ {xn,m : n, m ∈ N} be a T2 -space such that xn,m → x (m → ∞) for each n ∈ N, and for any ϕ ∈ NN , x ∈ / {xn,m : n ∈ N, m ≤ ϕ(n)}. Then Sω can be embedded into X. The following theorem may be regarded as a reformulation of [8, Theorem 76] which was proved under a strong additional condition, which was called Sakai’s condition. Theorem 3.2. For a space X, the following are equivalent:
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(1) Sω can be embedded into Cp (X); (2) There are sequences Un = {Un,m : m ∈ N} (n ∈ N) of covers of X consisting of cozero-sets in X and Zn = {Zn,m : m ∈ N} (n ∈ N) of covers of X consisting of zero-sets in X such that each Zn is a γ-cover of X, Zn,m ⊂ Un,m and for any ϕ ∈ NN , {Un,m : n ∈ N, m ≤ ϕ(n)} is not an ω-cover of X. Proof. (1)→(2). Let Sω = {0} ∪ {fn,m : n, m ∈ N} ⊂ Cp (X), where fn,m → 0 (m → ∞). For each n, m ∈ N, we put Un,m = {x ∈ X : |fn,m (x)| <
1 n },
Zn,m = {x ∈ X : |fn,m (x)| ≤
1 n+1 }.
Each Un,m (resp., Zn,m ) is a cozero-set (resp., zero-set) in X with Zn,m ⊂ Un,m . Let Un = {Un,m : m ∈ N} and Zn = {Zn,m : m ∈ N}. If I = {n ∈ N : X ∈ Un } is infinite, some sequence {fn,mn : n ∈ I} converges to 0 uniformly. This is a contradiction, so without loss of generality, we may assume Un,m = X for each n, m ∈ N. We can easily check that the condition fn,m → 0 (m → ∞) implies that Zn is a γ-cover of X. Let ϕ ∈ NN . Assume that Uϕ = {Un,m : n ∈ N, m ≤ ϕ(n)} is an ω-cover of X. By the condition 0∈ / {fn,m : n ∈ N, m ≤ ϕ(n)}, there are a finite set F ⊂ X and a k ∈ N such that [F, (− k1 , k1 )] ∩ {fn,m : n ∈ N, m ≤ ϕ(n)} = ∅. Since F is contained in infinitely many members of Uϕ , we have F ⊂ Un,m for some n > k and m ≤ ϕ(n). This implies |fn,m (x)| < n1 < k1 for all x ∈ F , in other words, fn,m ∈ [F, (− k1 , k1 )]. This is a contradiction. (2)→(1). For each n, m ∈ N, we take a continuous function fn,m : X → I such that fn,m (x) = 0 for all x ∈ Zn,m and fn,m (x) = 1 for all x ∈ X \ Un,m . Then fn,m → 0 (m → ∞). Let ϕ ∈ NN . Since Uϕ = {Un,m : n ∈ N, m ≤ ϕ(n)} is not an ω-cover of X, there is a finite subset F ⊂ X such that F is not contained in any member of Uϕ . Then we can easily check [Fϕ , (− 12 , 12 )] ∩ {fn,m : n ∈ N, m ≤ ϕ(n)} = ∅. By Lemma 3.1, Sω can be embedded into {0} ∪ {fn,m : n, m ∈ N} ⊂ Cp (X). 2 A space X is said to be strongly zero-dimensional if any disjoint zero sets in X can be separated by a clopen set in X. Using Theorem 3.2 and its proof, we have the following corollary immediately, where strong zero-dimensionality is used only for (1) → (2), the equivalence of (2) and (3) is true for any space X, and it is due to Bonanzinga, Cammaroto and Matveev [8, Theorem 78].2 Corollary 3.3. For a strongly zero-dimensional space X, the following are equivalent: (1) Sω can be embedded into Cp (X); (2) There is a sequence Un = {Un,m : m ∈ N} (n ∈ N) of γ-covers of X consisting of clopen sets in X such that for any ϕ ∈ NN , {Un,m : n ∈ N, m ≤ ϕ(n)} is not an ω-cover of X; (3) Sω can be embedded into Cp (X, D). Let X = Cp (Sω ). Then obviously Sω can be embedded into Cp (X). But Sω cannot be embedded into Cp (X, D), because X is connected. The author does not know if this corollary is true for a zero-dimensional space X. In general, if Sω can be embedded into j∈ω Xj , then it can be embedded into j
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Proof. We show only the case of Cp (X). Assume that Sω can be embedded into Cp (X)ω . Then Sω can be embedded into Cp (X)k for some k ∈ N. Since Cp (X)k is homeomorphic to Cp (Y ), where Y = X1 · · · Xk and each Xj is a homeomorphic copy of X, Y has sequences Un = {Un,m : m ∈ N} and Zn = {Zn,m : m ∈ N} (n ∈ N) in Theorem 3.2 (2). For each 1 ≤ j ≤ k, let idj : X → Xj be the identity map. For each n, m ∈ N, we put Un,m =
−1 {idj (Un,m ∩ Xj ) : 1 ≤ j ≤ k}, Zn,m = {id−1 j (Zn,m ∩ Xj ) : 1 ≤ j ≤ k}.
Then each Un,m (resp., Zn,m ) is a cozero-set (resp., zero-set) in X and Zn,m ⊂ Un,m is satisfied. Each Zn = {Zn,m : m ∈ N} is a γ-cover of X, because for each x ∈ X {idj (x) : 1 ≤ j ≤ k} is contained in Zn,m for all but finitely many m ∈ N. Assume that there is a ϕ ∈ NN such that {Un,m : n ∈ N, m ≤ ϕ(n)} is an ω-cover of X. Then for each finite subset F ⊂ X, there are n, m ∈ N such that F ⊂ Un,m . This implies {idj (F ) : 1 ≤ j ≤ k} ⊂ Un,m , hence {Un,m : n ∈ N, m ≤ ϕ(n)} is an ω-cover of Y . This is a contradiction. By Theorem 3.2, Sω can be embedded into Cp (X). 2
Finally, we solve Problem 1.4 in the first section in the affirmative under CH. Remark 3.5. Let (Z, +) be the usual discrete group of integers. In [13, Lemma 2.6], under CH the authors constructed a Lusin set L ⊂ Zω (i.e., L is uncountable and its intersection with every first category set in Zω is countable) such that L + L = Zω . Though L is a Lusin set in Zω , by a homeomorphism between Zω and the space of irrationals of R, it is a Lusin set in R. A space X is said to be Rothberger if for every sequence {Un : n ∈ ω} of open covers of X, there are some members Un ∈ Un such that {Un : n ∈ ω} is a cover of X. Every Rothberger space is trivially Menger. Since any Lusin set is Rothberger [13, Theorem 2.7], L is Menger. However, L × L is not Menger, because L + L = Zω implies that Zω is the continuous image of L × L. In [13, Theorem 2.8], using the condition L + L = Zω , the authors proved that this Lusin set L has a sequence Un = {Un,m : m ∈ ω} (n ∈ ω) of increasing covers of L consisting of clopen sets in L such that for each ϕ ∈ ω ω , {Un,ϕ(n) : n ∈ ω} is not an ω-cover of L. Applying the equivalence of (2) and (3) in Corollary 2.6 due to Bonanzinga, Cammaroto and Matveev, Sω can be embedded into Cp (L, D) ⊂ Cp (L). In general, if a space of countable tightness has a homeomorphic copy of Sω , then Sω can be embedded into X as a closed set [15, Corollary 3.9]. Therefore, since Cp (L) has countable tightness, Sω can be embedded into Cp (L, D) as a closed set. On the other hand, in [13, Theorem 2.13], under CH the authors constructed a Lusin set H ⊂ R such that every finite power of H is Rothberger. For this H, Sω cannot be embedded into Cp (H), because Cp (H) has countable fan-tightness. In contrast with a Lusin set, for any Sierpiński set S (i.e., S is an uncountable subset of R and its intersection with every Lebesgue measure zero set in R is countable), Sω cannot be embedded into Cp (S) by [19, Corollary 11]. 4. An embedding of Sω into Cp (X) and related properties In this section, we examine some connections among the projective Menger property, an embedding of Sω into Cp (X) and the Whyburn property. Proposition 4.1. If Sω cannot be embedded into Cp (X), then X is projectively Menger. Proof. Let Un = {Un,m : m ∈ N} (n ∈ N) be a sequence of covers of X consisting of cozero-sets in X. For each n, m ∈ N, we put Vn,m = Un,1 ∪ · · · ∪ Un,m . Since Vn = {Vn,m : m ∈ N} is an increasing cover of X consisting of cozero-sets in X, we can find zero-sets Zn,m in X such that Zn,m ⊂ Vn,m and
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Zn = {Zn,m : m ∈ N} is an increasing cover of X. Applying Theorem 3.2 to Vn and Zn , we can take some ϕ ∈ NN such that {Vn,ϕ(n) : n ∈ N} is an ω-cover of X. This implies that {Un,m : n ∈ N, m ≤ ϕ(n)} is a cover of X. 2 The converse of this proposition is not true. Recall the Lusin set L mentioned in Remark 3.5, L is Menger, but Sω can be embedded into Cp (L). In [8, Theorem 18], the authors noted that if every finite power of X is projectively Menger, then every countable subspace of Cp (X) has countable fan-tightness. We make clear when every countable subspace of Cp (X) has countable fan-tightness. It is known [13, Theorem 3.9] that every finite power of X is Menger if and only if for any sequence Un (n ∈ N) of open ω-covers of X, there are finite subfamilies Vn ⊂ Un such that {Vn : n ∈ N} is an ω-cover of X. This latter covering property is denoted by the symbol Sf in (Ω, Ω) in [13]. Definition 4.2. A space X is projectively Sf in (Ω, Ω) if every second countable continuous image of X satisfies Sf in (Ω, Ω). Theorem 4.3. For a space X, the following are equivalent: (1) Every countable subspace of Cp (X) has countable fan-tightness; (2) For any sequence Un = {Un,m : m ∈ N} (n ∈ N) of ω-covers of X consisting of cozero-sets in X, there is some ϕ ∈ NN such that {Un,m : n ∈ N, m ≤ ϕ(n)} is an ω-cover of X; (3) X is projectively Sf in (Ω, Ω). Proof. (1)→(2). Let Un = {Un,m : m ∈ N} (n ∈ N) be a sequence of ω-covers of X consisting of cozero-sets in X. We can put Un,m = {Zn,m,l : l ∈ N}, where each Zn,m,l is a zero-set in X and Zn,m,l ⊂ Zn,m,l+1 . −1 −1 Take a continuous function fn,m,l : X → I satisfying fn,m,l (0) = Zn,m,l and fn,m,l (1) = X \ Un,m . For each n ∈ N, let An = {fn,m,l : m, l ∈ N}. By the facts that each Un is an ω-cover of X and {Zn,m,l : l ∈ N} is increasing, we can see 0 ∈ {An : n ∈ N}. By our assumption (1), there is some ϕ ∈ NN such that 0 ∈ {fn,m,l : n ∈ N, m, l ≤ ϕ(n)}. Then {Un,m : n ∈ N, m ≤ ϕ(n)} is an ω-cover of X. Indeed, let F be a finite subset of X, and consider the basic open neighborhood [F, (− 21 , 12 )] of 0. Take n ∈ N and m, l ≤ ϕ(n) such that fn,m,l ∈ [F, (− 12 , 12 )]. This implies F ⊂ Un,m . (2)→(3). Let f : X → Y be a continuous map onto a second countable space Y . Let Un (n ∈ N) be a sequence of open ω-covers of Y . Every member of Un is a cozero-set in Y , and let us recall [10] that every finite power of a space is Lindelöf if and only if every open ω-cover of the space has a countable ω-subcover. Hence, we may assume that each Un is a countable ω-cover of Y consisting of cozero-sets in Y . Considering the ω-covers f −1 (Un ) = {f −1 (U ) : U ∈ Un } of X, we have (3). (3)→(1). Take any countable subset A = {fn : n ∈ N} ⊂ Cp (X), and consider the continuous map f : X → RN defined by f (x) = (fn (x))n∈N . By our assumption (3), every finite power of f (X) is Menger, hence Cp (f (X)) has countable fan-tightness by Arhangel’skii’s theorem mentioned in the first section. Consider the embedding f ∗ of Cp (f (X)) into Cp (X) induced by f . Let pn : f (X) → R be the projection of f (X) to the n-th coordinate. Then obviously f ∗ ({pn : n ∈ N}) = A. Thus, A has countable fan-tightness. 2 Thus, if X is projectively Sf in (Ω, Ω), then Sω cannot be embedded into Cp (X). But the converse is not true. Let S be a Sierpiński set in R such that S + S in R is the set of irrationals constructed in [13, p.250] under CH. As noted in Remark 3.5, Sω cannot be embedded into Cp (S), but the fact that S + S is the set of irrationals implies that S × S is not Menger. Hence, S is not projectively Sf in (Ω, Ω). The following proposition coincides with [8, Proposition 24]. Because the definition of Sf in (Ωcz , Ωcz ) in [8, Proposition 24] is just the condition (2) in Theorem 4.3.
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Proposition 4.4. If every finite power of X is projectively Menger, then X is projectively Sf in (Ω, Ω). Proof. Let Y be a second countable continuous image of X. Then each finite power Y n is a second countable continuous image of X n . Hence, Y n is Menger. 2 Remark 4.5. Bonanzinga et al. asked in [8, Question 25] whether Proposition 4.4 (i.e., Proposition 24 in [8]) can be reversed. The answer is trivially in the negative. Because, as mentioned in [8, p. 880], there is a space X which is the topological sum of two countably compact spaces such that every countable subspace of Cp (X) has countable fan-tightness, but X 2 is not projectively Menger. On the other hand, by Theorem 4.3, X is projectively Sf in (Ω, Ω). A space X is said to be Whyburn if whenever A ⊂ X and x ∈ A \ A, there is a subset B ⊂ A such that B = {x} ∪ B. This notion was considered in [24]. Every Fréchet–Urysohn T2 -space is obviously Whyburn. A sufficient condition for a space X to be Whyburn is: Lemma 4.6 ([7, Corollary 3.4]). If a space X has countable fan-tightness and each point of X is Gδ , then X is Whyburn. In particular, every countable space of countable fan-tightness is Whyburn. Definition 4.7 ([17]). A space X is ω-Whyburn if whenever A is a countable subset in X and x ∈ A \ A, there is a subset B ⊂ A such that B = {x} ∪ B. An ω-Whyburn space was called an APω -space in [8]. If a space X is ω-Whyburn, then every countable subspace of X is obviously Whyburn. But the converse does not hold. This fact was first observed by Angelo Bella. Example 4.8 (A. Bella). Let D be an infinite maximal almost-disjoint family consisting of infinite subsets of N. Let Ψ = N ∪ D be the Mrówka–Isbell space [11, 5I], and let Ψ∗ = {∞} ∪ Ψ be the one-point compactification of Ψ. Since every compact space of countable tightness has countable fan-tightness [2, Theorem 3], Ψ∗ has countable fan-tightness. By Lemma 4.6, every countable subspace of Ψ∗ is Whyburn. On the other hand, the point ∞ is in the closure of N, but every infinite subset of N has a limit point in D, thus Ψ∗ is not ω-Whyburn. In [8, Theorem 84 (2)], the authors showed that if Cp (X) is ω-Whyburn, then X is projectively Menger. We can show the following stronger assertion. Proposition 4.9. If every countable subspace of Cp (X) is Whyburn, then Sω cannot be embedded into Cp (X). Proof. Assume that Sω can be embedded into Cp (X). Then, as discussed in [17, Proposition 2.6], Sω ×(ω+1) can be embedded into Cp (X). However, the countable space Sω × (ω + 1) is not Whyburn: for example see [17, Proposition 1.1]. 2 Proposition 4.10. If X is projectively Sf in (Ω, Ω), then every countable subspace of Cp (X) is Whyburn. Proof. By Theorem 4.3, if X is projectively Sf in (Ω, Ω), then every countable subspace of Cp (X) has countable fan-tightness. Our conclusion follows from Lemma 4.6. 2 A family N of subsets of a space X is said to be a network for X if whenever x ∈ X and U is a neighborhood of x, there is some N ∈ N such that x ∈ N ⊂ U . A space X is said to be ℵ0 -stable [1] if
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whenever a space Y is a continuous image of X and Y has a continuous one-to-one map onto a second countable space, Y has a countable network. A σ-pseudocompact space is ℵ0 -stable [1, Propositions II.6.2 and II.6.6]. A space X is said to be ℵ0 -monolithic [1] if for every countable subset A ⊂ X, A has a countable network. Arhangel’skii [1] proved that a space X is ℵ0 -stable if and only if Cp (X) is ℵ0 -monolithic. Therefore, if a space X is σ-pseudocompact, the closure of each countable subset of Cp (X) has a countable network. Proposition 4.11. If X is σ-pseudocompact, then Cp (X) is ω-Whyburn. Proof. Let A be a countable subset of Cp (X) and assume that 0 ∈ A\A. Then, E = {0} ∪ A has a countable network, in particular each point of E is Gδ in E. Since a σ-pseudocompact space is projectively Sf in (Ω, Ω), by Theorem 4.3, {0} ∪A has countable fan-tightness. We use Bella and Yaschenko’s idea in [7, Corollary 3.4] (i.e., Lemma 4.6 in this paper). We put {x} = {Gn : n ∈ ω}, where each Gn is an open neighborhood of x in E, and Gn+1 ⊂ Gn is satisfied. Using countable fan-tightness of the space {0} ∪ A, we can take finite subsets Fn ⊂ Gn ∩A such that x ∈ {Fn : n ∈ ω}. Then we have {Fn : n ∈ ω} = {x} ∪ {Fn : n ∈ ω}. 2 We summarize implications observed in this section. X is σ-pseudocompact X n is projectively Menger for all n ∈ N ↓ ↓ X is projectively Sf in (Ω, Ω) Cp (X) is ω-Whyburn ↓ ↓ Every countable subset of Cp (X) is Whyburn ↓ Sω cannot be embedded into Cp (X) ↓ X is projectively Menger By this diagram, immediately we have: Proposition 4.12. Every finite power of X is projectively Menger if and only if for any n ∈ N, Sω cannot be embedded into Cp (X n ). Let us recall Proposition 2.9 (2): If Cp (X) is projectively Menger, then any finite power of Cp (X) is σ-pseudocompact. Hence, we can see that all conditions obtained by replacing X by Cp (X) in the diagram are equivalent. Corollary 4.13. For a space X, the following are equivalent: (1) (2) (3) (4)
Cp Cp (X) is ω-Whyburn; Every countable subset of Cp Cp (X) is Whyburn; Sω cannot be embedded into Cp Cp (X); X is pseudocompact and b-discrete.
Murtinová [14, Theorem 3] proved that Cp Cp (X) is Whyburn if and only if X is finite. 5. Remarks on selection principles Sf in (Γ, Ω) and Uf in (O, Ω) In this section, we remark that an embedding of Sω into Cp (X) is related with open problems on selection principles Sf in (Γ, Ω) and Uf in (O, Ω).
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A space X is said to satisfy Sf in (Γ, Ω) [13] if for every sequence {Un : n ∈ N} of open γ-covers of X, there are finite subfamilies Vn ⊂ Un such that {Vn : n ∈ N} is an ω-cover of X. A space X is said to satisfy Uf in (O, Ω) [13] if for every sequence {Un : n ∈ N} of countable open covers of X, there are finite subfamilies Vn ⊂ Un such that { Vn : n ∈ N} is an ω-cover of X. By a simple observation, we can see that Sf in (Γ, Ω) implies Uf in (O, Ω), and that every Lindelöf space with Uf in (O, Ω) is Menger. In addition, a space X is said to satisfy Uf in (O, Γ) [13] if for every sequence {Un : n ∈ N} of countable open covers of X, there are finite subfamilies Vn ⊂ Un such that every point of X is contained in Vn for all but finitely many n ∈ N. A Lindelöf space with Uf in (O, Γ) is called a Hurewicz space. We summarize the implications of these covering properties. Uf in (O, Γ) → Uf in (O, Ω) ↑ Sf in (Γ, Ω)
if Lindelöf
→
Menger
So far as the author knows, the following problems posed in [13, Problems 1 and 2] are still open, where the symbol Uf in (Γ, Ω) in [13, Problem 1] coincides with Uf in (O, Ω), and the symbol Uf in (Γ, Γ) in [13, Problem 2] coincides with Uf in (O, Γ). Problem 5.1 ([13, Problems 1 and 2], [23, Problem 2.1]). For a space of reals: (1) Is Uf in (O, Ω) = Sf in (Γ, Ω)? (2) And if not, does Uf in (O, Γ) imply Sf in (Γ, Ω)? Concerning these problems, we consider a special embedding of Sω into Cp (X). Definition 5.2. Sω can be embedded into Cp (X) (resp., Cp (X, D)) monotonically if there is an embedding e : Sω → Cp (X) (resp., e : Sω → Cp (X, D)) such that e(∞) = 0 and e((n, m)) ≥ e((n, m + 1)) for all n, m ∈ N. Proposition 5.3. For a space X, the following are equivalent: (1) Sω can be embedded into Cp (X) monotonically; (2) There is a sequence Un = {Un,m : m ∈ N} (n ∈ N) of increasing covers of X consisting of cozero-sets in X such that for any ϕ ∈ NN , {Un,ϕ(n) : n ∈ N} is not an ω-cover of X; (3) There is a sequence Un = {Un,m : m ∈ N} (n ∈ N) of covers of X consisting of cozero-sets in X such that for any ϕ ∈ NN , { m≤ϕ(n) Un,m : n ∈ N} is not an ω-cover of X. Proof. (1)→(2). Let e : Sω → Cp (X) be a monotonic embedding, and let e(Sω ) = {0} ∪ {fn,m : n, m ∈ N}. Note that each fn,m is non-negative. For each n, m ∈ N, we put Un,m = {x ∈ X : fn,m (x) < 1/2n }. The condition fn,m ≥ fn,m+1 implies Un,m ⊂ Un,m+1 , and the condition fn,m → 0 (m → ∞) implies X = {Un,m : m ∈ N}. Thus Un = {Un,m : m ∈ N} is an increasing cover of X consisting of cozero-sets in X. The rest of this proof can be done by the same arguments as in the proof of Theorem 3.2 (1) → (2). The equivalence of (2) and (3) is obvious. (2)→(1). Let Un = {Un,m : m ∈ N} (n ∈ N) be a sequence in the assertion (2). We take a zero-set Zn,m in X such that Zn,m ⊂ Un,m , Zn,m ⊂ Zn,m+1 and X = {Zn,m : m ∈ N} for n ∈ N. Let fn,m : X → I be a −1 −1 continuous function such that fn,m (0) = Zn,m and fn,m (1) = X \ Un,m . We put gn,m = fn,1 · fn,2 · · · fn,m . −1 Then obviously gn,m ≥ gn,m+1 , gn,m → 0 (m → ∞) and gn,m (1) = X \ Un,m . Let ϕ ∈ NN , and assume 0 ∈ {gn,m : n ∈ N, m ≤ ϕ(n)}. For any finite subset F ⊂ X, there is some gn,m ∈ [F, (−1/2, 1/2)] ∩ {gn,m :
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n ∈ N, m ≤ ϕ(n)}. This means F ⊂ Un,m ⊂ Un,ϕ(n) . Since {Un,ϕ(n) : n ∈ N} is not an ω-cover of X, this is a contradiction. By Lemma 3.1, Sω can be embedded into Cp (X) monotonically. 2 Thus, if X satisfies Uf in (O, Ω), Sω cannot be embedded into Cp (X) monotonically. The converse is true for a normal countably paracompact space X (in particular, for a space X of reals). Because, if {Un : n ∈ N} is an open cover of a normal countably paracompact space X, we can find a cover {Vn : n ∈ N} of X consisting of cozero-sets such that Vn ⊂ Un for all n ∈ N: for example see [9, Theorem 5.2.3]. Corollary 5.4. Let X be a normal countably paracompact space. Then, X satisfies Uf in (O, Ω) if and only if Sω cannot be embedded into Cp (X) monotonically. Corollary 5.5. If a normal countably paracompact space X satisfies Uf in (O, Γ), then Sω cannot be embedded into Cp (X) monotonically. Quite similarly to Proposition 5.3, we have: Proposition 5.6. For a space X, the following are equivalent: (1) Sω can be embedded into Cp (X, D) monotonically; (2) There is a sequence Un = {Un,m : m ∈ N} (n ∈ N) of increasing covers of X consisting of clopen sets in X such that for any ϕ ∈ NN , {Un,ϕ(n) : n ∈ N} is not an ω-cover of X; (3) There is a sequence Un = {Un,m : m ∈ N} (n ∈ N) of covers of X consisting of clopen sets in X such that for any ϕ ∈ NN , { m≤ϕ(n) Un,m : n ∈ N} is not an ω-cover of X. The Lusin set L in Remark 3.5 satisfies the condition (2) in Proposition 5.6, so Sω can be embedded into Cp (L, D) monotonically. By Propositions 5.3 and 5.6, we have: Corollary 5.7. For a strongly zero-dimensional space X, Sω can be embedded into Cp (X) monotonically if and only if it can be embedded into Cp (X, D) monotonically. In view of Theorem 3.2, we can see that if a space X satisfies Sf in (Γ, Ω), then Sω cannot be embedded into Cp (X). Therefore, we have the implications below for normal countably paracompact spaces X. Sω cannot be embedded into Cp (X) monotonically if Lindelöf Menger Uf in (O, Γ) → Uf in (O, Ω) → ↑ Sω cannot be embedded into Cp (X) ↑ Sf in (Γ, Ω) Concerning Problem 5.1, the following questions look interesting. Question 5.8. For a space X of reals: (1) If Sω can be embedded into Cp (X) (resp., Cp (X, D)), then can it be embedded into Cp (X) (resp., Cp (X, D)) monotonically? (2) Does Uf in (O, Γ) imply that Sω cannot be embedded into Cp (X)? (3) If Sω cannot be embedded into Cp (X), then does X satisfy Sf in (Γ, Ω)?
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If a space X of reals is a σ-set (i.e., every Gδ -subset of X is Fσ ), then every open γ-cover {Un : n ∈ N} of X has a closed γ-cover {Cn : n ∈ N} of X such that Cn ⊂ Un for all n ∈ N [16]. Hence, for a σ-set X of reals, Question 5.8 (3) is in the affirmative. Acknowledgements The author expresses gratitude to Angelo Bella and Boaz Tsaban. Angelo Bella kindly informed the author of Example 4.8. Boaz Tsaban pointed out that to solve Problem 1.4, the assumption cov(M) = cof (M) is enough instead of CH, in view of [20, Theorem 32]. References [1] A.V. Arhangel’skii, Factorization theorems and function spaces: stability and monolithicity, Sov. Math. Dokl. 26 (1982) 177–181. [2] A.V. Arhangel’skii, Hurewicz spaces, analytic sets and fan tightness of function spaces, Sov. Math. Dokl. 33 (1986) 396–399. [3] A.V. Arhangel’skii, Topological Function Spaces, Kluwer, 1992. [4] A.V. Arhangel’skii, Cp -theory, in: K. Kunen, J.E. Vaughan (Eds.), Handbook of Set-Theoretic Topology, Elsevier Sci. Pub., 1984, pp. 3–56. [5] A.V. Arhangel’skii, Projective σ-compactness, ω1 -caliber, and Cp -spaces, Topol. Appl. 104 (2000) 13–26. [6] M.O. Asanov, On cardinal invariants of spaces of continuous functions, Sov. Topol. Teor. Mnozhestv. Izhevsk 2 (1979) 8–12, in Russian. [7] A. Bella, I.V. Yaschenko, On AP and WAP spaces, Comment. Math. Univ. Carol. 40 (1999) 531–536. [8] M. Bonanzinga, F. Cammaroto, M. Matveev, Projective versions of selection principles, Topol. Appl. 157 (2010) 874–893. [9] R. Engelking, General Topology, revised and completed edition, Helderman Verlag, Berlin, 1989. [10] J. Gerlits, Zs. Nagy, Some properties of C(X). I, Topol. Appl. 14 (1982) 151–161. [11] L. Gillman, M. Jerison, Rings of Continuous Functions, reprint of the 1960 edition, Graduate Texts in Mathematics, vol. 43, Springer-Verlag, New York–Heidelberg, 1976. [12] W. Hurewicz, Über eine verallgemeinerung des Borelschen Theorems, Math. Z. 24 (1925) 401–421. [13] W. Just, A.W. Miller, M. Scheepers, P.J. Szeptycki, The combinatorics of open covers II, Topol. Appl. 73 (1996) 241–266. [14] E. Murtinová, On (weakly) Whyburn spaces, Topol. Appl. 155 (2008) 2211–2215. [15] T. Nogura, Y. Tanaka, Spaces which contains a copy of Sω or S2 and their applications, Topol. Appl. 30 (1988) 51–62. [16] M. Sakai, The sequence selection properties of Cp (X), Topol. Appl. 154 (2007) 552–560. [17] M. Sakai, Notes on strongly Whyburn spaces, Comment. Math. Univ. Carol. 57 (1) (2016) 123–129. [18] M. Sakai, M. Scheepers, The combinatorics of open covers, in: K.P. Hart, J. van Mill, P. Simon (Eds.), Recent Progress in General Topology III, Atlantic Press, 2014, pp. 751–799. [19] M. Scheepers, Cp (X) and Arhangel’skii’s αi -spaces, Topol. Appl. 89 (1998) 265–275. [20] M. Scheepers, B. Tsaban, The combinatorics of Borel covers, Topol. Appl. 121 (2002) 357–382. [21] D.B. Shakhmatov, A pseudocompact Tychonoff space all countable subsets of which are closed and C ∗ -embedded, Topol. Appl. 22 (1986) 139–144. [22] V.V. Tkačuk, The spaces Cp (X): decomposition into a countable union of bounded subspaces and completeness properties, Topol. Appl. 22 (1986) 241–253. [23] B. Tsaban, Selection principles and special sets of reals, in: E. Pearl (Ed.), Open Problems in Topology II, Elsevier B.V., 2007, pp. 91–108. [24] G.T. Whyburn, Mappings on inverse sets, Duke Math. J. 23 (1956) 237–240.