Tree duplicates, $G_\delta$-diagonals and Gruenhage spaces

We present an example in ZFC of a locally compact, scattered Hausdorff non-Gruenhage space $D$ having a $\G_delta$-diagonal. This answers a question posed by Orihuela, Troyanski and the author in a study of strictly convex norms on Banach spaces. In addition, we show that the Banach space of continuous functions $C_0(D)$ admits a $C^\infty$-smooth bump function.


Introduction
All topological spaces considered in this note will be Hausdorff. Recall that a norm on a Banach space is strictly convex if every element of the unit sphere is an extreme point of the unit ball. The authors of [8] introduced the following topological property to help understand the nature of strictly convex norms. Definition 1.1 ([8, Definition 2.6]). We say that a topological space X has ( * ) if there exists a sequence (U n ) ∞ n=1 of families of open subsets of X, with the property that given any x, y ∈ X, there exists n ∈ N such that (1) {x, y} ∩ U n is non-empty, and (2) {x, y} ∩ U is at most a singleton for all U ∈ U n .
If (U n ) ∞ n=1 satisfies properties (1) and (2) of Definition 1.1 then we will call it a ( * )-sequence. This notion can be regarded as a 'point-separation' property, in the sense that it specifies in advance a family of open sets which can separate pairs of distinct points in a controlled way. It generalises the extensively studied G δ -diagonal property.

Definition 1.2.
A space X has a G δ -diagonal if its diagonal is a G δ set in X 2 or, equivalently, if there is a sequence (G n ) ∞ n=1 of open covers of X, such that given x, y ∈ X, there exists n with the property that {x, y} ∩ U is at most a singleton for all U ∈ G n . See [1, Section 2] for a comprehensive introduction to spaces with G δ -diagonals. All spaces having a G δ -diagonal have ( * ), and if L is a locally compact space having ( * ) then so does its 1-point compactification L ∪ {∞}: simply adjoin to any ( * )sequence for L the singleton family {L}, which separates all points in L from ∞.
While compact spaces having G δ -diagonals are metrisable (cf. [1, Theorem 2.13]), compact spaces having ( * ) can be highly non-metrisable. The next definition presents another way in which points can be separated by a family of open sets, over which we have some control.
of families of open subsets of X, and sets R n , n ≥ 1, with the property that (1) if x, y ∈ X then there exists n ∈ N and U ∈ U n , such that {x, y} ∩ U is a singleton, and (2) U ∩ V = R n whenever U, V ∈ U n are distinct.
The relevance of spaces having ( * ) to the geometry of Banach spaces is partly explained by the next result. Recall that a topological space is scattered if every non-empty open subset admits a relatively isolated point. . Let K be a scattered compact space. Then C(K) admits an equivalent norm with a strictly convex dual norm if and only if K has ( * ). Moreover, the norm can be a lattice norm.
Here, C(K) denotes the Banach space of continuous real-valued functions on K. Gruenhage spaces were introduced in [2] for reasons other than Banach space geometry, but have found a place in this field nonetheless. Theorem 1.5 ([9, Theorem 7]). If K is Gruenhage compact then C(K) admits an equivalent lattice norm with a strictly convex dual norm. Theorem 1.4 is not a consequence of Theorem 1.5. Under the continuum hypothesis, there is a scattered compact, non-Gruenhage space having ( * ) [8,Example 4.10]. The purpose of this note is to show that such an example exists in ZFC. For more information about how these and related classes of topological spaces fit into Banach space theory, we refer the reader to [10,8].
It turns out that if X has cardinality at most the continuum c, then we have a much more straightforward description of Gruenhage's property available, which we will put to use in the next section. Proposition 1.6. [10, Proposition 2] Let X be a topological space with card X ≤ c. Then X is Gruenhage if and only if there is a sequence (U n ) ∞ n=1 of open subsets of X with the property that if x, y ∈ X, then {x, y} ∩ U n is a singleton for some n.
The basic idea behind the example remains the same as that of [8,Example 4.10]. We take a topological space X of cardinality c and endow a 'duplicate' D = X × {1, −1} with a new topology, the basic open sets of which use the existing structure of X to 'oscillate' rapidly between the levels +1 and −1. This oscillation induces a non-trivial interaction between the levels and will make it difficult to separate all the 'problem pairs' of the form (x, 1), (x, −1), x ∈ X, in the manner of Proposition 1.6. This will render the space non-Gruenhage. However, at the same time, we have construct D delicately enough to ensure that we don't lose the other properties that we want it to have.
We shall use a particular tree in its standard interval topology as our starting point. Before proceeding with the construction, we should point out that trees by themselves cannot furnish us with a desired example. According to [8,Theorem 4.6], if Υ is any tree which is Hausdorff in its standard interval topology, then Υ is Gruenhage if and only if it has ( * ). In particular, Υ has a G δ -diagonal in its standard interval topology if and only if it is R-embeddable [4], which in turn means that it is certainly Gruenhage; see e.g. [8, pp. 20 -21] for more details.

The Λ-Duplicate
Recall that a tree is a partially ordered set (Υ, ) with the property that, given any t ∈ Υ, the set of predecessors (0, t] = {s ∈ Υ : s t} is well ordered. For convenience, we shall regard 0 as an extra element, not in Υ, such that 0 ≺ t for all t ∈ Υ. Trees are natural generalisations of ordinal numbers. We will use standard interval notation throughout this note. For instance (r, t], where t ∈ Υ and r ∈ Υ ∪ {0}, is the set of all s ∈ Υ satisfying r ≺ s t. Other intervals such as [r, t) are defined accordingly. For further notation and details about trees, we refer the reader to e.g. [6].
The tree in question was first considered by Kurepa. Denoted by Λ in [6, Section 10], Kurepa's tree is the set of injective functions t : α −→ ω with (countable) ordinal domain and coinfinite range, and where s t if and only if t extends s. We shall regard functions in the usual set-theoretic sense, that is, as sets of ordered pairs, and with dom f and ran f the domain and range of a function f , respectively. In this note, we treat 0 as distinct from the empty function ∅, the latter being the least element of Λ.
A subset A ⊆ Λ ∪ {0} is an antichain if no two distinct elements of A are comparable in the tree order. We shall define Λ + to be the set of elements of Λ with successor ordinal domain. It is widely known and easy to show that {0, ∅} ∪ Λ + can be written as a countable union of antichains: if We set up a function τ on pairs (s, t), s ∈ Λ ∪ {0}, t ∈ Λ, s t, which will be central to the definition of our topology on D. Given s ≺ t, we define a finite sequence of As the β i are strictly decreasing, this process eventually stops at some finite stage k > 0, with β k = dom s. Let τ (s, t) = (β k , . . . , β 1 ).
For convenience, we also set τ (t, t) to be the empty sequence for each t ∈ Λ.
The next lemma will help when we use the τ sequences to define a basis for our topology. We let ⌢ denote concatenation of sequences.
Proof. If t is the empty function ∅ then let r = 0. If t ∈ Λ + then we let r = t↾ dom t−1 be the immediate predecessor of t in the tree order. Now suppose that dom t is a limit ordinal, and let τ (t, u) = (β k , . . . , β 1 ). By construction, we have By the choice of α, we have ensured that m ≥ k and γ i = β i whenever i ≤ k.
It is time to define the basic open sets. Let ℓ(s, t) denote the length of τ (s, t).
Observe that if π : D −→ Λ is the natural projection, then the restriction of π to any W (r, t, i) is injective. Moreover, the images π(W (r, t, i)) = (r, t] form the usual basis of the standard interval topology on Λ. Proposition 2.2. The W (r, t, i) form a basis for a locally compact scattered topology on D.
Proof. First, we show that these sets form a basis. If (t, k) ∈ W (r 1 , u 1 , i 1 ) ∩ W (r 2 , u 2 , i 2 ) then, by Lemma 2.1 and the fact that r 1 , r 2 ≺ t are comparable, we can find r ∈ [max{r 1 , r 2 }, t) such that τ (s, u j ) = τ (s, t) ⌢ τ (t, u j ) whenever s ∈ (r, t] and j = 1, 2. It follows that We conclude that the W (r, t, i) form a basis for a topology on D.
Now we show that this topology is Hausdorff and scattered. Let (t 1 , i 1 ), (t 2 , i 2 ) ∈ D be distinct. If t 1 = t 2 then we let r be the largest common predecessor of these elements. It is clear that W (r, t 1 , i 1 ) ∩ W (r, t 2 , i 2 ) is empty. Instead, if t 1 = t 2 then i 1 = −i 2 , so W (0, t 1 , i 1 ) ∩W (0, t 2 , i 2 ) is empty. To see that the topology is scattered, let E ⊆ D be non-empty and find minimal t ∈ Λ, subject to there being some i for Finally, we show that each W (r, t, i) is compact. Suppose that (u, k) ∈ W (r, v, i)∩ U, where U is some open set. Again from Lemma 2.1, we know that we can find s ∈ [r, u) such that τ (t, v) = τ (t, u) ⌢ τ (u, v) whenever t ∈ (s, u]. Moreover, we can choose s so that W (s, u, k) ⊆ U. If t ∈ (s, u] and (t, l) ∈ W (r, v, i), then we have and so (t, l) ∈ W (s, u, k) ⊆ U. This will allow us to show that W (r, v, i) is compact. The method follows that used to show that each (r, t] is compact in the usual interval topology of Λ. If W (r, v, i) is covered by a family of open sets U , we can find , where i 2 is the unique number satisfying (v 2 , i 2 ) ∈ W (r, v, i), and continue. The process stops at some finite k > 1, with v k = r and W (r, v, i) covered by U 1 , . . . , U k−1 .
Definition 2.3. We shall call D above, together with this topology, the Λ-duplicate.
Theorem 2.4. The Λ-duplicate has a G δ -diagonal but is not Gruenhage.
Consequently, if we are given distinct (u 1 , i 1 ), (u 2 , i 2 ) ∈ D, then by choosing p large enough, we can ensure that there is no V ∈ G p for which (u 1 , i 1 ), (u 2 , i 2 ) ∈ V . This establishes that the Λ-duplicate has a G δ -diagonal.
We shall suppose for a contradiction that D is Gruenhage. As card D = c, we can use Proposition 1.6 to find a sequence (U n ) ∞ n=1 of open subsets of D so that given any t ∈ Λ, there exists n for which We know that Λ = n,i E n,i . Now we are going to decompose each E n,i into countably many subsets. If t ∈ E n,i then (t, i) ∈ U n , so there exists some θ(t) ≺ t, where the A m are the antichains defined at the beginning of the section. Suppose that t, u ∈ E n,m,i and t ≺ u. Since θ(t), θ(u) ≺ u are comparable and θ(t), θ(u) ∈ A m , it follows that θ(u) = θ(t) ≺ t ≺ u. Now, we have is an even number.
To simplify the notation, we shall alter the indices and denote the E n,m,i by E n , n < ∞. In summary, we have shown that if D is Gruenhage then we can write Λ = ∞ n=1 E n , where each E n has the property that ℓ(t, u) is an even number whenever t, u ∈ E n and t ≺ u. In the final part of the proof, we use a Baire category type argument (cf. [6,Lemma 10.1]) to show that no decomposition of Λ into such sets E n is possible.
Set Λ 1 = Λ and let m 1 be minimal, subject to the condition that there exists some t 1 ∈ Λ 1 ∩ E m 1 . Let , by minimality of l 1 and the fact that k 1 / ∈ ran v. Therefore, τ (t 1 , v) = (dom t 1 ) and ℓ(t 1 , v) = 1. Since t 1 ∈ E m 1 and ℓ(t 1 , v) is not an even number, we have v / ∈ E m 1 . Continue by letting m 2 be minimal, subject to the condition that we can find some t 2 ∈ Λ 2 ∩ E m 2 . Necessarily m 2 > m 1 . Let As above, we find that Λ 3 ∩ E m 2 is empty because if v ∈ Λ 3 then τ (t 2 , v) = (dom t 2 ) and ℓ(t 2 , v) = 1, however t 2 ∈ E m 2 and ℓ(t 2 , v) must be even if v is to be an element of E m 2 as well.
Let m 3 > m 2 be minimal, subject to there being some t 3 ∈ Λ 3 ∩ E m 3 , and define It should be clear how to proceed. We obtain a decreasing sequence of sets (Λ j ) ∞ j=1 and corresponding least elements u j , with the property that Λ j+1 ∩ E m is empty whenever m ≤ m j . Moreover, if v ∈ Λ j+1 then k 1 , . . . k j / ∈ ran v. Let u = ∞ j=1 u j . Being the union of an increasing sequence of injective functions, u is also injective. By construction, we have ensured that k j / ∈ ran u for all j, whence ω \ ran u is infinite and u ∈ Λ. Moreover, u ∈ Λ j for all j. However, this means that u / ∈ E m for any m, because the m j form a strictly increasing sequence. This contradiction establishes the fact that D is not Gruenhage.
Corollary 2.5. The 1-point compactification K of D is a scattered compact non-Gruenhage space with ( * ). By Theorem 1.4 (but not Theorem 1.5), C(K) admits an equivalent lattice norm with a strictly convex dual norm.

The space C 0 (D) has a C ∞ -smooth bump
If L is locally compact and scattered then the Banach space C 0 (L) of continuous real-valued functions vanishing at infinity is an Asplund space. Recently, a consistent negative solution was given to the long-standing problem of whether every Asplund space admits a C 1 -smooth bump function, that is, a non-zero real-valued continuously Fréchet differentiable function which vanishes outside some norm-bounded set [7]. As far as the author is aware, the question of whether such an Asplund space can be found in ZFC, or whether an example of type C 0 (L) can exist, remains open. Thus, it makes sense to test C 0 (L) whenever a new locally compact scattered space L comes along. The purpose of this final section is to confirm that (unfortunately!) C 0 (D) does admit such a function. (1) whenever f ∈ C 0 (L) is non-zero then there exists (t, γ) ∈ L × Γ such that |f (t)| = f ∞ and (T f )(t, γ) = 0, and (2) for every pair (t, γ), the map f → (T f )(t, γ) is C ∞ -smooth, i.e., has Fréchet derivatives of all orders, on the set on which it is non-zero.
It follows from [5,Corollary 3] that if C 0 (L) admits such an operator then it admits a C ∞ -smooth bump function. We shall prove that C 0 (D) admits such an operator. Our method follows that of [6,Theorem 9.3], which shows that C 0 (Υ) admits a C ∞ -smooth bump function for every tree Υ. However, since the topology of D is slightly more complicated than that of ordinary trees, we present some of the details. Lemma 3.2. Suppose that U and V are open subsets of D such that the restrictions π ↾ U and π ↾ V of the natural projection π are injective, and π(U) = π(V ) = (r, t] for some r ∈ Λ ∪ {0}, t ∈ Λ. Then there exist basic open sets W 1 , . . . , W k and W ′ 1 , . . . , W ′ k such that U = W 1 ∪ · · · ∪ W k and V = W ′ 1 ∪ · · · ∪ W ′ k , and given any i ≤ k, The argument is similar to the one used to show that the basis elements are compact. Set t 1 = t and take p 1 , q 1 ∈ {1, −1} such that (t 1 , p 1 ) ∈ U and (t 1 , q 1 ) ∈ V . From Proposition 2.2, we can find t 2 ∈ [r, t 1 ) such that W (t 2 , t 1 , p 1 ) ⊆ U and W (t 2 , t 1 , q 1 ) ⊆ V . Set W 1 = W (t 2 , t 1 , p 1 ) and W ′ 1 = W (t 2 , t 1 , q 1 ). If p 1 = q 1 then W 1 = W ′ 1 , and if not then W 1 ∩ W ′ 1 is empty. If t 2 = r then stop. Otherwise, continue by finding p 2 , q 2 ∈ {1, −1} and t 3 ∈ [r, t 2 ) such that W (t 3 , t 2 , p 2 ) ⊆ U and W (t 3 , t 2 , q 2 ) ⊆ V . This process stops at a finite stage k. Since π is injective on U, we have U = W 1 ∪ · · · ∪ W k , and similarly for V .
In the next lemma, we gather together some properties of elements in C 0 (D) that we need in order to define our Talagrand operator. (1) Given (s, i) ∈ D, there are only finitely many (t, j) ∈ (s, i) + satisfying |f (t, j)| ≥ δ.
(2) If f is non-zero then there exists maximal s ∈ Λ, subject to there being Proof.
(1) Observe that is compact and discrete, hence finite.
(2) If f is non-zero then is compact, and thus there exist finitely many pairs (s k , i k ) ∈ M, k ≤ n, such that M ⊆ n k=1 W (0, s k , i k ). Take maximal s amongst the s k . (3) The intersection of any two basis elements of D is a finite union of pairwise disjoint basis elements. Hence, by a standard Stone-Weierstrass argument, C 0 (D) is equal to the closed linear span of the family of indicator functions 1 W , as W ranges over the basis elements. Thus we are done by uniform approximation if we can show that (3) applies to finite linear combinations of the 1 W . Let W 1 , . . . , W n be basis elements, a 1 , . . . , a n ∈ R and set f = n k=1 a k 1 W k . By splitting the W k into smaller basis elements if necessary, and by using Lemma 3.2, we can assume that whenever k = l, either W k = W l or W k ∩ W l is empty.
If π(W k ) = (r k , t k ], k ≤ n, then set F = {r 1 , . . . , r n } ∪ {t 1 , . . . , t n }. Take any (s, i) ∈ D satisfying s / ∈ F . Define E to be the set of k ≤ n such that (s, i) ∈ W k , so that f (s, i) = k∈E a i . From above, we know W k = W l and t k = t l whenever k, l ∈ E. If E is non-empty then let's denote this common set and endpoint by W and u, respectively. Because (s, i) ∈ W and s / ∈ F , we have s ≺ u. Take t ∈ s + such that t u, and then j ∈ {1, −1} such that (t, j) ∈ W . It is clear that (t, j) / ∈ W k whenever k / ∈ E, else W k = W ∋ (s, i).
In conclusion, f (t, j) = k∈E a k = f (s, i). If E is empty then pick any t ∈ s + . If (t, 1) / ∈ W k for all k then we are done because f (t, 1) = 0 = f (s, i). If (t, 1) ∈ W k for some k then we claim that (t, −1) / ∈ W l for any l. Certainly, (t, −1) / ∈ W k . We have r k ≺ t t k , meaning r k s, and as s / ∈ F we know that r k ≺ s. Because (s, i) / ∈ W k , we must have (s, −i) ∈ W k instead. Suppose that (t, −1) ∈ W l for some l. Then by the same argument we have r l ≺ s ≺ t l and (s, −i) ∈ W l . However, this implies (t, −1) ∈ W l = W k , which isn't so. Therefore (t, −1) / ∈ W l for all l and f (t, −1) = 0 = f (s, i). To verify that T is indeed a Talagrand operator of class C ∞ , we simply use Lemma 3.3 and follow the proof of [6, Theorem 9.3], replacing s by (s, i) and t by (t, j) throughout.
By [3], it follows that C 0 (D) also admits C ∞ -smooth partitions of unity.
School of Mathematical Sciences, University College Dublin, Belfield, Dublin 4, Ireland E-mail address: richard.smith@ucd.ie