Bulletin T.CXXXI de l’Académie serbe des sciences et des arts − 2005 Classe des Sciences mathématiques et naturelles Sciences mathématiques, N\underline{o} 30

REPRESENTING TREES AS RELATIVELY COMPACT SUBSETS OF THE FIRST BAIRE CLASS

S. TODORČEVIĆ

(Presented at the 7th Meeting, held on October 29, 2004)

A b s t r a c t. We show that there is a scattered compact subset K of the first Baire class, a Baire space X and a separately continuous mapping f : X × K ⟶ ℝ which is not continuous on any set of the form G × K, where G is a comeager subset of X. We also show that it is possible to have a scattered compact subset K of the first Baire class which does have the Namioka property though its function space C(K) fails to have an equivalent Fréchet-differentiable norm and its weak topology fails to be σ-fragmented by the norm.

AMS Mathematics Subject Classification (2000): 46B03, 46B05 Key Words: Baire Class-1, Function spaces, Renorming

Recall the well-known classical theorem of R.Baire which says that for every separately continuous real function defined on the unit square [0, 1]×[0, 1] is continuous at every point of a set of the form G×[0, 1] where G is a comeager subset of [0, 1]. This result was considerably extended by I.Namioka [12] who showed that the same conclusion can be reached for separately continuous functions on products of the form X × K where X is a Cech-complete space and where K is a compact space. A typical application of this result in functional analysis would say that a separately continuous group operation on a locally compact space is actually jointly continuous, so we in fact have a topological group. Another application would say that a weakly compact subset of a Banach space contains a comeager subset on which the weak and norm topologies coincide. Following [1], let us say that a compact space K has the Namioka property if for every Baire space X and every separately continuous mapping f : X × K ⟶ ℝ there is a comeager set G ⊆ X such that f is continuous at every point of the product G × K. Note that this is equivalent to saying that for every continuous map f from a Baire space X into (C(K), τ_{p}), where τ_{p} denotes the topology of pointwise convergence on K, there is a comeager subset G of X such that f : X ⟶ (C(K), norm) is continuous at every point of G. In this reformulation, the Namioka property enters into the theory of smoothness and renormings of Banach spaces [2] where it has been shown to be a useful principle of distinguishing various classes of spaces. For example, Deville and Godefroy [1] have shown that if C(K) admits a locally uniformly convex^{1} equivalent norm that is pointwise lower semicontinuous, then K has the Namioka property. The Namioka property is particularly interesting in the class of scattered compacta K which in terms of the function space C(K) is equivalent to saying that every continuous convex real-valued function defined on a convex open subset of C(K) is Frechet-differentiable at every point of a comeager subset of its domain^{2}. The class of strong differentiability function spaces C(K) is rich enough to distinguish many more smoothness requirements that one can have on a given Banach space. This has been shown by R. Haydon [3] by analyzing a particular kind of scattered locally compact space given by set-theoretic trees T with their interval topologies, i.e. topologies τ_{in} generated by basic open sets of the form (s, t] = {x ∈ T : s <_{T} x ≤_{T} t}, where s, t ∈ T ∪ {−∞} (see [17]). The purpose of this note is to show that there exist interesting trees T for which the corresponding locally compact space is homeomorphic to a relatively compact subset of the first Baire-class. Our first result in this direction was motivated by a conjecture of Haydon [4] and a question of A. Moltó [11] regarding a result from [5] which shows that if K is a separable compact set of Baire-class-1 functions, each of which have only countably many discontinuities, then C(K) admits an equivalent norm that is locally uniformly convex. Our example shows that this is no longer

1A norm ∥ · ∥ is locally uniformly convex if ∥ x_{n} ∥⟶∥ x ∥ and ∥ x + x_{n} ∥⟶ 2 ∥ x ∥ imply that ∥ x − x_{n} ∥⟶ 0.

2A Banach space with this property is usually called a strong differentiability space or an Asplund space.

true for an arbitrary (scattered) compact subset of the first Baire class. We use the K’s failure to satisfy the Namioka property in order to prevent its function space C(K) to have locally uniformly convex renorming. Our second example, however, shows that there might exist a compact scattered subset K of the first Baire class that has the Namioka property but still its function space C(K) failing to have an equivalent locally uniformly convex norm. This example should be compared with the example of Namioka and Pol[13] of a compact scattered space K whose function space C(K) distinguishes the same properties but which is far from being representable inside the first Baire class. Our results are obtained via a general procedure which represents countably branching trees admitting a strictly increasing map into the reals as relatively compact subsets of the first Baire class. So our results also bare on the possible structure theory of compact subsets of the first Baire class, a theory that already has isolated some of its critical examples [19].

1. Trees as relatively compact subsets of the first Baire class

The Helly space^{3} H, the split interval^{4} S(I), and the one-point compactification A(D) of a discrete space of cardinality at most continuum are some of the standard examples of pointwise compact sets of Baire-class-1 functions. Given a point x of some Polish space X, let δ_{X} : X ⟶ 2 be the corresponding Dirac-function δ_{X}(y) = 0 iff x ≠ y, then

A(X) = {δ_{X} : x ∈ X} ∪ {¯0}

is one of the representations of A(D) inside the first Baire class over X. The structure theory of compact subsets of the first Baire class developed in [19] unravels another critical example, the Alexandroff duplicate D(M) of a compact metric space M. It is the space on M × 2 where the points of M × {1} are taken to be isolated and where a typical open neighborhood of some (x, 0) has the form (U × 2) \ F for U an open neighborhood of x in M and F a finite subset of M × {1}. To see that D(M) is representable inside

3Recall that the Helly space is the set H = {f : [0, 1] ⟶ [0, 1] : f is monotonic} equipped with the topology τ_{p} of pointwise convergence on [0, 1]. The result of [5] shows that its function space C(H) admits a locally uniformly convex renorming.

4The split interval S(I) is the lexicographic product I × 2 of the unit interval I = [0, 1] and the 2-element ordering 2 = {0, 1}. Note that S(I) is homeomorphic to a subspace of H so its function space also admits a locally uniformly convex renorming.

the first Baire class, note that the subspace

{(x, 0) : x ∈ M} ∪ {(x, δ_{X}) : x ∈ M}

of M × A(M) is homeomorphic to D(M).

Let us identify the power set of the rationals with the Cantor cube 2^{ℚ}. For s, t ∈ 2^{ℚ}, let s ⊑ t denote the fact that s ⊆ t and

(∀x ∈ s)(∀y ∈ t \ s)x <_{ℚ} y.

For t ∈ 2^{ℚ}, let

[t] = {x ∈ 2^{ℚ} : t ⊑ x}.

Note that [t] is a compact subset of 2^{ℚ} which reduces to a singleton if sup(t) = ∞; otherwise if sup(t) < ∞, the set [t] is homeomorphic to 2^{ℚ} via a natural homeomorphism induced by an order-isomorphism of ℚ and ℚ ∩ [sup(t), ∞). Let 1_{[t]} : 2^{ℚ} ⟶ 2 be the characteristic function of the subset [t] of 2^{ℚ}, i.e.

1_{[t]}(x) = 1 iff t ⊑ x.

Note that 1_{[t]} is a Baire-class-1 function on 2^{ℚ} and that 1_{[t]} = δ_{t} iff sup(t) = ∞.

Let wℚ be the collection of all subsets of ℚ that are well-ordered under the induced ordering from ℚ. We consider wℚ as a tree under the ordering ⊑. The tree (wℚ, ⊑) has been indentified long time ago by Kurepa [9] who showed that, while there is a strictly increasing mapping from (wℚ, ⊑) into the reals, there is no strictly increasing mapping from (wℚ, ⊑) into the rationals. Since we are going to consider several different topologies some living on the same set, let us fix some notation for them. We have already reserved the notation τ_{in} for the locally compact topology on wℚ and its subtrees that is generated by subbasic clopen sets of the form

(−∞, t] = {x : x ⊑ t} (t ∈ wℚ).

When we consider the first Baire class B_{1}(X) over some separable space X we usually consider it equipped with the topology τ_{p} of pointwise convergence on X. The notation for few other topologies living typically on wℚ or its subtrees will be fixed as we go on.

Consider the following subset of the first Baire-class B_{1}(2^{ℚ}) over the Cantor set 2^{ℚ}:

K_{wℚ} = {1_{[t]} : t ∈ wℚ}.

Lemma 1. The set K_{wℚ} is a relatively compact^{5} subset of B_{1}(2^{ℚ}) with only the constantly equal to 0 mapping ¯0 as its proper accumulation point.

P r o o f. By Rosenthal’s theorem [16] characterizing the relative compactness of subsets of B_{1}(2^{ℚ}), it suffices to show that every sequence (t_{n}) of elements of wℚ has a subsequence (t_{n_{k}}) such that the sequence (1_{[t_{n_{k}}]}) converges pointwise to an element of K_{wℚ} ∪{¯0}. To see this, we apply Ramsey’s theorem and get a subsequence (t_{n_{k}}) of (t_{n}) such that either

1. t_{n_{k}} and t_{n_{l}} are incomparable in wℚ whenever k ≠ l or

2. t_{n_{k}} ⊑ t_{n_{l}} whenever k < l.

If (1) holds, (1_{[t_{n_{k}}]}) converges pointwise to ¯0, while if (2) holds then (1_{[t_{n_{k}}]}) converges pointwise to 1_{[t]} where t = ⋃_{k=0}^{∞} t_{n_{k}}. □

Lemma 2. The map t ↦ 1_{[t]} is a homeomorphism between (wℚ, τ_{in}) and (K_{wℚ}, τ_{p}).

P r o o f. Consider a subbasic clopen set

{1_{[t]} : t ∈ wℚ, x ∈ [t]}

of K_{wℚ}, where x ∈ 2^{ℚ}. Its preimage is equal to

{t ∈ wℚ : t ⊑ x},

a typical subbasic clopen set of the interval topology of wℚ. □ Combining Lemmas 1 and 2 we obtain the following.

Theorem 3. The one-point compactification of the tree wℚ is homeomorphic to a compact subset of the first Baire class. □

Our interest in the tree wℚ is partly based on its universality in the following sense.

Theorem 4. Suppose T is a Hausdorff ^{6} tree of cardinality at most continuum which admits a strictly increasing mapping into the reals. Then T is homeomorphic to a subspace of wℚ. If T is moreover countably branching then T is homeomorphic to an open subspace of wℚ.

5Recall that we take B_{1}(2^{ℚ}) equipped with the topology τ_{p} of pointwise convergence on 2^{ℚ} and that a subset S of B_{1}(2^{ℚ}) is relatively compact if its τ_{p}-closure is compact.

6A tree T is Hausdorff if two different nodes on the same level of T have different sets of predecessors.

P r o o f. Note that every tree T satisfying the hypothesis of the first part of Theorem 4 is isomorphic to a restriction to the form S ↾ Λ, where S is some countably branching tree and where Λ denotes the set of all countable limit ordinals. It follows that it suffices to prove that every countably branching Hausdorff tree T which admits a strictly increasing map f : T ⟶ ℝ is isomorphic to a downwards closed subtree of wℚ and therefore homeomorphic to an open subspace of wℚ when we view T and wℚ as locally compact spaces with their interval topologies. Let T^{0} denote the set of all nodes of T whose length is a successor ordinal, or topologically the set of all isolated points of (T, τ_{in}). Changing f if necessary, we may assume that f[T^{0}] ⊆ ℚ. In fact, we may assume that for every t ∈ T, if t_{0} ≠ t_{1} are two of its immediate successors, then f(t_{0}) and f(t_{1}) are two distinct rationals. To see that this can be arranged, let us assume as we may, that actually f[T_{0}] ⊆ ℚ_{d}, where ℚ_{d} denotes the set of all right-hand points of the complementary intervals to the Cantor ternary set. Since T is a well-founded ordering we can recursively change f to an f̄: T ⟶ ℝ in such a way that for all t ∈ T and q ∈ ℚ_{d}, the new mapping f̄ maps the subset

{x ∈ ImSuc(t) : f(x) = q}

of the set ImSuc(t) of immediate successors of t in T in a one-to-one fashion to the rationals of the complementary interval of the Cantor ternary set just left of q. Finally define φ : T ⟶ wℚ by

φ(t) = {f(x) : x ∈ T^{0}, x ≤_{T} t}.

Then φ is an isomorphic embedding of T into a downwards closed^{7} subtree of wℚ. □

Corollary 5. Every Hausdorff tree T of cardinality at most continuum admitting a strictly increasing real-valued function has a scattered compactification αT representable as a compact subset of the first Baire class. □

Remark 6. Note that in this representation the remainder αT \ T is either a singleton or homeomorphic to a one-point compactification of some discrete space. If one is willing to dispense with the remainder being scattered, one can easily have a compactification γT of T representable inside the first Baire class for which the remainder is homeomorphic to a subspace

7Recall that a subset T of wℚ is downwards closed if from s ⊑ t and t ∈ T we can conclude s ∈ T. Note that a downwards closed subset of wℚ is a τ_{in}-open subset of wℚ.

of the Alexandroff duplicate D(2^{ℕ}) of the Cantor set 2^{ℕ}. Hence one can have a first countable compactification of any tree satisfying the hypothesis of Corollary 5 that is still representable inside the first Baire class. To see this, referring to the above representation of D(2^{ℕ}) inside the first Baire class, it suffices to check that the subspace

{(x, ¯0) : x ∈ 2^{ℚ}} ∪ {(t, 1_{[t]}) : t ∈ wℚ}

of 2^{ℚ} ×A(2^{ℚ}) is a compactification of wℚ whose remainder is homeomorphic to a subspace of the duplicate D(2^{ℚ}). While this sort of Baire class-1 compactification γT fail to give us a strong differentiability space C(γT), their interest come from a recent work of W. Kubis and others about classes of metrizably fibered compacta (see [6]).

2. Separate versus joint continuity

Let

σℚ = {t ∈ wℚ : sup(t) < ∞}.

We consider σℚ as a subtree of wℚ and besides its locally compact interval topology τ_{in} we also consider the topology τ_{bc} generated by the following family of subbasic clopen sets

{x ∈ σℚ : t ⊑ x} and {x ∈ σℚ : sup(x) < q},

where t ∈ σℚ and q ∈ ℚ. In his numerous papers about trees of the form σℚ and wℚ, Kurepa has presented at least two different proofs that the two trees admit no strictly increasing mapping into the rationals. The argument from one of these two proofs, which is exposed in the author’s [17], is sufficient for proving the following property of the space (σℚ, τ_{bc}) though its present formulation was inspired by a similar fact about a closely related tree (also known to Kurepa) appearing in Haydon’s paper [3].

Lemma 7. (σℚ, τ_{bc}) is a Baire space.

P r o o f. It suffices to show that the player Nonempty has a winning strategy σ in the Banach-Mazur game (see [14]). Given a nonempty open set U ∈ τ_{bc}, we let σ(U) be any basic open subset of U of the form

[t, ∞)_{q} = {x ∈ σℚ : t ⊑ x, sup(x) < q},

where t ∈ σℚ is such that sup(t) ∉ ℚ and where q is a rational such that q > sup(t). Thus in any infinite run

U_{0} ⊇ [t_{0}, ∞)_{q_{0}} ⊇ U_{1} ⊇ [t_{1}, ∞)_{q_{1}} ⊇ …

of the Banach-Mazur game

t ∈ ⋂_{n=0}^{∞} U_{n} = ⋂_{n=0}^{∞}[t_{n}, ∞)_{q_{n}},

where t = ⋃_{n=0}^{∞} t_{n}. □ Now define f : σℚ × (wℚ ∪ {∞}) ⟶ {0, 1} by

f(s, t) = 1 iff s ⊒ t.

Note that for each s ∈ σℚ, the corresponding fiber-mapping f_{s} : wℚ ∪ {∞} ⟶ {0, 1} is continuous since

{t ∈ wℚ ∪ {∞} : f_{s}(t) = f(s, t) = 1} = {t ∈ wℚ : t ⊑ s}

is a τ_{in}-clopen subset of wℚ ∪ {∞}. Similarly for each t ∈ wℚ ∪ {∞} the corresponding fiber-mapping f^{t} : σℚ ⟶ {0, 1} is continuous since

{s ∈ σℚ : f^{t}(s) = f(s, t) = 1} = {s ∈ σℚ : s ⊒ t}

is by definition a τ_{bc}-clopen subset of σℚ.

Lemma 8. The mapping f is not continuous on any set of the form G × (wℚ ∪ {∞}), where G is a comeager subset of σℚ.

P r o o f. Note that every comeager G ⊆ σℚ relative to the topology τ_{bc} contains a point s and two sequences (s_{n}) and (t_{n}) such that s ⊑ t_{n} ⊑ s_{n} for all n, such that s_{n} ⟶ s relative to τ_{bc}, and such that t_{n} is incomparable with t_{m} whenever n ≠ m. It follows that t_{m} ⟶ ∞ in the one-point compactification wℚ∪{∞} of the interval topology of wℚ. Thus, (s_{n}, t_{n}) ⟶ (s, ∞) in the product space σℚ × (wℚ ∪ {∞}). However, f(s_{n}, t_{n}) = 1 for all n while f(s, ∞) = 0. □

The following result summarizes Lemmas 7 and 8 modulo the representation K_{wℚ} ∪ {¯0} of wℚ ∪ {∞} given above in Section 1.

Theorem 9. The compact scattered set K_{wℚ}∪{¯0} of Baire-class-1 functions on the Cantor cube 2^{ℚ} does not have the Namioka property about continuity of separately continuous functions on its products with Baire spaces.

□

Remark 10. In [5], Haydon, Moltó, and Orihuala have shown that if a compactum K can be represented as a compact set of Baire class-1 functions that have only countably many discontinuities, then the topology of C(K) of pointwise convergence on K is σ-fragmented by the norm, so in particular K has the Namioka property. In our representation of K_{wℚ}∪{¯0} inside the first Baire class over the Cantor cube 2^{ℚ} we use characteristic functions of closed subsets of 2^{ℚ} which most of the time have uncountably many discontinuities. Theorem 9 and the result of [5] show that the amount of discontinuities is necessary.

3. A tree that has the Namioka property

Recall that a subset A of a topological space X is a universally Baire subset of X if for every topological space Y, or equivalently for every Baire space Y, and every continuous mapping f : Y ⟶ X the preimage f^{−1}[A] has the property of Baire as a subset of Y. We say that A is a universally meager subset of X if for every Baire space Y and every continuous f : Y ⟶ X, the preimage f^{−1}[A] is a meager subset of Y, unless f is constant on some nonempty open subset of Y. We transfer these notions to subtrees of wℚ viewed as subsets of the Cantor set 2^{ℚ}. Recall that a subtree T of wℚ is an Aronszajn-subtree of wℚ (in short, A-subtree of wℚ) if

{t ∈ T : otp(t) = α}

is countable for every countable ordinal α. The first known A-subtree of wℚ was constructed by Kurepa [8]. In the next Section we shall however rely on a different construction of A-subtrees of wℚ discovered by the author in [18]. Our interest in these trees is based on the following observation.

Lemma 11. The one-point compactification T ∪ {∞} of a universally Baire A-subtree T of wℚ has the Namioka property about continuity of separately continuous functions defined on its products with Baire spaces.

P r o o f. Let X be a given Baire space and let f : X × (T ∪ {∞}) ⟶ ℝ be a given separately continuous mapping. Note that f can be identified with the mapping F : X ⟶ C(T ∪ {∞}) sending x ∈ X into the fiber f_{x} : T ∪{∞} ⟶ ℝ. The mapping F is continuous when we take C(T ∪{∞}) equipped with the topology τ_{p} of pointwise convergence on T ∪ {∞}. The conclusion that there is a comeager subset G of X such that f is continuous at every point of the product G × (T ∪ {∞}) is equivalent to the existence of a G ⊆ X such that F viewed as a mapping from X into C(T ∪ {∞}) with the norm topology is continuous at every point of G. To obtain such a G, it suffices to show that for every ε > 0 every nonempty open set U ⊆ X contains a nonempty open subset V such that the image F [U] has normdiameter ≤ ε. Working towards a contradiction, let us assume that for some U and ε such a V ⊆ X cannot be found. Replacing X by U, we may assume that U = X. Since T ∪ {∞} is a scattered space, standard arguments (see [3], [13]) show that by changing X and F we may assume that

F : X ⟶ C_{0}(T, 2),

where C_{0}(T, 2) is the family of all continuous {0, 1}-valued functions on T that vanish at ∞. Thus, C_{0}(T, 2) can be identified with the family of all compact clopen subsets of T. Since every compact clopen subset of T is a finite union of intervals of the form (s, t] (s, t ∈ T). So changing X and F again, we may assume that for every x ∈ X the image F (x) is an interval of the form (−∞, t] for t ∈ T. Thus we can replace (C_{0}(T, 2), τ_{p}) with (T, τ_{c}) as the range of F, where τ_{c} is the topology on T generated by subbasic open sets of the form

[t, ∞) = {x ∈ T : t ⊑ x} (t ∈ T).

Our assumption about F -images of open subsets of X amount to the assumption that

F : X ⟶ (T, τ_{c})

is not constant on any nonempty open subset of X. Note that for every q ∈ ℚ,

{t ∈ T : q ∈ t}

is a τ_{c}-open subset of T, so the function F viewed as a function from X into T with the separable metrizable topology τ_{m} of T induced by the Cantor cube 2^{ℚ} is a Borel map. So going to a comeager subset of X, we may assume that

F : X ⟶ (T, τ_{m})

is continuous. Let βF : βX ⟶ 2^{ℚ} be its extension to the Cech-Stone compactification βX of X. Our assumption that T is a universally Baire subset of 2^{ℚ} gives us that the preimage βF^{−1}[T] has property of Baire in βX. So pick an open set U and a meager set M such that

βF^{−1}[T] = U △ M.

Note that since X ⊆ βF^{−1}[T] is a Baire space, the open set U ⊆ βX is not empty. Let M = ⋃_{n=0}^{∞} M_{n}, where each M_{n} is nowhere dense in βX. Using the assumption that F is nowhere constant, which amounts to the assumption that βF is nowhere constant, it is straightforward to produce a Cantor scheme U_{σ} (σ ∈ 2^{<ℕ}) of nonempty open subsets of U such that:

(i) \overline{U}_{σ} ⊆ U \ M_{|σ|},

(ii) \overline{U}_{σ^{⌢}i} ⊆ U_{σ} for all i < 2 and

(iii) βF [\overline{U}_{σ^{⌢}0}] ∩ βF [\overline{U}_{σ^{⌢}1}] = ∅.

Let Z = ⋂_{n=0}^{∞} ⋃_{σ∈2^{n}} \overline{U}_{σ}. Then Z is a compact subset of U \M, so the image P = βF [Z] is a compact subset of T of size continuum. This contradicts the fact that an A-subtree T of wℚ, viewed as a subspace of the Cantor cube 2^{ℚ}, has universal measure 0 (see [19]). This finishes the proof. □

Theorem 12. Suppose T is a universally Baire A-subtree of wℚ which admits no strictly increasing mapping into the rationals. Then the onepoint compactification of T has the Namioka property but the weak topology of C_{0}(T) is not σ-fragmented ^{8} by the norm.

P r o o f. Note that if (C_{0}(T), τ_{p}) is σ-fragmented by the norm, then its subspace (C_{0}(T, 2) is σ-scattered and so in particular (T, τ_{c}) is σ-scattered. So the theorem will be proved once we show that (T, τ_{c}) is not σ-scattered. To see this, call a subset X of T special if it is a countable union of antichains of T. By our assumption, T is not special. So in order to show that (T, τ_{c}) is not a σ-scattered space, it suffices to show that for every nonspecial subset X of T and every well-ordering <_{w} on X there is t ∈ X and a sequence (x_{n}) ⊆ X converging to t relative to the topology τ_{c} such that t <_{w} x_{n} for all n. Otherwise, for every t ∈ X there will be a finite set F_{t} ⊆ ℚ∩[sup(t), ∞) such that min(x \ t) ∈ F_{t} for every x ∈ X such that t ⊑ x and t <_{w} x. Find a nonspecial Y ⊆ X and a finite F ⊆ ℚ such that F_{t} = F for all t ∈ Y. Since Y is nonspecial, it must contain an infinite sequence (t_{n}) such that t_{n} ⊏ t_{m} whenever n < m. Since <_{w} is a well-ordering, there must exist n < m such that t_{n} <_{w} t_{m}. By the choice of F_{t_{n}} = F, we conclude that

8Recall that a topological space (X, τ) is σ-fragmented by a metric ρ on X if for every ε > 0 there is a decomposition X = ⋃_{n=0}^{∞} X_{n}^{ε} such that for every n and A ⊆ X_{n}^{ε} there is U ∈ τ such that U ∩ A ≠ ∅ and ρ-diam(U ∩ A) < ε.

min(t_{m} \ t_{n}) ∈ F, contradicting the fact that F_{t_{m}} is also equal to F. This completes the proof. □

The following result follows from Theorem 12 and the results of Section 1.

Theorem 13. Under the assumption of Theorem 12, there is a scattered compact subset K of the first Baire class satisfying the Namioka property though the weak topology of its function space C(K) is not σ-fragmented by the norm, and so in particular, C(K) admits no locally uniformly convex renorming. □

Remark 14.

1. Recall that in the context of function spaces C_{0}(T) over a tree T, the norm σ-fragmentability of the weak topology of C_{0}(T) is equivalent to the existence of an equivalent norm on C_{0}(T) on whose unit sphere the weak and the norm topologies coincide (see [3]).

2. Regarding Theorem 13, we should also note that the existence of a scattered compactum K with the Namioka property such that C(K) is not σ-fragmented by the norm was first established by Namioka and Pol [13] assuming the existence of a co-analytic set of reals of cardinality continuum containing no perfect subset.

4. The hypothesis of Theorem 12

We finish the paper with comments about the consistency of assumption of Theorem 12. First of all, we should mention that the hypothesis of Theorem 12 is satisfied in the constructible universe. However, it’s consistency is considerably easier to show using the following construction based on the ideas from [18]. We start by fixing a C-sequence C_{α} (α < ω_{1}) such that C_{α+1} = {α} while for a limit ordinal α, C_{α} is a set of ordinals < α such that otp(C_{α}) = ω and sup(C_{α}) = α. Let C_{α}(0) = 0 and for 0 < n < ω, let C_{α}(n)) denote the n’th element of C_{α} according to its increasing enumeration with the convention that C_{α+1}(n) = α for all n > 0. From C_{α} (α < ω_{1}) one easily obtains a sequence e_{α} : α → ω (α < ω_{1}) of one-to-one mappings which is coherent in the sense that

{ξ < min{α, β} : e_{α}(ξ) ≠ e_{β}(ξ)}

is finite for all α and β (see [18]) though the reader can take this simply as an additional parameter in the definition of the functor r ↦ T (ρ_{1}^{r}) that we are now going to give. For each r ∈ ([ω]^{<ω})^{ω}, we associate another sequence C_{α}^{r} (α < ω_{1}) by letting C_{α}^{r} = ⋃_{n∈ω} D_{α}^{r}(n), where

D_{α}^{r}(n) = {ξ ∈ [C_{α}(n), C_{α}(n + 1)) : e_{α}(ξ) ∈ r(n)}.

(This definition really takes place only when α is a limit ordinal; for successor ordinals we put C_{α+1}^{r} = {α}.) Note that in general C_{α}^{r} is a subset of α of order type ≤ ω which in general may not be unbounded in α if α is a limit ordinal. However, for every r ∈ ([ω]^{<ω})^{ω} for which C_{α}^{r} is unbounded in α for every limit ordinal α, using C_{α}^{r} (α < ω_{1}), we can recursively define ρ_{1}^{r} : [ω_{1}]^{2} ⟶ ω as follows (see [18]):

ρ_{1}^{r}(α, β) = max{ρ_{1}^{r}(α, min(C_{β}^{r} \ α)), |C_{β}^{r} ∩ α|}.

Thus, given two countable ordinals α < β, the integer ρ_{1}^{r}(α, β) is simply the maximal of the ’weights’ |C_{β_{i}}^{r} ∩ α| of the ’minimal walk’ β = β_{0} > β_{1} > ... > β_{k} = α from β down to α along the C-sequence C_{α}^{r} (α < ω_{1}) determined by the condition that β_{i+1} = min(C_{β_{i}}^{r} \ α) for every i < k. It is known (see [18]) that the corresponding fiber mappings

(ρ_{1}^{r})_{β} : β ⟶ ω (β < ω_{1})^{9}

are all finite-to-one maps satisfying the coherence property saying that

{ξ < min{α, β} : ρ_{1}^{r}(ξ, α) ≠ ρ_{1}^{r}(ξ, β)}

is finite for all α and β. It follows that the corresponding tree

T (ρ_{1}^{r}) = {(ρ_{1}^{r})_{β} ↾ α : α ≤ β < ω_{1}}

is an Aronszajn-tree that admits a strictly increasing mapping into the real line. Hence, as we have seen it above, the tree T (ρ_{1}^{r}) is isomorphic to a downwards closed subtree of wℚ. Using the corresponding arguments from Section 6 of [18], we shall show the following property of the functor r ↦ T (ρ_{1}^{r}).

Theorem 15. If r is a Cohen real, then for every antichain A ⊆ T (ρ_{1}^{r}) there is a closed and unbounded set Γ ⊆ ω_{1} such that length(t) ∉ Γ for all t ∈ A.

9Defined by, (ρ_{1}^{r})_{β}(α) = ρ_{1}^{r}(α, β).

P r o o f. First of all, let us define what we mean by a ’Cohen real’ refereing the reader to some of the sources like [7] for more details if necessary. To express this, we consider the family [ω]^{<ω} of finite subsets of ω equipped with the discrete topology and its power ([ω]^{<ω})^{ω} with the corresponding product topology. Thus ([ω]^{<ω})^{ω} is just another topological copy of the Baire space ω^{ω}. A real r ∈ ([ω]^{<ω})^{ω} is Cohen over a given universe of sets (typically the one in which we put ourselves) if r belongs to every denseopen subset of ([ω]^{<ω})^{ω} belonging to the universe. Thus in particular, if r is a Cohen real then C_{α}^{r} is unbounded in α for every limit ordinal α, so the corresponding tree T (ρ_{1}^{r}) is well-defined. What we need to show is that for every subset A of T (ρ_{1}^{r}) for which the set {lenght(t) : t ∈ A} is stationary^{10} contains two comparable nodes. Since nodes of the tree T (ρ_{1}^{r}) are pairwise coherent this amounts to showing that for every stationary Γ ⊆ ω_{1} there exists γ, δ ∈ Γ such that (ρ_{1}^{r})_{γ} ⊏ (ρ_{1}^{r})_{δ}. This in turn amounts to showing that for every stationary Γ ⊆ ω_{1}, the set

G = {x ∈ ([ω]^{<ω})^{ω} : (∃γ, δ ∈ Γ) (ρ_{1}^{x})_{γ} ⊏ (ρ_{1}^{x})_{δ}}

is an subset of ([ω]^{<ω})^{ω} that is comeager relative to the set G_{ω_{1}} of all x ∈ ([ω]^{<ω})^{ω} for which C_{α}^{x} is unbounded in α for every limit ordinal α. So, given a finite partial function p from ω into [ω]^{<ω}, it is sufficient to find a finite extension q of p such that the basic open subset of ([ω]^{<ω})^{ω} determined by q is included in G modulo the set G_{ω_{1}}. Let n be the minimal integer that is bigger than all integers appearing in the domain of p or any set of the form p(j) for j ∈ dom(p). For γ ∈ Γ, set

F_{n}(γ) = {ξ < γ : e_{γ}(ξ) ≤ n}.

Applying the Pressing Down Lemma, we obtain a finite set F ⊆ ω_{1} and a stationary set Δ ⊆ Γ such that F_{n}(γ) = F for all γ ∈ Δ and such that if α = max(F) + 1 then e_{γ} ↾ α = e_{δ} ↾ α for all γ, δ ∈ Δ. A similar application of the Pressing Down Lemma will give us an integer m > n and two ordinals γ < δ in Δ such that C_{γ}(j) = C_{δ}(j) for all j ≤ m, C_{δ}(m + 1) > γ, and

e_{γ} ↾ (C_{γ}(m) + 1) = e_{δ} ↾ (C_{γ}(m) + 1).

Consider the graph H on the vertex-set ω where two different integers i and j are connected by an edge if we can find ξ < γ such that i = e_{γ}(ξ) and

10Recall that a subset Γ of ω_{1} is stationary if it intersects every closed and unbounded subset of ω_{1}.

j = e_{δ}(ξ), or vice versa i = e_{δ}(ξ) and j = e_{γ}(ξ). Since the mappings e_{γ} and e_{δ} are one-to-one and have only finite disagreement on ordinals < γ, one easily shows that the maximal proper H-path

P = {i_{0}, i_{2}, ...., i_{k}}

that starts from i_{0} = e_{δ}(γ) is finite (and in fact included in the e_{δ}-image of the finite set {ξ < γ : e_{γ}(ξ) ≠ e_{δ}(ξ)} ∪ {γ}) . The maximality of P means that i_{k} is not in the range of e_{γ}. This and the fact that P is an H-path gives that

e_{γ}^{−1}(P) = e_{γ}^{−1}(P \ {i_{k}}),

and that if we let D = e_{γ}^{−1}(P) , then e_{δ}^{−1}(P) = D ∪ {γ}. Let l > m be an integer such that D ⊆ C_{γ}(l + 1). Extend the partial function p to a partial function q with domain {0, 1, ..., l} such that:

(1) q(j) = P for all j such that m ≤ j ≤ l, and

(2) q(j) = ∅ for any j < m not belonging to dom(p).

Choose any x ∈ ([ω]^{<ω})^{ω} extending the partial mapping q and having the property that C_{β}^{x} is unbounded in β for every limit ordinal β. Then from the choices we made above about the objects n, Δ, F,m, γ, δ and q, we conclude that

(3) γ ∈ C_{δ}^{x}, and

(4) C_{δ}^{x} ∩ γ is an initial segment of C_{γ}^{x}.

It follows that, given a ξ < γ, the walk from δ to ξ along the C-sequence C_{β}^{x} (β < ω_{1}) either leads to the same finite string of the corresponding weights as the walk from γ to ξ, or else it starts with the first step equal to γ and then follows the walk from γ to ξ. Since ρ_{1}^{x}(ξ, δ) and ρ_{1}^{x}(ξ, γ) are by definitions maximums of these two strings of weights, we conclude that ρ_{1}^{x}(ξ, δ) ≥ ρ_{1}^{x}(ξ, γ). On the other hand, note that by (4), in the second case, the weight |C_{δ}^{x} ∩ ξ| of the first step from δ to ξ is less than or equal to the weight |C_{γ}^{x} ∩ ξ| of the first step from γ to ξ. It follows that we have also the other inequality ρ_{1}^{x}(ξ, δ) ≤ ρ_{1}^{x}(ξ, γ). Hence we have shown that

(∀β < γ) ρ_{1}^{x}(β, γ) = ρ_{1}^{x}(β, δ),

or in other words, that (ρ_{1}^{x})_{γ} ⊏ (ρ_{1}^{x})_{δ}. This finishes the proof. □

It follows that if r is a Cohen real, then T (ρ_{1}^{r}) admits no strictly increasing mapping into the rationals. This gives us one part of the hypothesis of Theorem 12. The other part is obtained assuming that 𝔭 > ω_{1} and the existence of an uncountable co-analytic set of reals without a perfect subset. (Recall that 𝔭 is the minimal cardinality of a centered family F of infinite subsets of ℕ for which one cannot find an infinite set M ⊆ ℕ such that M \N is finite for all N ∈ F.) Note that these two assumptions are preserved when a single Cohen real is added [15]. Recall also that these two assumptions imply that every set of reals of size at most ℵ_{1} is co-analytic and therefore that every A-subtree of wℚ is co-analytic (see [10]). Finally, note that coanalytic sets of reals are universally Baire. Thus, we have established the following

Theorem 16. If there is an uncountable co-analytic set of reals without a perfect subset , if 𝔭 > ω_{1}, and if r is a Cohen real, then T (ρ_{1}^{r}) is a universally Baire A-subtree of wℚ that admits no strictly increasing mapping into the rationals, and therefore its one-point compactification has the Namioka property and is homeomorphic to a compact subset of the first Baire class though the corresponding function space C_{0}(T (ρ_{1}^{r})) cannot be renormed by an equivalent locally uniformly convex norm.

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Matematicki Institut Kneza Mihaila 35 11001 Beograd Serbia and Montenegro e-mail:stevo@mi.sanu.ac.yu