1 Introducction

In this article, we take as reference the classic book Gâteaux Differentiability of Convex Functions and Topology: Weak Asplund Spaces ¹, in which Marián J. Fabian talks about the Asplund spaces. First, we define the multivalued functions as follows: Let 𝑋 and 𝑌 be sets. A multivalued function from 𝑋 to 𝑌 is a relation which assigns to each 𝑥 ( 𝑋 a subset of 𝑌, denoted 𝑓(𝑥); 𝑓(𝑥) = ø for some 𝑥 ∊ 𝑋 is admitted. The graph of the multivalued functions 𝑓 is the set Graph(𝑓): = {(𝑥, 𝑦) ∊ 𝑋 ( 𝑌: 𝑦 ( 𝑓(𝑥)}. If for each 𝑥 ∊ 𝑋, 𝑓(𝑥) has only one element, we say the function is single-valued. A single-valued function from 𝑋 to 𝑌, is a relation 𝑥 ↦ 𝑦 = 𝑓(𝑥) which assigns to each 𝑥 ∊ 𝑋 a unique 𝑦 ∊ 𝑌. The graph of such functions is Graph(𝑓): = {(𝑥, 𝑦) ∊ 𝑋 ( 𝑌: 𝑦 = 𝑓(𝑥)}.

The objective of this article is the study of the properties of the usco functions. A usco function is a multivalued function 𝑓: 𝑋 → 𝑌 between Banach spaces which is upper semicontinuous and 𝑓(𝑥) is compact for each 𝑥 ∊ 𝑋. A usco function 𝑓: 𝑋 → 𝑌 is minimal if for each usco function 𝑔: 𝑋 → 𝑌 such that

𝑔(x)⊆f(x) we have that f=𝑔.

One of the most important examples of the usco functions appears in the study of differentiability of the convex functions defined on an open convex subset Ω of a Banach space 𝑋. It is convenient to note that the subdifferential of a function 𝑓: Ω → ℝ ∪ {+∞} in 𝑥 ∊ Ω is the multivalued function ∂𝑓: 𝑋 → 𝑋´ defined as follows:

x'∈∂f(x) if and only if f(y) - f(x) ≥ ⟨y - x, x'⟩ for every y∈Ω

It is also important to observe that in the completely regular topological spaces, singletons and closed sets can be separated by open sets. If 𝑋 is a Banach space, the topological dual with the weak topology is a completely regular space. Let us remember that a subset 𝐺 of a topological space 𝑋 is residual if there exists a countable family (𝓤n)n∈ℕ of open dense subsets such that G⊇∩n𝓤n .

The Asplud 𝐺-spaces are Banach spaces in which every continuous convex function, defined on an open convex set Ω, is differentiable in the Gâteaux sense in a residual subset of Ω. We denote by 𝖀G the set of Asplund 𝐺-spaces. The Asplund 𝐹-spaces are Banach spaces in which every continuous convex function, is differentiable in the Fréchet sense in a residual subset of Ω. We denote by 𝖀F the set of Asplund 𝐹-spaces. A Banach space 𝑋 is a Stegall space if for every Baire space 𝑍, every usco minimal function 𝑓: 𝑍 → 𝑋´ (with the weak topology) is single-valued in a residual subset. We denote by 𝔖 the set of Stegall spaces.

2 𝛽-Differentiability

Let us start with the following definition:

Definition 2.1 Let 𝑋 be a vector space on a field 𝕂. A vector bornology in 𝑋 is a subset 𝛽 of parts of 𝑋, denoted by ℘(𝑋), satisfying the following axioms:

𝑏𝑜𝑟𝑣 1. The union of all sets of 𝛽 is 𝑋:

∪B∈β B = X

𝑏𝑜𝑟𝑣 2. 𝛽 is stable under inclusions. That is, if 𝐵 ( 𝛽 and 𝐷 ⊂ 𝐵, then 𝐷 ( 𝛽.

𝑏𝑜𝑟𝑣 3. 𝛽 is stable under finite union. That is, if

{Sk: 1 ≤k≤n}⊂β , then

∪k=1n‍Sk∈β.

𝑏𝑜𝑟𝑣 4. 𝛽 is stable under the sum operation. That is, if Sk:1≤k≤n⊂β then

∑k=1n‍Sk∈β.

𝑏𝑜𝑟𝑣 5. 𝛽 is stable under the scalar multiplication operation. That is, if 𝑆 ∊ 𝛽 and λ ( 𝕂, then λ𝑆 ∊ 𝛽.

𝑏𝑜𝑟𝑣 6 is stable under the formation of balanced envelope. That is, if 𝑆 ∊ 𝛽, then 𝑏𝑎𝑙 (𝑆) ∊ 𝛽. Where 𝑏𝑎𝑙 (𝑆) denote the balanced hull of a set 𝑆 defined by

balS=∩λ≤1‍λS.

In the context of a topological vector space (𝑋, τ), we have some natural bornologies.

Examples 2.2 Let (𝑋, τ) be an 𝑡.𝑣.𝑠., then the following collections of subsets of 𝑋 are vector bornologies.

1. The 𝐹-bornology βF is the collection of all the τ-bounded subsets of 𝑋.

2. The 𝐻-bornology βH is the collection of all compact subsets of 𝑋.

3. The 𝐺-bornology βG is the collection of all finite subsets of 𝑋.

From the definitions of these bornologies, it is easy to see that

βG⊆βH⊆βF.

For a proof of this result see ².

In the sequel, 𝑋 and 𝑌 denote Banach spaces, Ω is an open subset of 𝑋 and ℒ(𝑋, 𝑌) denotes the space of linear transformation from 𝑋 to 𝑌 that are continuous.

Definition 2.3 Let 𝛽 be a bornology in 𝑋. We say that a function 𝑓: Ω → 𝑌 is 𝛽-differentiable in 𝑎 ∊ Ω if there exists a function 𝑢 ∊ ℒ(𝑋, 𝑌) such that for every 𝑆 ∊ 𝛽

The linear and continuous function 𝑢 is called 𝛽-derivative of 𝑓 at the point 𝑎 ∊ Ω and is denoted by u:=dβ(x) .

The following result provides a differentiability criterium with respect to a given bornology:

Theorem 2.4 Let 𝑋 and 𝑌 be 𝑡.𝑣.𝑠., Ω an open subset of 𝑋, and 𝛽 be a bornology in 𝑋. A necessary condition for 𝑓: Ω → 𝑌 to be 𝛽-differentiable in 𝑎 ∊ Ω is that for every 𝑆 ∊ 𝛽

This result follows doing a suitable modification in the proof of proposition 1.23 in ³.

In the case of a convex continuous functions 𝑓:Ω ⊂ 𝑋 → 𝑌, we have the following characterization of the 𝛽-differentiable in 𝑎 ∊ Ω:

Theorem 2.5 Let 𝑋 be an 𝑡.𝑣.𝑠., Ω an open convex subset of 𝑋, and 𝛽 be a bornology in 𝑋. A necessary and sufficient condition for a convex and continuous function 𝑓:Ω → ℝ be 𝛽-differentiable in 𝑎 ∊ Ω is that for every 𝑆 ∊ 𝛽

Similar considerations as theorem 2.4.

3 USCO Functions

In this section we will go deeper into the study of the class of usco functions. We start giving some basic definitions.

Definition 3.1 (Multivalued function) Let 𝑋 and 𝑌 be topological spaces. A multivalued function of a set 𝑋 in a set 𝑌 is a correspondence 𝑥 ↦ (𝑥), which assigns to each 𝑥 ∊ 𝑋 a subset 𝑓(𝑥) of the set 𝑌. It is possible that the set (𝑥) is the empty set. The effective domain of this function is the set of the 𝑥 ∊ 𝑋 such that (𝑥) ≠ Ø.

Definition 3.2 (Upper/Lower semicontinuous multivalued function) Let 𝑋 and 𝑌 be topological spaces. A multivalued function b: 𝑋 → 𝑌 is upper semicontinuous in 𝑎 ( 𝑋 if for every open set 𝑊 in 𝑌 such that 𝑊 ( (𝑎) (open neighborhood of 𝑓(𝑎)) there exists an open set 𝑉 of 𝑋 such that 𝑎 ∊ 𝑉 (neighborhood of 𝑎) and 𝑓(𝑥) ( 𝑊 for all 𝑥 ∊ 𝑉. In the case that 𝑓 is upper semicontinuous for any 𝑎 ∊ 𝑋, we say that 𝑓 is upper semicontinuous in 𝑋.

A multivalued function 𝑓: 𝑋 → 𝑌 is lower semicontinuous in 𝑎 ∊ 𝑋 if for every open set 𝑊 in 𝑌 such that 𝑓(𝑎) ∩ 𝑊 ≠ Ø, there exists an open set 𝑉 of 𝑋 such that 𝑎 ∊ 𝑉 (neighborhood of 𝑎) and 𝑓(𝑥) ∩ 𝑊 ≠ Ø for all 𝑎 ∊ 𝑉. In the case that 𝑓 is lower semicontinuous for any 𝑎 ∊ 𝑋, we say that 𝑓 is lower semicontinuous in 𝑋.

The first remark is that 𝑓 is continuous in 𝑎 ∊ 𝑋 if it is both upper semicontinuous and lower semicontinuous at the point 𝑎.

Definition 3.3 (Graph of a multivalued function) Let 𝑋 and 𝑌 be topological spaces. The graph of a multivalued function 𝑓: 𝑋 → 𝑌 is the subset Graph (𝑓) of 𝑋 ( 𝑌 of the pairs (𝑥, 𝑦) ∊ 𝑋 ( 𝑌 such that 𝑦 ∊ (𝑥).

Definition 3.4 The limit values of a net (yt)t∈T in a topological space (𝑌, 𝛼) are the elements of the set

Where ≽ is the partial order relation in 𝛵.

We note that 𝑦 is a limit value of the indicated net if and only if there exists a subnet that converges to 𝑦.

Theorem 3.5 Let 𝑋 and 𝑌 be topological spaces. A multivalued function 𝑓: 𝑋 → 𝑌 is upper semicontinuous if and only if for any closed set 𝐶 ⊂ 𝑌, the set

is closed on 𝑋.

Proof. Suppose that 𝑓 is upper semicontinuous, we want to show that

Let 𝐶 be a closed subset of 𝑌 and 𝐴 = 𝐶^𝐶, then 𝐴 is an open set in 𝑌 and if (𝑥) ∩ 𝐶 = Ø, then 𝑓(𝑥) ( 𝐴. Since 𝑓 is upper semicontinuous, there is a neighborhood open 𝑉 of 𝑥 such that (𝑧) ( 𝐴 for all 𝑧 ∊ 𝑉. But this means that (𝑧) ∩ 𝐶 = Ø for all 𝑧 ( 𝑉 and, therefore,

is open set and so x∈X:fx∩C≠∅ is closed on 𝑋. Suppose now that

is closed with 𝐶 is closed, we want to establish that 𝑓 is upper semicontinuous. Let 𝑎 ∊ 𝑋 and 𝑊 is an open subset of 𝑌 such that (𝑎) ( 𝑊. Then, 𝑊^𝐶 is a closed set and by hypothesis

is closed in 𝑋 and its complement 𝑉 is an open set. Now, 𝑎 ∊ 𝑉 since (𝑎) ∩ 𝑊^𝐶 = Ø and 𝑓(𝑥) ∩ 𝑊^𝐶 = Ø for all 𝑥 ∊ 𝑉, so that 𝑓(𝑥) ⊆ 𝑊 for all 𝑋 ∊ 𝑉.

Observe that now we are able to introduce a relation of order on the set of multivalued functions (𝑋, 𝑌) with 𝑋 and 𝑌 being topological spaces. For 𝑓, 𝑔 in (𝑋, 𝑌), se define the order ≼ as

Definition 3.6 (Usco Function) Let 𝑋 and 𝑌 be topological spaces. A multivalued function 𝑓: 𝑋 → 𝑌 is a usco function if 𝑓 is upper semicontinuous such that 𝑓(𝑥) ≠ Ø and is compact for any 𝑥 ∊ 𝑋. We denote the set of usco functions from 𝑋 to 𝑌 by 𝒰 (𝑋,𝑌).

We say that 𝑓 ∊ 𝒰 (𝑋,𝑌) is a minimal usco function if 𝑓 is a minimal element of the ordered set (𝒰 (𝑋,𝑌), ≼). This means that, if 𝑔 ∊ 𝒰 (𝑋, 𝑌) and 𝑔≼ 𝑓, then 𝑓 = 𝑔.

Now, we will establish some results about usco functions.

Theorem 3.7 For every 𝑢 ∊ 𝒰 (𝑋,𝑌), there exists a minimal usco function 𝑓 ∊ 𝒰 (𝑋,𝑌) such that 𝑓≼ 𝑢.

Proof. Let 𝑢 ∊ 𝒰 (𝑋,𝑌) and let ℋ be the collection of usco functions ℎ such that ℎ≼𝑢. Let us show that every chain ℒ contained in ℋ is bounded below. If ℱ is a finite subset in ℒ, then ∩f∈𝓕 (fx) is a closed and nonempty subset contained in the compact set 𝑢(𝑥) and which we can order ℱ linearly. Then ∩h∈𝓛h(x) is a closed and nonempty set contained in the compact set 𝑢(𝑥) and is therefore compact. We apply theorem 3.5 to the chain {ℎ: ℎ ∊ ℒ} to conclude that the function

is upper semicontinuous and thus, a usco function that minorizes ℒ. By Zorn’s lemma ℋ has a minimal element.

Lemma 3.8 Let (𝑋, 𝛼) and (𝑌, τ) be topological spaces and 𝑓: 𝑋 → 𝑌 be a usco function. If (𝑥_𝑡,𝑦_𝑡)_𝑡∊𝛵 is a net in 𝐺𝑟𝑎𝑝ℎ(𝑓) and 𝑥_𝑡 → 𝑥, then (𝑦_𝑡)_𝑡∊𝛵 has at least one limit value in 𝑓(𝑥).

Proof. By contradiction, we assuming that the net (𝑦_𝑡)_𝑡∊𝛵 has not limit values in 𝐺𝑟𝑎𝐺ℎ(𝑓). That is, for all 𝑧 ∊ 𝑓(𝑥), there exists a 𝑡_𝑧 such that z∉ys:s≽tz¯ . Let 𝑊_𝑧 be an open neighborhood of 𝑧 such that Wz∩ys:s≽tz¯=∅ . Since 𝑓(𝑥) is compact, there exists a finite subset F⊆f(x) such that f(x)⊆∪z∈FWz=A . For the upper semicontinuity of 𝑓, there exists an open neighborhood 𝑉 ( 𝑋 of 𝑥 such that

As 𝑥_𝑡 → 𝑥, there exists a t¯∈T (where 𝛵 is a directed set) such that 𝑥_𝑡 ∊ 𝑉 provided that t≽t¯ . We can suppose that t≽t¯≽suptz:z∈F . Consequently, fxt⊆A if  t≽t¯ . By hypothesis, we have (xt,yt)∈Graph(f) , i.e., 𝑦_𝑡 ∊ 𝐴 and 𝑦_𝑡 ∊ 𝑊𝑧 for some 𝑧 ∊ 𝐹, which implies that Wz∩ys:s≽tz¯≠∅ , which is a contradiction.

Theorem 3.9 (Characterization of usco functions) Let (𝑋, 𝛼) and (𝑌, 𝛽) be topological spaces. A multivalued function 𝑓: 𝑋 → 𝑌 is a usco function if and only if its graph 𝐺𝑟𝑎𝑝ℎ(𝑓) is a closed set. Moreover, there exists a usco function 𝑢 ∊ 𝒰 (𝑋, 𝑌) such that 𝑓 ≼ 𝑢.

Proof. Assume that 𝑓 is a usco function. If prove that x,y∈Graphf¯ , then we will show that (𝑥, 𝑦) ∊ 𝐺𝑟𝑎𝑝ℎ(𝑓), i.e., 𝑦 ∊ 𝑓(𝑥). There is a net (𝑥_𝑡, 𝑦_𝑡)_𝑡∊𝛵 such that 𝑥_𝑡 → 𝑥, 𝑦_𝑡 → 𝑦 and yt∈f(xt) . By lemma 3.8, (𝑦_𝑡)_𝑡∊𝛵 has at least one adhesion value in 𝑓(𝑥). That is, there is a subnet ytss∈S such that yts→y'∈fx and by the uniqueness of the limits, y=y'∈fx . It is clear that 𝑓 ≼ 𝑢 = 𝑓.

In the other direction, if Graph (𝑓) is closed and there is a usco function 𝑢 such that 𝑓 ≼ 𝑢, then 𝑓 is a usco function. By theorem 3.5, it suffices to show that 𝑓(𝑥) is closed for all 𝑥 ∊ 𝑋. Let y∈fx¯ and we will show that 𝑦 ∊ 𝑓(𝑥). Then there is a net (𝑦_𝑡)_𝑡∊𝛵 in 𝑓(𝑥) such that 𝑦_𝑡 → 𝑦. Let 𝑥_𝑡 = 𝑥 for all 𝑡 ∊ 𝛵. Then (𝑥_𝑡, 𝑦_𝑡)_𝑡∊𝛵 is a net in 𝐺𝑟𝑎𝑝ℎ(𝑓) that fulfills the conditions of lemma 3.8 and, therefore, a subnet (𝑦_𝑡)_𝑡∊𝛵 converge to 𝑦′ ∊ 𝑓(𝑥) and as 𝑦_𝑡 → 𝑦, it follows that 𝑦′ = 𝑦 ∊ 𝑓(𝑥).

The following lemma will be helpful for the main result in this section.

Lemma 3.10 Let (𝑋, τ) be the topological space, (𝑌, σ) Hausdorff space and 𝑓: 𝑋 → 𝑌 a usco function. The following are equivalent:

(i)𝑓 is a minimal usco function.

(ii) If 𝐴 is an open subset of 𝑋, 𝑊 is an open subset of 𝑌 and 𝑓(𝑎) ∩ 𝑊 ≠ Ø for some 𝑎 ∊ 𝐴, then there exists an open nonempty subset 𝑉 ( 𝐴 such that 𝑓(𝑥) ( 𝑊 for all 𝑥 ∊ 𝑉.

(iii)If 𝐴 is an open subset of 𝑋 and 𝐶 is a closed subset of 𝑌 such that 𝑓(𝑎) ∩ 𝐶 ≠ Ø for all 𝑎 ∊ 𝐴, then 𝑓(𝑎) ( 𝐶 for all 𝑎 ∊ 𝐴.

Proof. We will show that (𝑖) ⇒ (𝑖 𝑖). Let 𝑎 ( 𝑋 and 𝑊 ( 𝑌 open subsets as in (ii). We need to establish that exists an 𝑎₀ ∊ 𝐴 such that 𝑓(𝑎₀) ( 𝑊 because the fact that 𝑓 is upper semicontinuous implies that there is an open neighborhood 𝑉 in 𝐴 of 𝑎₀ such that

We argue by contradiction, let us assume that this statement is false and let 𝐶:= 𝑊^𝐶 closed in 𝑌. Then, 𝑓(𝑥) ∩ 𝐶≠ Ø for any 𝑥 ∊ 𝐴. We define the function ℎ: 𝑋 → 𝑌 by

It is clear that ℎ(𝑥) ≠ Ø, closed for all 𝑥 ∊ 𝑋 and ℎ ≼ 𝑓. Then, we conclude that ℎ ∊ 𝒰 (𝑋, 𝑌) by theorem 3.9. Since 𝑓 is a minimal usco function by hypothesis, then ℎ = 𝑓 and consequently, ℎ(𝑥)= 𝑓(𝑥) ( 𝐶 for all 𝑥 ∊ 𝐴. This is a contradiction, since by hypothesis 𝑓(𝑎) ∩ 𝑊 ≠ Ø for some 𝑎 ∊ 𝐴.

Now we show that (𝑖 𝑖) ⇒ (𝑖) . By hypothesis 𝑓: 𝑋 → 𝑌 is a usco function. From theorem 3.7, there exists a minimal usco function 𝑔 ≼ 𝑓 and 𝑔 = 𝑓, as we will show below. If 𝑓 and 𝑔 are not equal, there exists 𝑥₀ ∊ 𝑋 such that 𝒈(x0)≠f(x0) . As 𝒈(x0)⊂f(x0) there exists 28 ∊ 𝑓(𝑥₀) such that 𝑧 ∉ 𝑔(𝑥₀). Thus, there exists an open 𝛵 in 𝑌 such that 𝑧 ∊ 𝛵 and 𝑔x0∩T¯=∅ since 𝑔x0 is compact. From this, it follows that 𝑔x0⊆T¯c . Let W:=T¯c⊆Y be an open neighborhood of 𝑔x0 . By the upper semicontinuity of 𝑔, there exists an open neighborhood 𝐴 ( 𝑋 of 𝑥₀ such that

This construction allows us to see that 𝑥₀ ∊ 𝐴 and 𝑓(𝑥₀) ∩ 𝛵 ≠ Ø. Then, there is a subset open nonempty 𝑉 ( 𝐴 such that

which is a contradiction, since 𝑔x⊆fx⊆T, 𝑔x⊆W and T∩W=∅ if x∈V .

Now we see that (𝑖𝑖) ⇒ (𝑖𝑖𝑖). Let 𝐴 be a subset open in 𝑋 and 𝐶 be a subset closed in 𝑌 as in (iii). Suppose there exists 𝑎 ∊ 𝐴 such that 𝑓(𝑎) is not contained in 𝐶. Then, 𝑓(𝑎) ∩ 𝑊≠ Ø for some 𝑎 ∊ 𝐴, where 𝑊:= 𝐶^𝐶 is an open subset in 𝑌. For (ii) there exists a nonempty open subset 𝑉 ( 𝐴 such that 𝑓(𝑥) ( 𝑊 for all 𝑥 ∊ 𝑉, i.e., 𝑓 (𝑉) ∩ 𝐶 = Ø , which is a contradiction, since by hypothesis 𝑓(𝑥) ∩ 𝐶≠ Ø for all 𝑥 ( 𝐴.

We will show that (𝑖𝑖𝑖) ⇒ (𝑖𝑖). Let 𝐴 ( 𝑋 and 𝑊 ( 𝑌, with 𝑓(𝑎₀) ∩ 𝑊≠ Ø such that 𝑎₀ ∊ 𝐴. Let us prove that there exists 𝑎 ∊ 𝐴 such that 𝑓(𝑎) ( 𝑊. Otherwise, we would have 𝑓(𝑎) ∩ 𝑊^𝐶 ≠ Ø for all 𝑎 ∊ 𝐴, where 𝐶:= 𝑊^𝐶 is a closed subset in 𝑌. Then, for (iii) we have 𝑓(𝑎) ( 𝑊^𝐶 for all 𝑎 ∊ 𝐴 and hence 𝑓(𝐴) ∩ 𝑊= Ø, which is a contradiction. Therefore, 𝑊 is an open neighborhood in 𝑌 of 𝑓(𝑎₀) and by the upper semicontinuity of 𝑓, there exists an open neighborhood V⊆X  of a0 contained in 𝐴 such that 𝑓(𝑉) ( 𝑊. In other words, 𝑓(𝑥) ( 𝑊 for all 𝑥 ( 𝑉.

We also have the following theorem of minimal usco functions:

Theorem 3.11 (Characterization of minimal usco functions) Let (𝑋, τ) and (𝑌,σ) be topological spaces and 𝑓: 𝑋 → 𝑌 be a usco function. Then the following statements are equivalent:

Proof. We will show 1) ⇒ 2).

(i) 𝑔 ∘ 𝑓 is compact and nonempty for all 𝑥 ∊ 𝑋. Indeed, as 𝑓 is a usco function, then 𝑓(𝑥) is compact for all 𝑥 ∊ 𝑋. Then 𝑔(𝑓(𝑥)) is compact since 𝑔 is continuous.

(ii) 𝑔 ∘ 𝑓 is upper semicontinuous. In fact, let 𝑥₀ ∊ 𝑋 and 𝑊 be an open neighborhood in 𝑍 such that 𝑔(𝑓(𝑥₀)) ⊆ 𝑊. By continuity, we have 𝐴:= 𝑔^-1 (𝑊) is open in 𝑌 such that 𝑓(𝑥₀) ⊆ 𝐴. Since 𝑓 is upper semicontinuous, there is a neighborhood open 𝑉 in 𝑋 of 𝑥₀ such that 𝑓(𝑥) ⊆ 𝐴 for all 𝑥 ∊ 𝑉 and, therefore, 𝑔(𝑓(𝑥)) ⊆ 𝑊 for any 𝑥 ∊ 𝑉. So, we have shown that (𝑔 ∘𝑓) ∊ 𝒰 (𝑋, 𝑍).

(iii) It remains to show that 𝑔 ∘ 𝑓 is a minimal usco function. Using equivalences established in lemma 3.10, we must verify that 𝑔 ∘ 𝑓 satisfies (ii). Let 𝐴 be an open subset in 𝑋 and 𝑊 subset open in 𝑍 such that ((𝑔 ∘ 𝑓(𝑎)) ∩ 𝑊≠ Ø for some 𝑎 ∊ 𝐴. For continuity of 𝑔, we have 𝑔^-1(𝑊) is an open subset in 𝑌 such that 𝑓(𝑎) ( 𝑔^-1 (𝑊) (this is because ((𝑔 ∘ 𝑓)(𝐴)) ∩ 𝑊≠ Ø if and only if 𝑓(𝐴) ∩ 𝑔^-1 (𝑊) ≠ Ø. Since 𝑓 is upper semicontinuous, there is an open neighborhood 𝑉 ⊆ 𝐴 of 𝑎 such that

𝑓(𝑥) ⊆ 𝑔^-1 (𝑊) for any 𝑥 ∊ 𝑉

and, therefore, 𝑔(𝑓(𝑥)) ⊆ 𝑊 for any 𝑥 ∊ 𝑉 .

We will show 2) ⇒ 1). Let 𝑍=𝑌 and 𝑖_𝑌: 𝑌→𝑌 be the identity function 𝑖_𝑌(𝑦)=𝑦.

We observe that 𝑖_𝑌 ∘ 𝑓 = 𝑓, which by hypothesis is a minimal usco function.

Definition 3.12 (Residual space) Let (𝑋, 𝛼) be a topological space. We said that 𝐺 ( 𝑋 is a residual set, if there is a family of open sets (𝒰_𝑛)_𝑛(ℕ which are dense in 𝑋 such that ⋂_𝑛∊ℕ 𝒰_𝑛 ⊆ 𝐺.

Definition 3.13 (Baire space) A Baire space is a topological space (𝑋, 𝛼) with the property that any residual is dense in 𝑋.

Regarding Baire space, we have the following well known examples (see ⁴):

1. Any locally compact topological space is a Baire space.

2. Any open subset of a Baire space is a Baire space.

3. Any complete metric space is a Baire space.

Theorem 3.14 Let (𝑋, 𝛼) be a Baire space and (𝑌, 𝑑) be a complete metric space. Then any minimal usco function 𝑓: 𝑋 → 𝑌 is single-valued in some residual subset 𝑅 of 𝑋.

Proof. Let 𝑓: 𝑋 → 𝑌 be a minimal usco function, we will see that, for every ε > 0, there exists an open set 𝑉 such that diameter of 𝑓(𝑉) is less than ε (𝑑𝑖𝑎𝑚(𝑓(𝑉)) < ε. In fact, let 𝑦 ∊ 𝑓(𝑋) and By,ε2 . There is at least one 𝑎 ∊ 𝑋 such that f(a) ∩ By,ε2 ≠∅ , otherwise f(x) ⊆ By,ε2c for all 𝑥 ∊ 𝑋. By the lemma 3.10.(ii), there exists an open nonempty subset 𝑉 of 𝑋 such that f(x) ⊆ By,ε2c for all 𝑥 ∊ 𝑉 and consequently

diamfV≤ε2

Let 𝐴_ε = ⋃{ 𝑉 ⊆ 𝑉 : 𝑋 𝑜p𝑒𝑛 and 𝑑𝑖𝑎𝑚(𝑓(𝑉)) < ε}. It is clear that this set is open. We will show that 𝐴_ε is dense. Let 𝑥 ∊ 𝑋 and 𝑈 be an open neighborhood of 𝑥. If y ∈ f(x) and By,ε2 then f(x) ∩ By,ε2 ≠∅ and again by lemma 3.10.(ii), there is an open nonempty subset 𝑉 of 𝑈 such that f(v) ⊆ By,ε2 for all 𝑣 ∊ 𝑉, i.e.,

diamf(v) ≤ε2

and, therefore, 𝑉 ( 𝐴_ε. We have shown that 𝑈 ∩ 𝐴_ε ≠ Ø. If ε = 𝑛^-1, we will write 𝐴_𝑛 instead of A1n . Then {𝐴_𝑛: 𝑛 ∊ ℕ} is a countable collection of open subsets which are dense in 𝑋. Since this is a Baire space, then 𝑅: = ⋂ _𝑛∊ℕ 𝐴_𝑛 is a subset residual dense. If 𝑥 ∊ 𝑅, then

diamf(x) <1n for any n ∈ℕ,

and, therefore, 𝑓(𝑥) consists of only one element. That is, 𝑓|_𝑅 is a single-valued function.

4 The Asplund Spaces

We recall that a subset 𝐺 of a topological space (𝑋, 𝛼) is a set 𝐺_δ if it can be expressed as a countable intersection of open and dense subsets. In complete metric spaces and so, in Banach spaces, any residual subset is dense by the Baire’s theorem.

Definition 4.1 (Asplund 𝐹-space/𝒢-space/𝒢𝒟-space) An Asplund 𝐹-space is a Banach space with the following property: If 𝑓 is a continuous convex function defined in open and convex subset 𝑈 ⊂ 𝑋, then 𝐹-differentiable in a dense subset 𝐺_δ of 𝑈.

A 𝒢-Asplund is a Banach space with the following property: If 𝑓 is a continuous convex function defined in open and convex subset 𝑈 ⊂ 𝑋, then 𝐺-differentiable in a dense subset 𝐺_δ of 𝑈.

An Asplund 𝒢𝒟-space is a Banach space with the following property: If 𝑓 is a continuous convex function defined in open subset 𝑈 ⊂ 𝑋, then 𝐺-differentiable in a dense subset 𝐺_δ of 𝑈.

We note that a 𝒢-Asplund space is also an Asplund 𝒢𝒟-space. We want to point out that in the usual literature, the Asplund 𝐹-spaces are called the Asplund spaces, the 𝒢-Asplund are called weak Asplund spaces and Asplund 𝒢𝒟-spaces are called weak differenciable spaces. We could talk about the 𝐻-Asplund spaces, but it does not make much sense because it has been shown that continuous convex function wich are 𝐺-differentiables are 𝐻-differentiables. In the original article by E. Asplund ⁵, a 𝒢-Asplund is a weak differentiability spaces (𝑊𝐷𝑆) and a Asplund 𝐹-space is called strong differentiability spaces (𝑆𝐷𝑆). The definition we use comes from R. Phelps ⁶.

We will denote by 𝔘_𝐺 the class of the 𝒢-Asplund and 𝔘_𝐹 the class of Asplund 𝐹-spaces.

4.1 Asplund 𝐹-Spaces and Dentables Subset of a Banach Space

In this section we present the proof of the Namioka-Phelps theorem, which characterizes Asplund 𝐹-spaces by means of the geometric condition of dentability. First, we discuss the notion of the dentable set. To do that, if 𝑀 is a subset of a Banach space 𝑋, we define the support function of this set 𝑀 as follows:

x'↦pMx':=supx∈Mx,x'

of 𝑋′ in ℝ ∪{∞}. If we replace 𝑀 by T=adhσX',X∘cvM , then 𝛵 is convex, closed and bounded in the norm topology and 𝑝_𝑀 = 𝑝_𝛵. This function is sublineal and lower semicontinuous considering in 𝑋′ the weak-* topology denoted by σ (𝑋′, 𝑋) because it is the upper lower bound of lower semicontinuous functions.

Suppose now that 𝑋 is a Banach space and 𝑀′ is a subset of 𝑋′. Proceeding as before, we define the support function of the set 𝑀′ as follows:

x'↦pM'x':=supx'∈M'x', x

of 𝑋′ in ℝ ∪{∞}. If T'=adhσX,X'∘cvM' , then 𝛵′ is convex, σ (𝑋′, 𝑋)-closed (topology in 𝑋) and bounded in the norm topology and 𝑝_𝑀′ = 𝑝_𝛵′.

Definition 4.1.2 (𝑋-slice) Let 𝑋 be a Banach space, 𝛼 > 0 and 𝑥 ∊ 𝑋. For a nonempty bounded subse t 𝑀 of 𝑋, we define the (𝛼, 𝑥′)-slice of 𝑀 as the set

𝓡α,x';M:=x∈M:x,x'>pMx'-α.

Is clear that every slice is a relatively open set of 𝑀 with the topology σ (𝑋′, 𝑋), i.e., σ (𝑋, 𝑋′)|_𝑀 (restriction of σ (𝑋, 𝑋′) to 𝑀).

Definition 4.1.3 (𝑋′-slice) Let 𝑋 be a Banach space, 𝛼 > 0 and 𝑥 ∊ 𝑋. For a nonempty bounded subset 𝑀′ of 𝑋′, we define the weak*- (𝛼, 𝑥)-slice of 𝑀′ as the set

𝓡'α,x;M':=x'∈M':x',x>pM'x-α.

We note that ℛ′(𝛼, 𝑥; 𝑀′) is open in the topology σ (𝑋′, 𝑋)|_𝑀′. On the other hand, if 𝛼 < 𝛽, then

𝓡'α,x;M'⊆𝓡' (β, x; M').

Moreover, if T'=adhσ(x,x') o cv(M'), then 𝓡'(α,x;M')⊆𝓡'(α,x;T') since 𝑝_𝑀′ = 𝑝_𝛵′.

Definition 4.1.3 (Dentable set) Let 𝑋 be a Banach space. A subset 𝑀 ⊂ 𝑋 is called dentable if it admits slices of arbitrarily small diameter. We say that a subset 𝑀′ ⊂ 𝑋′ is weak *-dentable if it admits weak *-(𝛼, 𝑥)-slice of arbitrarily small diameter. A Banach space is dentable if any bounded subset is dentable.

Theorem 4.1.4 (Namioka - Phelps). A Banach space 𝑋 is an Asplund 𝐹-space if and only if its dual 𝑋′ is weak *- dentable.

Proof. Suppose 𝑋 is an Asplund 𝐹-space and show that all bounded subset of 𝑋′ admits slices of arbitrarily small diameter, otherwise, out so there would be a bounded subset 𝑀′ of 𝑋′ in which all (𝛼, 𝑥)-slice has a diameter 𝑟 > 0.

Let T'=adhσX',X∘cvM' . This set is convex, σ (𝑋′, 𝑋)-closed and bounded. By the Banach-Alaoglu theorem, 𝛵′ is σ (𝑋′, 𝑋)-compact. Since 𝑀′ ( 𝛵′, all (𝛼, 𝑥)-slice of 𝛵′ has a diameter ≥ 𝑟, we will show that the continuous sublinear function

x'↦px=pM'x:=supx'∈M'x',x

Of 𝑋 in ℝ is not 𝐹-differentiable in any point, which contradicts that 𝑋 is an Asplund 𝐹-space. With this purpose in mind, let ε=r4 and δ > 0 arbitrary. We will choose 𝛼 and 𝛽 appropriately. For now, we assume that 𝑟 − 2 𝛽 > 0 , 𝛼 > 0 and 𝛽 > 0. If 𝑥 ∊ 𝑋, then 𝑑𝑖𝑎𝑚ℛ′(𝛼, 𝑥; 𝛵′) ≥ 𝑟, so that there are 𝑎′, 𝑏′ ∊ ℛ′(𝛼, 𝑥; 𝛵′) such that ‖ 𝑎′− 𝑏′‖_𝑋′ ≥ 𝑟 − 𝛽,

a',x  >  px-α    and    b',x  >  px-α

There exists a y∈X such that yx= 1 and a'-b', y>r-2β Now, if 𝑡 > 0, then

px+ty+px-ty-2px  ≥  a',x+ty+b',x-ty-a'+b',x-2α =  ta'-b',y-2α  ≥  tr-2β-2α.

We can now choose 𝛼 and 𝛽 appropriately. Let β=r4 and α=rδ4 . So

Since inequality (5) holds for all 𝑡 > 0, in particular if t=δ2 , so we have shown that if ε=r4 and δ is arbitrary, then there is always a 𝑡 and a 𝑦 such that

0<t<δ,  yX=1    and    px+ty+px-ty-2pxt  ≥  r4.

In virtue of theorem 2.4 with the bornology 𝛽_𝐹, we conclude that 𝑝 is not 𝐹-differentiable in 𝑥, which is a contradiction.

Now, we will show that 𝑋 is an Asplund 𝐹-space. Let Ω be an open subset of 𝑋 and 𝑓: Ω → ℝ a convex function. Theorem 2.5 makes possible to locate the points of differentiability of 𝑓. The bornology that we are going to consider is 𝛽_𝐹, of all bounded subsets 𝑋 since here the 𝐹-derivative is defined. Let 𝐷(𝑓, ε) be the set of 𝑥 ∊ Ω for which there is a δ (𝑥, ε) > 0 such that

fx+th+fx-th-2fxt < ε  if  0<t<δ  for any  h∈X  with  h  ≤ 1

Then:

(i) 𝐷(𝑓, ε) is open for all ε > 0.

(ii) ∩ε>0‍Df,ε is the set of 𝐹-differentiability of function 𝑓.

The proof of (ii) is obvious. We will proof (i), i.e., 𝐷(𝑓, ε) is open. Since 𝑓 is locally Lipschitzian by theorem 1.1, there exists a 𝑟 > 0 and a constant 𝑘 > 0 such that 𝐵 (𝑥, 𝑟) ( Ω and

a,b∈Bx,r    implies    fb-fa  ≤  kb-aX.

By the definition of 𝐷(𝑓, ε), there exists δ > 0 such that δ<r2 and

fx+th+fx-th-2fxt  ≤  η  <  ε    if    hX≤1    and    0<t<δ.

Let z∈Bx,r2. Then z+th,  z-th∈Bx,r since

z±th-xX  ≤  r2+thX  <  r2+t  <  r2+δ  <  r.

Now let us observe that if 𝑡 = δ, then

fz+δh+fz-δh-2fzδ  ≤  fx+δh+fx-δh-2fxδ+fz+δh-fx+δhδ

+fz-δh-fx-δhδ+2fz-fxδ

≤  η+4kδz-xX.

Let α=minδ/4kε-η,r2 . Consequently,

fz+th+fz-th-2fzt  <  ε,  if  z-xX<α,  hX≤1  and  0<t<δ.

Once the points of 𝐹-differentiability of 𝑓 are located, it remains to show that 𝐴_ε: = 𝐷(𝑓, ε) is dense for every ε > 0. On the one hand, we have that Ω is a Baire space. If ε = 𝑛⁻¹, we will write An=Df,1n instead of A1n . Then {𝐴_𝑛: 𝑛 ∊ ℕ} is a countable collection of open subsets in Ω. Since Ω is a Baire space, if we define

R:=∩n∈ℕ‍An

(set of differentiability of 𝑓) is a set 𝐺_δ.

We will show that 𝐷(𝑓, ε) is dense for every ε > 0. That is, if 𝑥₀ ( Ω (fixed but arbitrary) any open neighborhood 𝑊 of 𝑥₀ intersects 𝐷(𝑓, ε). As the subdifferential function is locally bounded, we can suppose that ∂ 𝑓(𝑊) = 𝛵′ is a bounded subset of 𝑋′. As 𝑋′ is weak *-dentable, there exists a weak- *(𝛼, 𝑧)-slice of 𝛵′ of arbitrarily small diameter, let say < ε. If 𝑎′∊ ℛ(𝛼, 𝑧; 𝛵′), then 𝑎′∊ ∂ 𝑓(𝑎) for some 𝑎 in 𝑊, since

𝓡α,z;T'=x'∈T':x',z≥supx'∈T'x',z-α.

Since 𝑊 is open, there exists 𝑟 > 0 (sufficiently small) such that 𝑏:= 𝑎 + 𝑟𝑧 ∊ 𝑊. If 𝑏′ ∊ ∂ 𝑓(𝑏), therefore,

b',a-b  ≤  fa-fb    and    a',b-a  ≤  fb-fa,

consequently,

As 𝑎 − 𝑏 = −𝑟𝑧 and 𝑏 − 𝑎 = 𝑟𝑧 by replacing in (6)

0  ≥  b',-rz+a',rz=ra',z-b',z

and how 𝑟 > 0

b',z≥a',z>supa'∈T'a',z-α.

We have, thus, shown that ∂fb⊆R'α,z;T' . Now, the set ℛ(𝛼, 𝑧; 𝛵′) is σ (𝑋′, 𝑋) -open in 𝑋′ and as the subdifferential function is  X,σX',X -continuous, there exists a δ > 0 such that

Suppose that hX≤1. Then ∂fb+th ∂fb-th⊆𝓡'α,z;T' if 0<t<δ since

b+th-bX=thX ≤t<δ and b-th-bX=thX ≤ t<δ.

If u∈∂fb,v∈∂fb+th,w∈∂fb-th and 0<t<δ , then

ub+th-ub=ub+th  ≤  fb+th-fbvb-ub+th=v-th        ≤  fb+th-fbwb-wth=wb+th  ≤  fb-fb-th.

Now, as 0 < 𝑡 < δ and by the linearity of 𝑢, 𝑣, 𝑤 we have

uh ≤ fb+th-fbt ≤ vh-uh ≤fb-th-fbt≤-wh.

By (7) we know that 𝑣, 𝑤 ∊ ℛ′(𝛼, 𝑧; 𝛵′) and that the diameter of this set is < ε, then we obtain

0≤ fb+th+fb-th-2fbt≤vh-wh≤vh-whX' <ε.

This last inequality is true for every ℎ ∊ 𝑋 such that ‖ℎ‖_𝑋 ≤ 1 and every 0 < 𝑡 < δ. Therefore, 𝑏 ∊ 𝐷(𝑓, ε), i.e., 𝑊∩ 𝐷(𝑓, ε) ≠ Ø. This means that 𝐷(𝑓, ε) is dense and in consequence

R:=∩n∈ℕ‍An

(set of differentiability of 𝑓) is a 𝐺_δ-dense set. We have, thus, proved that 𝑋 is an Asplund 𝐹-space.

5 The Class of Stegall 𝕾 on Topological Spaces

This section is the central part of the article, since it introduces an intermediate class between the Asplund 𝐹-spaces and 𝒢-Asplund, called the class of Stegall 𝔖^ . This class establishes sufficient conditions for a Banach space to be a 𝒢-Asplund.

Let us recall that a topological space (𝑋, τ) is completely regular, if it is Hausdorff and for each closed set 𝐶 and every point 𝑝 that does not belong to 𝐶, there is a continuous function 𝑓: 𝑋 → [0,1] such that 𝑓|_𝐶 = 0 and 𝑓(𝑝) = 1. The Urysohn’s lemma, every metric space is completely regular.

Theorem 5.1 If 𝑋 is a separable Banach space and 𝐾 ⊂ 𝑋′ is bounded, then the topology σ (𝑋′, 𝑋)|_𝐾 is metrizable (the topology of the dual is metrizable on the bounded sets).

Theorem 5.2 Let 𝑋 be a Banach space. Then (𝑋′, σ (𝑋′, 𝑋)) is a space completely regular.

For the proof of previuos theorems see ¹.

Definition 5.3 (𝑆-space) An 𝑆-space is a completely regular topological space 𝑋 that satisfies the following condition:

If 𝑍 is a Baire space and 𝑓: 𝑍 → 𝑋 is a minimal usco function, then 𝑓 is single-valued in a residual subset of 𝑍. We will denote by 𝔖, the set of all the 𝑆-spaces.

Even though Marian J. Fabian in ¹ proves the following theorems, some important details were left out. We present here complete proofs.

Theorem 5.4 Every metric space (𝑋, 𝑑) is an 𝑆-space.

Proof. Let (𝑋, 𝑑) be a metric space, 𝑍 be a Baire space and 𝑓: 𝑍 → 𝑋 is a minimal usco function. For each 𝑛 ∊ ℕ, we define the open set

We will show that G=∩n∈ℕ‍Un Is residual. Note that each one 𝒰_𝑛 is open and we will prove that each 𝒰_𝑛 is dense for all 𝑛 ∊ ℕ. Let 𝑥 ∊ 𝒰_𝑛 and 𝑉_𝑥 ⊂ 𝑍 be an open neighborhood of 𝑥 with

diamfVx  <  1n

If y∈fx and By,13n, then fx∩By,13n≠∅ and by lemma 3.10.(ii), there is a nonempty open subset Ω of 𝑉_𝑥 such that fx' ⊆ By,13n for each 𝑥′∊ Ω. So

diamfΩ ≤ 13n

Then, Ω ⊂ 𝒰_𝑛 and, therefore, Ø ≠ 𝑉_𝑥 ∩ 𝒰_𝑛, which proof the density of the 𝒰_𝑛. So {𝓤n:n∈ℕ} is a countable collection of open and dense subsets in 𝑍 and, therefore, the intersection 𝐺 of the 𝒰_𝑛 is a residual subset. Finally, by vertue of theorem 3.14, the usco function 𝑓 is single-valued in 𝐺 and, consequently, 𝑋 ( 𝔖.

Theorem 5.5 Let 𝑋 be a completely regular space. If (Xn)n∈ℕ is a sequence of closed subsets of 𝑋_𝑛 , with 𝑋 ( 𝔖 for every 𝑛 ∊ ℕ and 𝑋=⋃_𝑛∊ℕ 𝑋_𝑛, then 𝑋 ∊ 𝔖.

Proof. Let 𝑍 be a Baire space and 𝑓: 𝑍 → 𝑋 a minimal usco function. For each 𝑛 ∊ ℕ, we define the set

Zn:=z∈Z:fz∩Xn≠∅.

Since 𝑓 is a upper semicontinuous function and by hypothesis each 𝑋_𝑛 is closed in 𝑋, then each 𝑍_𝑛 is a closed set in 𝑍 by theorem 3.5. If An=int(Zn), then A=∪n∈ℕAn is open and dense in 𝑍, since 𝑍 is a space of Baire and 𝑍 = ⋃_𝑛∊ℕ 𝑍_𝑛 (see ⁷, p.63). It is clear that

𝐴_𝑛 ≠ for some 𝑛 ∊ ℕ

Let 𝑛 ∊ ℕ be such that 𝐴_𝑛 ≠ Ø, and let fn=fAn . We will show that 𝑓_𝑛 is a minimal usco function and 𝑓_𝑛(𝐴_𝑛) ( 𝑋_𝑛.

1) In general, if 𝑓: 𝑍 → 𝑋 is a usco function and 𝑔: 𝑋 → 𝑌 (with 𝑌 a regular space) is a continuous single-valued function, by theorem 3.11.2, we have that 𝑔 ∘ 𝑓 is a usco function. From this it follows that if 𝑋 is a subspace of 𝑌, then 𝑓|_𝑋 is a usco function since fx=ix°f and 𝑖_𝑋 is continuous.

2) We will show that 𝑓_𝑛 is minimal. For this we will use lemma 3.10 several times. Let V⊆An open and C⊆X closed such that

3) fnz∩C≠∅    for any z∈V

Then

fz∩C≠∅    for any z∈V

and as 𝑓 is minimal, by lemma 3.10.(iii) then 𝑓(𝑧) ( 𝐶 for every 𝑧 ∊ 𝑉. So

fnz⊆C for any z∈V

and, hence, 𝑓_𝑛 is minimal (using lemma 3.10 again).

1) 𝑓_𝑛(𝐴_𝑛) ( 𝑋_𝑛. By the definition 𝑍_𝑛, we have fnz∩C≠∅ for every 𝑧 ∊ 𝐴_𝑛 and by lemma 3.10.(iii), we obtain that fnAn⊆Xn since 𝑋_𝑛 is closed by hypothesis.

Let

𝑅: = {𝑧 ∊ 𝑍: 𝑓(𝑧) is a singleton 𝑓 or all 𝑧}

And 𝑅_𝑛: = 𝑅 ∩ 𝐴_𝑛.

We note that 𝑅_𝑛 is the set of 𝑧 ∊ 𝐴_𝑛 such that fn(Z) consists of a single element. By hipothesis 𝑋_𝑛 ∊ 𝔖, so 𝐴_𝑛 is a Baire space because it is an open subset of a Baire space and 𝑓_𝑛: 𝐴_𝑛→ 𝑋_𝑛 is a minimal usco function. Then we conclude that 𝑅_𝑛 is a residual set in 𝐴_𝑛 for each 𝑛∊ℕ. Now we can show that 𝑅 is residual in 𝑍. We have shown that there exists a countable collection {An}n∈ℕ of open sets of 𝑍 such that 𝑅 ∩ 𝐴_𝑛 is residual in 𝐴_𝑛 and A=∪n∈ℕAn is dense in 𝑍. This implies that 𝑅 is residual as we shall proof continuation. We define

W1=A1  y  Wn=An-adh∪k=1n-1‍Ak  if  n≥2.

Then Wnn∈ℕ is an open and disjoint collection of open sets and 𝑊_𝑛 ( 𝐴_𝑛 for each 𝑛∊ℕ. Let us show that 𝑊 = ⋃_𝑛∊ℕ 𝑊_𝑛 is dense in 𝑍. Let 𝑧 ∊ 𝑍 and 𝑉 be a open neighborhood of 𝑧 contained in 𝑍. As 𝑉 ∩ 𝐴 ≠ Ø, then 𝑉 ∩ 𝐴_𝑛 ≠ Ø for some 𝑛∊ℕ. Let 𝑝 be the first ordinal such that 𝑉 ∩ 𝐴_𝑝 ≠ Ø. Then 𝑉 ∩ 𝑊_𝑝 ≠ Ø and, therefore, 𝑉 ∩ 𝑊 = 𝑉 ∩ ⋃_𝑛∊ℕ 𝑊_𝑛 ≠ Ø which shows that 𝑊 is dense at 𝑍.

Since 𝑅 ∩ 𝐴_𝑛 is residual in 𝐴_𝑛, there exist a countable collection {Tkn:k∈ℕ} of open and dense subsets of 𝐴_𝑛 such that R∩An ⊇∩k∈ℕTkn. Now since Wn⊆An , then

The set Wn∩Tkn is open and dense in 𝑊_𝑛. In fact, let 𝑧 ( 𝑊_𝑛 and 𝑉 be a open neighborhood of 𝑧 contained in 𝑊_𝑛. Then 𝑉 is an open neighborhood of 𝑧 contained in 𝐴_𝑛. Since 𝛵_𝑘𝑛 is dense in 𝐴_𝑛, then V∩Tkn≠∅ and, therefore, V∩(Wn∩Tkn)≠∅ . Since 𝑊_𝑛 is a Baire space, then

Wn∩∩k∈ℕ‍Tkn=∩k∈ℕ‍Wn∩Tkn≠∅.

This is the final part of the proof. Let Tk=∅∪n∈ℕ(Tk∩Wn) . Then we will see that 𝛵_𝑘 is open and dense in 𝑍. To prove this statement, let 𝑉 be an open and nonempty subset of 𝑍. Since 𝑊 is dense in 𝑍, then Wn∩V≠∅ and, therefore, V∩Wn≠∅ for some 𝑛(ℕ (since 𝑊 is defined as the union of 𝑊_𝑛). As Wn∩Tk is open and dense in 𝑊_𝑛, it follows that

V∩Wn∩Tkn∩Wn=V∩Tkn∩Wn≠∅.

We have, thus, shown that V∩Tk≠∅ for any open and nonempty of 𝑍, i.e., 𝛵_𝑘 is open and dense in 𝑍. As 𝑍 is a Baire space, then T:=∩k∈ℕTk is dense in 𝑍. Now we remark that

T=∩k∈ℕ‍Tk=∩k∈ℕ‍∪n∈ℕ‍Tkn∩Wn.

As

Wn∩Tk=∪m∈ℕ‍Wn∩Tkm∩Wm=Tkn∩Wn

Since Wn∩Wm=∅ if n≠m , then

∩k∈ℕ‍Tk=∩k∈ℕ‍∪n∈ℕ‍Wn∩Tkn=∩k∈ℕ‍Tkn∩∪n∈ℕ‍Wn=∪n∈ℕ‍∩k∈ℕ‍Wn∩Tkn.

Using (9), we obtain that

∩k∈ℕ‍Tk⊆∪n∈ℕ‍R∩Wn⊆R.

In other words, we have shown that 𝑅 is residual, since each 𝛵_𝑘 is open and dense, accordingly 𝑋 = ⋃_𝑛∊ℕ 𝑋_𝑛 ∊ 𝔖.