# A Characterization of the Existence of a Fundamental Bounded Resolution for the SpaceCc(X) in Terms of X

Authors: Juan Carlos Ferrando
Journal: Journal of Function Spaces
 (2018)

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## Abstract

We characterize in terms of the topology of a Tychonoff spaceX the existence of a bounded resolution for CcX that swallows the bounded sets, where CcX is the space of real-valued continuous functions on X equipped with the compact-open topology.


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## Body

## 1. Preliminaries

In the sequel, unless otherwise stated,X is a nonempty completely regular Hausdorff space. We represent by CpX the ring CX of real-valued continuous functions defined on X equipped with thepointwise topology τp. As usual, we denote by LpX the weak∗ dual of CpX. When CX is equipped with thecompact-open topology τc we write CcX. As in [1], we denote by C∗X the linear space of real-valued continuous and bounded functions defined on X. If C∗X is regarded as a subspace of CcX, we denote this space by Cc∗X. Since C∗X is dense in CcX, both Cc∗X and CcX have the same dual. The Banach space C∗X equipped with the supremum norm has recently been studied in [2]. Let us recall that a family {Aα:α∈NN} of subsets of a set X is called aresolution for X if it covers X and Aα⊆Aβ whenever α≤β; i.e., αi≤βi for every i∈N (see [3, Chapter 3]). If E is a topological vector space, a resolution {Aα:α∈NN} consisting ofbounded sets (see [4, Definition 1.4.5]) is referred to as abounded resolution. A bounded resolution {Aα:α∈NN} on E thatswallows the bounded sets, i.e., such that for each bounded set Q in E there is γ∈NN with Q⊆Aγ, is referred to as afundamental bounded resolution. The existence of such a resolution in a locally convex space E has shown to be equivalent to the existence of a so-called G-base of absolutely convex neighborhoods of the origin in the strong dual Eβ′ of E. Besides, fundamental bounded resolutions are essential in order to get a proper extension of the class of DF-spaces (see [5] for details). Fundamental compact resolutions, i.e., resolutions consisting of compact sets which swallow the compact sets, have been widely used, even in Banach space theory [6], since they were introduced in [7]. A well-known result of Christensen [8, Theorem 3.3] asserts that a metrizable space X is Polish if and only if X has a fundamental compact resolution. Moreover, it has been shown in [9, Theorem 2] that CcX has a G-base of neighborhoods of the origin if and only if X has a fundamental compact resolution. In this note we provide two characterizations, in terms of the domain space X, of the existence of a fundamental bounded resolution for CcX, one by means of certain uniformity for X and the other purely topological. Our main motivation is the two following Cp-theoretic results.Theorem 1 ([7, Theorem 3.7] or [10, Problem 216]).
The spaceCpX has a fundamental compact resolution if and only if X is countable and discrete.Theorem 2 ([11, Theorem 3.3]).
The spaceCpX has a fundamental bounded resolution if and only if X is countable.A spaceX is called K-analytic if there is an upper semicontinuous compact-valued map T from the product space NN, where N is equipped with the discrete topology, into X such that ⋃Tα:α∈NN=X. A family F of functions from a uniform space X,N into a uniform space Y,M is calleduniformly equicontinuous [12, Chapter 7, Problem G] if for each V∈M there is U∈N such that fx,fy∈V whenever f∈F and x,y∈U. Let N be a uniformity for a (nonempty) set X and denote by τN the uniform topology defined by N. A base {Uα:α∈NN} of the uniformity N is called a G-base if Uβ⊆Uα whenever α≤β. There is no loss of generality by assuming that each Uα is a symmetric vicinity.


## 2. Fundamental Bounded Resolutions forCcX

Although it can be easily seen that each metrizable locally convex spaceE has a fundamental bounded resolution, if the locally convex space CcX has a fundamental bounded resolution, unlike what happens with CpX, the space X needs not be countable and moreover CcX needs not be metrizable.Proposition 3.
LetX be a metrizable space. Then CcX has a fundamental bounded resolution if and only if X is σ-compact.Proof.
IfX is σ-compact then CcX is weakly K-analytic by [13, Proposition 2.2], or it has a compact resolution that swallows the compact sets by [14, Corollary 2.10]. In any case CcX has a bounded resolution. Since X is a kR-space then CcX is complete, hence locally complete. So, it follows from Valdivia’s [3, Theorem 3.5] that there exists a resolution {Aα:α∈NN} of CcX consisting of Banach disks that swallows the bounded sets of CcX, which shows that CcX has a fundamental bounded resolution. Conversely, if CcX has a fundamental bounded resolution then CpX has a bounded resolution, so [3, Corollary 9.2] shows that X is σ-compact.Example 4.
According to the previous propositionCcR has a fundamental bounded resolution, but R is neither countable nor discrete. On the other hand, CcQ has also a fundamental bounded resolution, but Q, although is countable, is not discrete. Observe that CcR is metrizable, but CcQ is not. Of course, if X is hemicompact, or even compact, then CcX is metrizable, or even a Banach space, and in this case CcX has obviously a fundamental bounded resolution.Theorem 5.
The spaceCcX has a fundamental bounded resolution if and only if X,M, where M is the uniformity for X generated by the pseudometrics (1)dAx,y=supf∈A⁡fx-fyfor each bounded set A of CcX, has a G-base.Proof.
LetE be the topological dual of CcX and denote by B the family of all bounded sets of CcX and by βE,CX the strong topology on E. By identifying X with its canonical homeomorphic embedding in LpX, note that X⊆LX⊆E. The strong topology βE,CX generates a unique admissible translation-invariant uniformity N on E, so that τN=βE,CX. By considering also f∈CX as a linear functional on E, observe that for each N∈N there is A∈B such that (2)u,v∈E×E:supf∈A⁡f,u-v≤1⊆N.Hence M⊆X×X belongs to the relative uniformity M of N to X×X if and only if there exists A∈B such that (3)x,y∈X×X:supf∈A⁡fx-fy≤1⊆M.If {Aα:α∈NN} is a fundamental bounded resolution for CcX, by setting (4)Uα=x,y∈X×X:supf∈Aα⁡fx-fy≤1we obtain a G-base {Uα:α∈NN} of M. If α≤β then Uβ⊆Uα, and if M∈M there is A∈B such that x,y∈M whenever supf∈Afx-fy≤1, so if γ∈NN is such that A⊆Aγ, clearly Uγ⊆M.
Conversely, if the uniform structure forX generated by the family of pseudometrics dA:A∈B, where (5)dAx,y=supf∈A⁡fx-fy,has a G-base {Uα:α∈NN}, this entails that for each A∈B there is δ∈NN such that supf∈Afx-fy≤1 for every x,y∈Uδ. Setting (6)Aα=f∈CX:supx,y∈Uα⁡fx-fy≤1,for each α∈NN, clearly Aα⊆Aβ if α≤β and A⊆Aδ. Consequently {Aα:α∈NN} is a fundamental bounded resolution for CcX.LetX be a nonempty completely regular Hausdorff topological space and let KX denote the family of all compact sets of X.Lemma 6.
A subsetA of CX is bounded for the compact-open topology τc if and only if there exists a sequence FnA:n∈N of subsets of KX such that(1)
for eachn∈N, if K∈FnA then supf∈Asupx∈Kfx≤n(2)
FnA⊆Fn+1A for each n∈N(3)
⋃n=1∞FnA=KX(4)
If a setQ∈KX is such that supx∈Qfx≤n for each f∈CX with supx∈Kfx≤n for all K∈FnA, then Q∈FnAProof.
IfA is τc-bounded and n∈N, define (7)FnA=K∈KX:supf∈A⁡supx∈K⁡fx≤n.Clearly FnA⊆Fn+1A for each n∈N and if K∈KX there is m∈N with (8)supf∈A⁡supx∈K⁡fx≤m,so that K∈FmA, which shows that ⋃n=1∞FnA=KX. In addition, if K∈FnA the relation supf∈Asupx∈Kfx≤n holds by construction. Finally, if we set (9)Bn=f∈CX:supx∈K⁡fx≤nforeveryK∈FnAthen A⊆Bn. Therefore, if Q∈KX verifies that supf∈Bnsupx∈Qfx≤n then (10)supf∈A⁡supx∈Q⁡fx≤n,so that Q∈FnA. Hence FnA:n∈N satisfies the required conditions.
Conversely, if there exists a sequenceFnA:n∈N of KX satisfying the four conditions of the statement of the lemma (actually, only the first and the third conditions are required) then clearly A is τc-bounded on X, for if P∈KX there is k∈N with P∈FkA such that supf∈Asupx∈Pfz≤k.In what follows the fourth condition above, which is independent ofA, will be referred to as theclosure condition of FnA, and we shall say that the family FnA isclosed.Definition 7.
A collection{Fα,n:α,n∈NN×N} of closed subsets of KX will be called acovering net of KX if Fα,n:n∈N is an increasing covering of KX for each α∈NN such that Fβ,n⊆Fα,n whenever α≤β for all n∈N.Theorem 8.
The spaceCcX has a fundamental bounded resolution if and only if there is a covering net {Fα,n:α,n∈NN×N} such that if Fn:n∈N is an increasing covering of KX by closed sets, there exists γ∈NN such that Fγ,n⊆Fn for all n∈N.Proof.
If there exists a covering net{Fα,n:α,n∈NN×N} for X which satisfies the property of the statement of the theorem, the sets (11)Aα=f∈CX:supx∈K⁡fx≤nforeveryK∈Fα,nandeachn∈Ncompose a fundamental bounded resolution for CX. Indeed, each set Aα is τc-bounded by virtue of the previous lemma, since Fα,n:n∈N is an increasing covering of KX consisting of closed sets such that supf∈Aαsupx∈Kfx≤n for all K∈Fα,n. Besides Aα⊆Aβ whenever α≤β since Fβ,n⊆Fα,n, and if A is a τc-bounded subset of CX, according to the previous lemma there exists an increasing covering FnA:n∈N of KX consisting of closed sets. Therefore, by the condition in the statement of the theorem, there exists γ∈NN such that Fγ,n⊆FnA for all n∈N. Now, if f∈A then supk∈Kfx≤n for all K∈FnA and n∈N, in particular for each K∈Fγ,n and all n∈N, which shows that f∈Aγ. Hence A⊆Aγ, which proves that {Aα:α∈NN} is a fundamental bounded resolution for CcX.
Conversely, ifCcX has a fundamental bounded resolution {Bα:α∈NN} and we set (12)Fα,n=K∈KX:supf∈Bα⁡supx∈K⁡fx≤nthen Fα,n:n∈N is an increasing covering of KX such that Fβ,n⊆Fα,n whenever α≤β for all n∈N. In addition supf∈Bαsupx∈Kfx≤n for K∈Fα,n and n∈N by the definition of Fα,n. Moreover, Fα,n satisfies the closure condition, for if (13)Bα,n=f∈CX:supQ∈Fα,n⁡supx∈Q⁡fx≤nthen Bα⊆Bα,n, so if supf∈Bα,nsupx∈Kfx≤n then supf∈Bαsupx∈Kfx≤n, which means that K∈Fα,n.
Now, ifFn:n∈N is any increasing covering of KX consisting of closed sets, define (14)P=f∈CX:supx∈K⁡fx≤nforeveryK∈Fα,nandeachn∈Nand observe that if K∈Fn then supf∈Psupx∈Kfx≤n, which according to the preceding lemma ensures that P is a τc-bounded set. Since {Bα:α∈NN} is a fundamental bounded resolution for CcX there exists δ∈NN such that P⊆Bδ. Now if Q∈Fδ,n then supf∈Bδsupx∈Qfx≤n so that, in particular, supf∈Psupx∈Qx≤n. We claim that Q∈Fn. Indeed, since supx∈Qfx≤n for each f∈P, we have that supx∈Qfx≤n holds for each f∈CX such that supx∈Kfx≤n for every K∈Fn by virtue of the definition of P. Therefore, the closure property of Fn yields K∈Fn. This shows that Fδ,n⊆Fn for every n∈N, which, bearing in mind the properties of the family {Fα,n:α,n∈NN×N} established before, guarantees that {Fα,n:α,n∈NN×N} is a covering net for X consisting of closed sets that satisfies the required property.In what follows we shall refer to a Tychonoff spaceX satisfying the conditions of the statement of Theorem 8 as a cn-space. It is shown in [5, Proposition 3.2] that if X is a cn-space or, which is equivalent, if CcX has a fundamental bounded resolution, then CcX has acountablecs∗-network at the origin [15, 16]. The next theorem shows that in order for X to be a cn-space it suffices that Cc∗X have a fundamental bounded resolution.Theorem 9.
LetX be completely regular. The space Cc∗X has a fundamental bounded resolution if and only if X is a cn-space.Proof.
IfX is a cn-space, Theorem 8 ensures that CcX has a fundamental bounded resolution, which implies that Cc∗X, as a subspace of CcX, also has a fundamental bounded resolution. Conversely, if Cc∗X has a closed fundamental bounded resolution {Aα:α∈NN}, for each α∈NN let Bα denote the closure of Aα in Cc(X). Let us show that B={Bα:α∈NN} is a fundamental bounded resolution for Cc(X). Indeed, let us denote by KX the family of all compact sets of X and pick an arbitrary bounded subset B of CcX. For each K∈KX and f∈B, set M(f,K)=sup{fx:x∈K} and define fK(x)=fx if fx≤Mf,K and fK(x)=signfx·Mf,K if fx>Mf,K. Clearly fK∈C∗(X), besides fK(x)=fx for x∈K and fKx≤fx for all x∈X. Define PB={fK:f∈B,K∈KX} and note that f is an adherent point of PB in Cc(X). Therefore, B is contained in the closure of PB in Cc(X). If Q is any compact subset of X, the fact that B is τc-bounded guarantees that (15)supg∈PB⁡supx∈Q⁡gx=supf∈B,K∈KX⁡supx∈Q⁡fKx≤supf∈B⁡supx∈Q⁡fx<∞,which shows that PB is a τc-bounded subset of C∗X. Hence, there is γ∈NN such that PB⊆Aγ, so that B⊆Bγ. Consequently B is a fundamental bounded resolution for Cc(X), as stated. Another application of Theorem 8 shows that X is a cn-space.The existence of a bounded resolution on a locally convex spaceE does not imply the existence of a fundamental bounded resolution for E, as the following example shows.Example 10.
IfX is an infinite Talagrand compact set, the weak∗ dual LpCp(X) of CpCpX has a bounded resolution but it has no fundamental bounded resolution.Proof.
SinceCp(X) is K-analytic, LpCp(X) is also K-analytic by [17, Proposition 0.5.13]. So LpCp(X) has even a compact resolution by [3, Theorem 3.2]. Suppose by contradiction that LpCp(X) has a fundamental bounded resolution {Aα:α∈NN}. Identifying Cp(X) with its homeomorphic copy in LpCp(X), for each α∈NN set Bα=Aα∩Cp(X) and consider the family B={Bα:α∈NN}. We claim that Bα is a functionally bounded subset of Cp(X). Indeed, if F∈CCp(X) according to [17, Proposition 0.5.11] there exists a (unique) continuous functional uF of LpCp(X) such that uFCp(X)=F. Since Aα is bounded, there is C>0 such that uFa<C for all a∈Aα. In particular, Fb=uFb<C for every b∈Bα. So B is a functionally bounded resolution in Cp(X). If B is a functionally bounded subset of Cp(X), then B, considered as a subset of LpCp(X), is bounded. Therefore, there is α∈NN with B⊆Bα. Hence B swallows the functionally bounded subsets of Cp(X). Since Cp(X) is Lindelöf and hence a μ-space, the family {Bα¯τp:α∈NN} consists of compact subsets and swallows the compact sets of Cp(X). So Cp(X) has a fundamental compact resolution. But according to Theorem 1, the space X should be countable and discrete. Therefore, X being compact is finite, a contradiction.



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*Source: 8219246-2018-09-10.xml*