Interpolation sets in spaces of continuous metric-valued functions

Let $X$ and $M$ be a topological space and metric space, respectively. If $C(X,M)$ denotes the set of all continuous functions from X to M, we say that a subset $Y$ of $X$ is an \emph{$M$-interpolation set} if given any function $g\in M^Y$ with relatively compact range in $M$, there exists a map $f\in C(X,M)$ such that $f_{|Y}=g$. In this paper, motivated by a result of Bourgain in \cite{Bourgain1977}, we introduce a property, stronger than the mere \emph{non equicontinuity} of a family of continuous functions, that isolates a crucial fact for the existence of interpolation sets in fairly general settings. As a consequence, we establish the existence of $I_0$ sets in every nonprecompact subset of a abelian locally $k_{\omega}$-groups. This implies that abelian locally $k_{\omega}$-groups strongly respects compactness.


Introduction
Let G be a locally compact abelian group and let G denote its Pontryagin dual group.
We say that a subset E of G is Sidon if for every bounded function f on E there corresponds a Borel regular measure on G, µ, such that µ(γ) = f (γ) for all γ ∈ E (here µ denotes the Fourier transform of µ). If, in addition, µ is assumed to be discrete (it has a countable support) then it is said that E is an I 0 -set. Therefore, each I 0 -set is Sidon. For instance, lacunary (or Hadamard) sets (i.e. sequences (z n ) n ⊆ N such that inf z n+1 /z n > 1) are perhaps the simplest examples of I 0 -sets. The search for interpolation sets is a main goal in harmonic analysis and the monograph by Graham and Hare [25] contains most of the recent results in this area.
In this paper, this question is approached from a topological viewpoint that is based on the equivalent formulation of this notion given by Hartman and Ryll-Nardzewski [26] (in fact, the term I 0 -set is due to them). According to their (equivalent) definition a subset E of a locally compact abelian group G is an I 0 -set if for each f ∈ l ∞ (E) (that is, for each complex-valued, bounded function defined on E) there exists an almost periodic function f b on G such that f (γ) = f b (γ) for all γ ∈ E. Furthermore, since every almost periodic function on a topological group G is the restriction of a continuous function defined on the Bohr compactification bG of G, it follows that E ⊆ G is an I 0 -set if each f ∈ l ∞ (E) can be extended to a continuous function f b defined on bG. The latter property implies that E bG is canonically homeomorphic to βE, the Stone-Čech compactification of the set E equipped with the discrete topology. This equivalent definition of I 0 -set and the duality methods obtained from Pontryagin-van Kampen duality allows us to apply topological techniques in the investigation of this family of interpolation sets. Thus, we can prove the existence of I 0 -sets for much larger classes of groups than locally compact abelian groups. Several applications of our results to different questions related to the Bohr compactification and topology of topological abelian groups are also obtained. Last but not least, we deal with the topological properties of sets of continuous functions.
Indeed, if X and M are a topological space and a metrizable space respectively, given a subset G ⊆ C(X, M), we look at the possible existence of copies of βω (the Stone-Čech compactification of the natural numbers) within G M X . This property, or its absence, has deep implications on the topological structure of G as a set of continuous functions on X and has found many applications in different settings (for instance, see [20,22,17,25] where there are applications to topological groups, dynamical systems, functional analysis and harmonic analysis, respectively).
The starting point of this paper stems from a celebrated theorem by Bourgain, Fremlin and Talagrand about compact subsets of Baire class 1 functions [7], that we present in the way it is formulated by Todorčević in [35]. A variant of this result is due to Pol [33, p. 34], that again was formulated in different terms (cf. [9]). Here B 1 (X) denotes the set of all Baire class 1 functions defined on X.
is uniformly bounded, and K = G the closure of G in [−1, 1] X . Then the following are equivalent: (b) G contains a sequence whose closure in R X is homeomorphic to βω.
In both cases we have a dichotomy result that basically characterizes two crucial properties about sets of continuous functions defined on a Polish and metric complete space, respectively. In this paper we look at this question in terms of the set of continuous functions G ⊆ C(X, M) alone, when M is a general metric space. We first extend the notion of I 0 -set, given by Hartman and Ryll-Nardzewski for complex-valued functions, to a more general setting, which will be needed later on when we apply it to topological groups. Definition 1.3. Let X and M be a topological space and metric space, respectively. If C(X, M) denotes the set of all continuous functions from X to M, we say that a subset Y of X is an M-interpolation set (or, we can simply say an Interpolation set for C(X, M)) when for each function g ∈ M Y , which has relatively compact range in M, there exists a map f ∈ C(X, M) such that f |Y = g.
A main goal in this paper is the understanding of the key (topological) facts that characterize the existence of interpolation sets. Thereby, this research continues the task accomplished in previous projects [19,20] and [15]. Here, we introduce a crucial property stronger than the mere non-equicontinuity, that provides sufficient conditions for the existence of Interpolation sets in different settings. We refer to [6] for its motivation, where this notion implicitly appears.
Definition 1.4. Let X be a topological space and let M be a metric space. We say that B-family such that each g ∈ G factors through Φ (that is, for each g ∈ G, there is a map , then there is a nonempty compact subset ∆ of X and a countable infinite subset L of G such that L is separated by ∆. As a consequence, if M is a Banach space, G contains a countably infinite M-interpolation set.
Remark 1.8. From Theorem 1.2, one can deduce the existence of an interpolation subset in a set G of real-valued continuous functions defined on a complete metric space X, when G X contains a function that is not Baire one. The main difference in our approach is that this property is isolated within the set G.
Theorem B. Let X be aČech-complete group and K a compact group. If G ⊆ CHom(X, K) is not equicontinuous, then G contains a countable subset L such that L K X is canonically homeomorphic to βL, when L is equipped with the discrete topology.
In case K = U(n), the unitary group of degree n, it follows that L is an C n 2 -interpolation set.
A consequence of this result is a variation of a well-know Theorem by Corson and Glicksberg [13] asserting that if a subset G of continuous homomorphisms defined on a hereditarily Baire group has a compact, metric closure, then it is equicontinuous. In case X isČech-complete and K is a compact group, these constraints can be relaxed considerably.
Theorem C. Let X be aČech-complete group, K be a compact group and G be an infi- open neighbourhood which is a k ω -space in the induced topology. It is clear that every k ω -space is a k-space (see [24]). A k ω -group (resp. locally k ω -group) is a topological group where the underlying topological space is a k ω -space (resp. locally k ω ).
The class of abelian locally quasiconvex, locally k ω -groups includes, in addition to all locally compact abelian groups: all free abelian groups on a compact space, indeed on any k ω space; all dual groups of countable projective limits of metrizable (more generally,Čech-complete) abelian groups; all dual groups of abelian pro-Lie groups defined by countable systems [24,32]. Moreover, this class is preserved by countable direct sums, closed subgroups, and finite products [24].
Theorem D. Let G be an abelian locally quasiconvex, locally k ω -group. If {g n } n<ω is a sequence in G that is not precompact in G, then {g n } n<ω contains an I 0 -set.
The Bohr compactification of a topological group G, can be defined as a pair (bG, b) where bG is a compact Hausdorff group and b is a continuous homomorphism from G onto a dense subgroup of bG such that every continuous homomorphism h : G → K into a compact group K extends to a continuous homomorphism h b : bG → K, making the following diagram commutative: The topology that b induces on G will be referred to as the Bohr topology. A topological group G is said to be maximally almost periodic (MAP, for short) when the map b is oneto-one, which implies that the Bohr topology will be Hausdorff.
The duality theory can be used to represent the Bohr compactification of an abelian group as a group of homomorphisms. Indeed, if G is an abelian topological group and Γ d denotes its dual group equipped with the discrete topology then bG coincides with the dual group of Γ d .
Given a topological group G, let G + denote the algebraic group G equipped with the Bohr topology. Glicksberg [23] has shown that in a locally compact abelian (LCA, for short) group G, every compact subset in G + is compact in G. This result concerning LCA groups is one of the pivotal results of the subject, often referred to as Glicksberg's theorem.
Given a topological group G and a property P, we say after Trigos-Arrieta [36] that G respects the property P when G and G + have the same sets satisfying P. Taking this terminology, Glicksberg's theorem asserts that locally compact Abelian groups respect compactness. Trigos-Arrieta considered some properties (pseudocompactness, countable compactness, functional boundedness) obtaining that they are respected by locally compact Abelian groups. Several authors have dealt with this question subsequently (cf. [3,5,29,18]). Glicksberg result was extended in a different direction by Comfort, Trigos-Arrieta and Wu [12] by the following remarkable result.
Let G be a LCA group and let N be a closed metrizable subgroup of its Bohr compactification bG. Denote by π the canonical projection from bG onto bG/N and set b N making the following diagram commutative: In the same paper, the following classes of topological grous is introduced: A group G strongly respects compactness if satisfies the thesis in Theorem 1.13. The authors also propose the question of clarifying the relation between these two classes of groups and furthermore the characterization of the groups that strongly respect compactness. Using the techniques studied in this paper, we can prove that every abelian locally quasiconvex, locally k ω group respects any compact-like property P that implies functional boundedness and, furthermore, strongly respects compactness, improving the results obtained by Gabriyelyan [18] for locally k ω -groups. As a matter of fact, this result has been already applied to solve Question 4.1 in [12] (see [30]).
Definition 1.14. Let X be a topological space. A subset A of X is functionally bounded when every real-valued continuous function defined on X is bounded on A. We say that a topological property P on X is stronger than or equal to functional boundedness if for each A ⊆ X that satisfies P (A ∈ P for short), it holds that A is functionally bounded.
Theorem E. Let G be an abelian, locally quasiconvex, locally k ω , group. Then the following holds: (i) G respects any compact-like property P stronger than or equal to functional boundedness.

Interpolation sets in topological spaces
Definition 2.1. Let X and M be a topological space and metric space, respectively, and let C(X, M) denote the space of continuous functions of X into M. Given a subset L ⊆ C(X, M), we say that K ⊆ X separates L if for every subset A ⊆ L there are two closed subsets in M, say D 1 and D 2 , and In the sequel, we are going to apply the definition of M-interpolation set to subsets L ⊆ C(X, M) ⊆ M X , where X and M are a topological and a metric space, respectively.
That is to say, we will look at L as an Interpolation set for C(M X , M). First, we need a lemma, whose proof is known. However, we include it here for the reader's sake. We refer to [14,21,34] for further information. (a) There is a nonempty subset ∆ of X such that L is separated by ∆.
are equivalent.
Proof. That (b) implies (c) is folklore. It is also clear that (d) implies (c). For (a) implies (b), let B 1 and B 2 two disjoint subsets of L, which is separated by ∆. Then, there are two closed sets D 1 and D 2 in M and x 0 ∈ ∆ ⊆ X such that d(D 1 , D 2 ) ≥ ǫ 0 , for some and let f ∈ M X . We say that f is totally discontinuous if there are two subsets N 0 and We may assume that N 0 and N 1 are open sets because, otherwise, we would replace ∆ of X and a countable infinite subset L of G, which is separated by ∆. Furthermore, by Proof. Since X isČech-complete, it is a G δ -subset of its Stone-Cech compatification βX. By induction on n = |t|, t ∈ 2 (ω) (i.e. the set of finite sequences of 0's and 1's), we define a family {U t : t ∈ 2 (ω) } of non-empty open subsets in βX and a sequence of functions {h n : n < ω} ⊆ G, satisfying the following conditions for all t ∈ 2 (ω) : Construction: If n = 0, by regularity we can find U ∅ a nonempty open set in βX such because V t is a relatively open subset of X and the sets A 0 and A 1 are dense in X. Since Let h β n be the continuous extension of h n , then we can select two open disjoint neighborhoods in βX, O t0 and O t1 of a t and b t , respectively, satisfying: and U t1 satisfies the conditions (ii), (iii) and (iv). Moreover, using a continuity argument, we can adjust the two open sets to satisfy (v).
which is a closed subset of βX and, as a consequence, ∆ is compact. On the other hand, we also have ∆ = Clearly ϕ is an onto and continuous map.
Let us see that {h n } n<ω is separated by ∆. Indeed, for any arbitrary subset S ⊆ ω, it suffices to select σ ∈ 2 ω such that σ(0) = 0 and σ(n for every n ∈ S and h n (z) ∈ N 0 for every n ∈ S. Finally, in case M is a Banach space, We need the following compact space K that is defined as in [10].
Definition 2.6. Let (M, d) be a metric space that we always assume equipped with a bounded metric. We set Being pointwise closed and equicontinuous by definition, it follows that K is a compact and Given any element f ∈ M X , we associate a mapf ∈ R X×K defined by f (x, α) = α(f (x)) for all (x, α) ∈ X × K.
In like manner, given any subset G of M X we setǦ We are now in position of proving Theorem A.
Proof of Theorem A. We may assume wlog that the map Φ is surjective because otherwise we would deal with the separable and metrizable space Φ(X). Due to the fact thatČech-completeness is hereditary for closed subsets, we may assume, from here on, In order to simplify the notation below, we say that a set of X is Using an inductive argument, for every integer n < ω, we find f n ∈ G, α n ∈ K and a finite collection {U n,k } 1≤k≤n of nonempty open sets in X satisfying the following conditions (for each n < ω and each k = 1, . . . , n): Construction: If n = 1, since V 1 is an open subset in X there exists f 1 ∈ G such that diam(f 1 (V 1 )) ≥ ǫ. By the continuity of f 1 , it follows that there exists a nonempty open subset W 1,1 such that: is not empty. By regularity, we can find a nonempty open subset U 1,1 such that U 1,1 ⊆ Assume now that f n , α n and {U n,k } 1≤k≤n have been obtained, with n ≥ 1. By hypothesis, there exists f n+1 ∈ G such that diam(f n+1 (U n,k )) ≥ ǫ for all k ∈ {1, . . . , n} and By the continuity of f n+1 , we can find nonempty open subsets {W n+1,k } 1≤k≤n+1 satisfying: (1) W n+1,k ⊆ U n,k , for all 1 ≤ k ≤ n; (2) W n+1,n+1 ⊆ V ′ n+1 (therefore W n+1,n+1 is A j -small for 1 ≤ j ≤ n); for all x ∈ W n+1,k and 1 ≤ k ≤ n + 1.
We claim that α n+1 ∈ K. Indeed, if m 1 , m 2 ∈ M, then and choose k 0 ∈ {1, . . . , n + 1} such that Then, On the other hand, for all x ∈ W n+1,k ′ and 1 ≤ k ′ ≤ n + 1: Take the open covering A n+1 of X. Then, for each k ∈ {1, . . . , n+1} there is A k ∈ A n+1 such that A k ∩ W n+1,k is a nonempty open subset of X. By regularity we can find an open set U n+1,k such that: This completes the construction. Now, for each k < ω, the intersection ∞ n=k U n,k is nonempty byČech-completeness.
Therefore, we can fix a point z k ∈ ∞ n=k U n,k for all k < ω. Note that Φ(x k ) ∈ V k and Φ(z k ) ∈ V k for all k ∈ ω.
For each m < ω, define It is easily seen that F m is closed and, consequently, Observe that, since { V k } k<ω is an open basis in Y , it follows that Y = m<ω F m and, hence X = m<ω F m . Being XČech-complete, it is a Baire space. Therefore, there is some m 0 < ω such that F m 0 has nonempty interior U in X. Since Φ is a quasi-open map, we have that Φ(U) has nonempty interior U included in F m 0 . It follows that inf α • f ( U) < r m 0 and Set F = U 0 , r def = r m 0 and δ def = δ m 0 and consider the following sets: where I 0 = [−1, r) and I 1 = (r + δ, 1]. Note that A 0 and A 1 are dense subsets in F . , which are disjoints. Moreover, since α ∈ K, it follows that d(N 0 , N 1 ) ≥ δ and f (A j ) ⊆ N j for j = 0, 1. Therefore f is totally discontinuous on F . It now suffices to apply Lemma 2.5 Remark 2.7. Note that the result remains valid if we assume that for each residual subset R of X there is a separable metrizable space Y and a continuous and quasi-open map Φ : R → Y such that for all g ∈ G there is a g ∈ C(Y, M) satisfying g(x) = ( g • Φ)(x) for all x ∈ R.

Interpolation sets in topological groups
In this section, we apply the results obtained previously in the setting of topological groups. Our first result clarifies the relevance of the notion of B-family when we deal with topological groups. From here on, we assume, wlog, that every metrizable topological group M is equipped with a left-invariant metric. Furthermore, if M is in addition compact, then we assume that M is equipped with a bi-invariant metric. From here on, if X and Y are topological groups, we let Hom(X, M) (resp. CHom(X, M)) denote the set of all homomorphisms (resp. continuous homomorphisms) of X into M. Proof. It is clear that, if G is a B-family, then it may not be equicontinuous. So, assume that G is not a B-family. Taking V = X and ǫ > 0 arbitrary, there exists a finite family {U 1 , . . . , U n } open subsets in X (wlog, we assume that U j = x j V j , where V j is a neighbourhood of the neutral element) such that for every g ∈ G there is V j , with 1 ≤ j ≤ n, satisfying that diam(g(x j V j )) < ǫ. Now, since g is a group homomorphism and d is left-invariant, it follows that diam(g(V j )) < ǫ as well. Set V 0 = V 1 ∩ . . . ∩ V n , then diam(g(xV 0 )) < ǫ for all g ∈ G and x ∈ X. Consequently G is equicontinuous.
The next result is a direct consequence of Lemma 3.1, Theorem A and Lemma 2.2.
Previously, we need the following definition. Recall that U(n) denotes the unitary group of degree n. Proof. By Troallic [37], we may assume wlog that G is countable. By Lemma 3.1, G is a B-family. Define an equivalence relation on X by x ∼ y if and only if g(x) = g(y) for all g ∈ G. Since G is countable and consists of group homomorphisms, it follows that the quotient space X = X/∼ is a compact metrizable group. Therefore, if p : X → X denotes the canonical quotient map, each g ∈ G factors through a map g defined on CHom( X, M); that is g(p(x)) def = g(x) for any x ∈ X. Since every quotient group homomorphism is automatically open, Theorem A implies that there is a nonempty subset ∆ of X and a subset L of G such that L is separated by ∆. In case M is a Banach space, applying Lemma 2.2, we obtain that L is a M-interpolation set.
Next result is folklore but we include its proof for the sake of completeness. Proof. It suffices to prove that Gh is equicontinuous if G is equicontinuous. Let x 0 be an arbitrary but fixed point in X. Since right translations are continuous mappings on a topological group, and G (resp. h) is equicontinuous (resp. continuous) on X, given ǫ > 0, there is a neighbourhood U of x 0 such that d(g(x 0 )h(x 0 ), g(x)h(x 0 )) < ǫ/2 and d(h(x 0 ), h(x)) < ǫ/2 for all x ∈ U and all g ∈ G. Thus, applying the left invariance of the group metric, we obtain for all x ∈ U, which completes the proof.
With the hypothesis of the previous lemma, if g ∈ CHom(X, M), the symbol g −1 denotes the map defined by g −1 (x) = g(x) −1 = g(x −1 ) for all x ∈ X. Combining Lemmata 3.1 and 3.3, we obtain: such that for all g ∈ G there is j ∈ {1, . . . , n} with diam(gg −1 0 (U j )) < ǫ. We can assume V j and an arbitrary element x 0 ∈ X.
Given g ∈ G, there is j ∈ {1, . . . , n} such that We can now prove Theorem B.
Proof of Theorem B. Since K is compact, there is a representation π : K → U(n) such that {π • g : g ∈ G} is not equicontinuous. Therefore, we assume that K = U(n) wlog.  2]), since G ⊆ CHom(X, U(n)) is not equicontinuous, there exists a separable compact subset F of X and a countable subset L ⊆ G such that L| F is not equicontinuous. Set H as the smallest closed subgroup generated by F , it follows that H ≤ X is closed and separable and L ⊆ G countable such that L| H is not equicontinuous. So we can assume wlog that X is separable and G is countable. On the other hand, byČech-completeness of X, there must be a compact subgroup C of X such that X/C is separable, complete and metrizable [8], thereby , a Polish space.
-Case (1): We may suppose wlog that all elements of G| C are pairwise inequivalent. By [11,Th. 1], it follows that G| C is discrete as a subset of CHom(C, U(n)) in the compact open topology on C, which implies that G| C may not be equicontinuous on C. Applying Corollary 3.2, there is a nonempty subset ∆ of C and a countable subset L of G such that L is separated by ∆. Thus, by Lemma 2.2, L U(n) X is canonically homeomorphic to βL (where L is equipped with the discrete topology) and we are done.
-Case (2): Set H def = {ϕ 1 , . . . , ϕ m } ⊆ G such that every g ∈ G is equivalent to an element in H when they are restricted to C. If we define G i = {g ∈ G : g| C ∼ ϕ i | C }, . . m} such that G i is not equicontinuous. So, we may assume wlog that there is g 0 ∈ G such that g| C ∼ g 0 | C for all g ∈ G. Therefore, for each g ∈ G, there is U g ∈ U(n) with (U −1 g gU g )| C = g 0 | C . Denote by g the map U −1 g gU g and set G def = {U −1 g gU g : g ∈ G}, which is a subset of CHom(X, U(n)). It is easily seen that G is not equicontinuous on This would imply that G is equicontinuous, which is a contradiction). Hence, Gg −1 0 is a B-family on X by Lemmas

and 3.4.
Let π C : X → X/C the canonical quotient map, which is open and continuous. Since X/C is Polish and each gg −1 0 factors through X/C, we apply Theorem A and Lemma 2.2 in order to obtain ∆ ⊆ X and L ⊆ G such that Set L def = {g : g ∈ L} ⊆ G and consider the map The map ψ is continuous because L is discrete. Moreover, using that L U(n) X is canonically homeomorphic to βL (L with the discrete topology), there is a continuous extension Using a compactness argument on the group U(n), it is not hard to verify that if p, q ∈ L U(n) X and ψ(p) = ψ(q) then p and q are equivalent. Since Orbit(p) = {U −1 pU : U ∈ U(n)} has the cardinality of continuum c and | L U(n) X | = |βL| = |βω| = 2 c , we obtain: Applying [15,Cor. 2.16], it follows that L contains a subset P such that P U(n) X is canonically homeomorphic to βP (with P equipped with the discrete topology). This completes the proof.
Corollary 3.5. Let X be aČech-complete abelian group. If an infinite subset G of X is not equicontinuous, then G contains a countably infinite I 0 -set.
Next follows the proof of Theorem C.
Proof of Theorem C. If for every countable subset L ⊆ G and compact separable may not contain any copy of βω. By Theorem B, this implies that L| Y is equicontinuous on Y . Applying [16,Theorem B], it follows that G is hereditarily equicontinuous on X, which implies that G is equicontinuous because G consists of group homomorphisms.

I 0 -sets in abelian locally k ω groups
In this section, we study the existence of I 0 -sets for abelian locally k ω groups, a large family of topological groups that includes, for example, all LCA groups, the free abelian groups on a compact space and all countable direct sum of compact groups. The proof of our main results are obtained using methods of Pontryagin-van Kampen duality. Therefore, we first recall some basic defintions and facts about the Pontryagin duality of abelian groups. From here on, all groups are supposed to be abelian and, therefore, we will use additive notation to deal with them. In particular, we identify T with the additive group [−1/2, 1/2) by identifying ±1/2.
If G is a topological group, a character on a topological abelian group G is a continuous group homomorphism from G to the torus group T. The set of all characters on G, with pointwise addition, is a group. defined by E(g)(χ) = χ(g) for all g ∈ G and χ ∈ G, is a topological isomorphism. By the Pontryagin-van Kampen theory, we know that every LCA group is reflexive. Furthermore, the dual of a compact group is discrete and the dual of a discrete group is compact. In general, the dual of a LCA group is also locally compact. As a consequence, every compact abelian group is equipped with the topology of pointwise convergence on its dual group. Similarly, for X ⊆ G, let , for all χ ∈ X .
The following facts are well known (see [4]).  For each set A ⊆ G, the set A ✄ is a quasiconvex subset of G. Thus, the topological group G is locally quasiconvex for each topological abelian group G. Moreover, local quasiconvexity is hereditary for arbitrary subgroups.
The set A ✄✁ is the smallest closed, quasiconvex subset of G containing A.
In the case where G is a topological vector space, G is locally quasiconvex in the present sense if, and only if, G is a locally convex topological vector space in the ordinary sense.
If G is locally quasiconvex, its characters separate points of G, and thus the evaluation is a compact subset of G (Lemma 4.2), and thus U ✄✄ is a neighborhood of 0 in G. As The following theorem of Glöckner, Gramlich and Hartnick [24] states that there exists a relation between the abelian locally k ω groups and the abelianČech-complete groups.
Theorem 4.4. If G is an abelian locally k ω group, the G isČech-complete.
Conversely, G is locally k ω , for each abelianČech-complete topological group G.
Using this duality and Theorem B we obtain: Proof of Theorem D. Consider the abelianČech-complete group G. By means of the evaluation map E : G → G ⊆ C( G, T), we can look at the sequence {g n } n<ω as a subset of C( G, T). Furthermore, since {g n } n<ω is not precompact in G, it follows that {g n } n<ω is not equicontinuous on G. Indeed, if it were equicontinuous on G, by Arzelà-Ascoli's theorem, then {g n } n<ω would be precompact in C c ( G, T), the group C( G, T) equipped with the compact open topology. Now, since G is a locally quasiconvex k-space, the evaluation map E : G → G is a topological isomorphism onto its image (see [28]). Thus {g n } n<ω would also be precompact in G, which is a contradiction.
Therefore, the sequence {g n } n<ω is not an equicontinuous set on G and, by Theorem B, contains an I 0 -set.
The next result was proved in [19,Lemma 4.11].
Lemma 4.5. Let G be a maximally almost periodic abelian group, A a subset of G and let N be a subset of bG containing the neutral element such that A + N is compact in bG.
If F is an arbitrary subset of A, there exists A 0 ⊆ A with |A 0 | ≤ |N| such that We are now in position of proving Theorem E.
Proof of Theorem E. Let G be a locally quasiconvex, locally k ω group and let bG denote its Bohr compactification.
(i) Let P be a topological property implying functional boundedness and let A be any subset of G satisfying P in G + , which (by definition) is equipped with the weak topology generated by G; that is to say G + ⊆ T G . Reasoning by contradiction assume that A does not satisfy P in G. We claim that A may not be a precompact subset of G. Indeed, if it were, since every locally k ω -group is complete [24,Remark 7.3], it would follow that A G would be compact in G. Therefore, it would also be compact in G + that is equipped with a weaker topology. Since any compact topology is (Hausdorff) minimal, this would imply that the Bohr topology would coincide with the original topology of G on the compact subset A G and, as a consequence, on A. Thus A would have property P in G.
So, assume wlog that A is not precompact in G. As in the proof of Theorem D, if we take the abelianČech-complete group G and inject G in C( G, T) by means of the evaluation map E : G → G ⊆ C( G, T), it follows that A is not equicontinuous on G. By [16,Cor. 2.4], it follows that there exists a countable subset F ⊆ A and a separable compact subset X ⊆ G such that F is not equicontinuous on X. Taking the closure in G of the subgroup generated by X, we may assume wlog that X is a separable closed subgroup of G. On the other hand, since A is functionally bounded in G + and X ⊆ G, it follows that F is also functionally bounded in G, when the latter is equipped with weak topology generated by X.
Set X ⊥ def = {g ∈ G : χ(g) = 0 for all χ ∈ X} and take the quotient G/X ⊥ , which clearly is a maximally almost periodic group whose dual is X. Furthermore, the group G/X ⊥ is locally k ω and G/X ⊥ ∼ = X, which isČech-complete. If p : G → G/X ⊥ denotes the open quotient map and bG/X ⊥ denotes the Bohr compactification of G/X ⊥ , it follows that there is a canonical extension p b : bG → bG/X ⊥ . Therefore, we have that p b (F ) is a functionally bounded subset of p b (G + ) = G + /X ⊥ that is not equicontinuous on X. Indeed, if p b (F ) were equicontinuous on X, then it would follow that F would be equicontinuous on X, which is not true. In other words, we may assume wlog that A is a countable, functionally bounded subset of G + that is not equicontinous on G, which is separable.
As in the proof of Theorem B, by theČech-completeness of X, there must be a compact subgroup C of X such that X/C is separable, complete and metrizable [8], thereby, a Polish space.
Let A| C def = {g| C : g ∈ A} ⊆ C. We have two possible cases: (1) A |C contains infinitely many different elements.
(2) A |C only contains a finite number of different elements. -Case (2) is not equicontinuous. So, we may assume wlog that there is g 0 ∈ A such that g |C = g 0|C for all g ∈ A. By Lemma 3.3, we know that Ag −1 0 is not equicontinuous on X and, by Lemma 3.4, it follows that Ag −1 0 is a B-family on X. Let π C : X → X/C the canonical quotient map, which is open and continuous. Since X/C is Polish and each gg −1 0 factors through X/C, we apply Theorem A and Lemma 2.2 in order to obtain ∆ ⊆ X and L ⊆ A such that which again yields On the other hand, repeating the same argument as in (1), we deduce that |L T ∆ |∆ | = |L Cp(∆,T) |∆ | ≤ c, which again is in contradiction with (II). This completes the proof of (i).
(ii) The proof of this part replays some of the steps followed to prove (i). For the reader's sake, we will avoid unnecessary repetitions as much as possible.
Let N be a closed metrizable subgroup of bG and let A be a subset G. It is obvious In order to prove the non-trivial converse implication, again reasoning by contradiction, assume that b N (A) is compact in bG/N but A + (N ∩ G) is not compact in G. As G is complete [24,Remark 7.3], this means that A + (N ∩ G), being closed in G, is not precompact in the topology inherited from G. As in the proof of (i), it follows that there exists a countable subset F ⊆ A + (N ∩ G) and a separable compact subset X ⊆ G such that F is not equicontinuous on X. Taking the closure in G of the subgroup generated by X, we may assume that X is a separable closed subgroup of G.
Again, the quotient group G/X ⊥ is a MAP, locally k ω group whose dual group X iš Cech-complete. If p : G → G/X ⊥ denotes the open quotient map and p b : bG → bG/X ⊥ is the canonical extension to their Bohr compactifications, we have that p(A + (N ∩ G)) is contained in p b (A + N) = p(A) + p b (N), which is compact in bG/X ⊥ . Applying Lemma 4.5 to p(F ) and p b (N), we obtain that there exists A 0 ⊆ p(A) with |A 0 | ≤ |p b (N)| ≤ c such that cl bG/X ⊥ p(F ) ⊆ A 0 + p b (N) + cl (G/X ⊥ ) + p(F − F ). Now, being the group X separable, it follows that G/X ⊥ can be equipped with a metrizable precompact topology. As a consequence |G/X ⊥ | ≤ c. All in all, we obtain that |cl bG/X ⊥ p(F )| ≤ c.
On the other hand, p(F ) is not equicontinuous as a subset of C(X, T) and, by Theorem D, this means that it contains an I 0 -set, which yields |cl bG/X ⊥ p(F )| = |βω| = 2 c > c. This is a contradiction that completes the proof.