Notes on the g-closure
In this note, we prove that H is ag-closed subgroup of the compact group K if and only if every sequentially continuous character of the dual of H is continuous in other words, if the dual ofH is a so-calledsk-group. We conclude that every countable subgroup of a compact group isg-closed, and thus give a positive answer to a problem of Dikra njan, Milan and Tonolo. We also show that the relationship between precompact sk-groups and precompact groups is similar to that between (Hausdorff) k-spaces and (Hausdorff) topological spaces. 1. PRELIMINARIES: PRECOMPACT GROUPS A topological groupG is precompactif for every neighborhoodU of the identity inG, there exists a finite subset F ⊂ G such thatG = FU . Throughout this note, K is a fixed abelian compact Hausdorff topological group, and A = K̂ is its Pontryagin dual. It is a well-known result of Comfort a nd Ross [6] that every precompact Hausdorff group topology τ onA coincides with the initial topology on Awith respect to characters χ in a dense subgroup H ≤ K. In other words, one has τ = τH , whereτH is the coarsest (group) topology onA such that eachχ : A → T in H is continuous. In fact, H ≤ K separates the points of A if and only ifH is dense inK, andH = (̂A, τ) (cf. [6]). ForH ≤ K, considerH := ⋂ χ∈H kerχ. It follows from the Pontryagin duality theory that the dual ofA/H is the closureH̄ of H in K, andH = H̄. Thus, althoughH need not separate points in A, it always separates the points of A/H. Therefore, in the context of characters and duality, the precompact (but not necessarily Hausdorff ) g oup (A, τH) is equivalent to the precompact Hausdorff group (A/H, τH), because the latter is the maximal Hausdorff quotient of the earlier (see [12, 2.10]). There is an close relationship between properties of (A, τH) andH. To mention only two basic ones, the identity map (A, τH1) → (A, τH2) is continuous (i.e., τH1 ⊇ τH2) if and only ifH1 ≥ H2 (cf. [12, 2.6]), and(A, τH) is metrizable if and only ifH is countable (cf. [12, 2.12]). For a detailed account of such results, we refer the reader to [12] and [4] (more references???). To shorten notations, we put aα H −→ a0 whenever the net {aα} converges to a point a0 in A in the topologyτH . 2 G. Luḱacs / Notes on the g-closure 2. THE GALOIS-CLOSURE Following Dikranjan, Milan and Tonolo [9, 2.1], given an abe lian topological groupG, for u = (un) ∈ Ĝ , we putsu(G) = {x ∈ G | un(x) −→ 0 in T }, and for a subgroup H ≤ G, we set gG(H) = ⋂ {su(G) | u ∈ Ĝ , H ≤ su(G)}. (1) In other words,g is the largest closure operator with respect to which each b sicg-closedsetsu(G) is closed. In what follows, we focus onG = K, our fixed compact abelian group. Our aim is to present a slightly different and new approach to gK by studying the relationship between (A, τH) and (A, τg(H)). For χ ∈ K, put Sχ(ĜN) = {(an) ∈ A | χ(an) −→ 0}, and forH ≤ K, set SH(A ) = ⋂ χ∈H Sχ(A ) = {(an) ∈ A N | an H −→ 0}. One says that a map f : X → Y between topological spaces is s-continuousif whenever xn −→ x0 for a sequence{xn} in X, f(xn) −→ f(x0) in Y . (Since we do not assume any separation axiom, the limit of{xn} need not be unique.) Note that in the case of homomorphisms between topological groups, it suffices to check sequences c onverging to the neutral element. We say that the topologies τ1 andτ2 on a setX ares-equivalentif both (X, τ1) → (X, τ2) and (X, τ2) → (X, τ1) ares-continuous, or equivalently, if they have the same converg ent sequences. Theorem 2.1. LetH ≤ K. Then: (a) gK(H) = {χ ∈ K | χ(an) −→ 0 for all a ∈ SH(AN)}; (b) gK(H) = {χ ∈ K | χ : (A, τH) → T is s-continuous}; (c) H is gK closed if and only if every s-continuous character of (A, τH) is continuous; (d) The topologiesτH andτg(H) ares-equivalent for everyH ≤ K. PROOF. (a) Fora = (an) ∈ A,H ≤ sa(K) if and only ifχ(an) −→ 0 for everyχ ∈ H. This is equivalent toan H −→ 0, which is the same as a ∈ SH(AN). Therefore, gK(H) = ⋂ {sa(K) | a ∈ SH(A )} = {χ ∈ K | χ(an) −→ 0 for all a ∈ SH(A )}, (2) as desired. (b) Let χ ∈ gK(H), and suppose that an H −→ 0. Thena = (an) ∈ SH(AN), and thus χ(an) −→ 0 by (a). This shows that χ is s-continuous. Conversely, suppose that χ : (A, τH) → T is s-continuous, and let (an) ∈ SH(AN). Thenan H −→ 0, and sinceχ is s-continuous, this implies χ(an) −→ 0. Hence,χ ∈ gK(H), by (a). (c) follows from (b). (d) Sinceg(H) ≥ H, the topologyτg(H) is finer thanτH (cf. [12, 2.6]), so it suffices to show that (A, τH) → (A, τg(H)) is s-continuous. Ifan H −→ 0, thena ∈ SH(AN), and thus, by (a), χ(an) −→ 0 for everyχ ∈ g(H), which means that an g(H) −→ 0, as desired. The following corollary is known, and also appears [1, 5.11] : G. Luḱacs / Notes on the g-closure 3 Corollary 2.2. If H is a g-dense subgroup of K, then (A, τH) has non non-trivial convergent sequences. PROOF. By Theorem 2.1(d), if H is g-dense, then(A, τH) is s-equivalent to the Bohr compactification(A, τK) of the discrete group A. It is well-known that the latter has no convergent sequence s (cf. [10] and [11]). Following Noble [13], we say that a group G is ans-group if every s-continuous homomorphismψ : G→ L into a Hausdorff groupL is continuous. Corollary 2.3. For H ≤ K, if (A, τH) is ans-group, thenH is g-closed. PROOF. Since(A, τH) is ans-group, itss-continuous characters are continuous ( T is certainly Hausdorff). Therefore, by Theorem 2.1(c), H is g-closed inK. Corollary 2.4. LetH be a countable subgroup of K. ThenH is g-closed. Corollary 2.4 is a positive solution to a problem of Dikranja n, Milan and Tonolo [9, Problem 5.1], and it is a far-reaching generalization of a result by Bı́ró, Deshouillers and Sós [3]. Independently, Dikranjan and Kunen [8, 1.9] and Mathias Beiglb ̈ ock, Christian Steineder and Reinhard Winkler [2] have also obtained Corollary 2.4. PROOF. LetH = {h1, h2, . . .} be countable. The topology τH is the initial one with respect to the homomorphism(hi) : A→ T. Thus, the maximal Hausdorff quotient (A/H, τH) is metrizable, and in particular, (A, τH) is ans-group. Therefore, by Corollary 2.3, H is g-closed. Theorem 2.5. Everyg-closed subgroup of a compact group is realcompact. Recall that a mapf : X → Y between topological spaces is said to be countably continuous if f is continuous on every countable subset of X. PROOF. Let H be ag-closed subgroup of K. Sinceg is hereditary, without loss of generality we may assume that H is dense inK, and so(A, τH) is Hausdorff. By Theorem 2.1(c), every scontinuous character of (A, τH) is continuous. In particular, every countably continuous c haracter of (A, τH) is continuous. By a result of Hernández and Macario [12, 3.2 ], this is equivalent toH being realcompact. Proposition 2.6. LetB be a discrete abelian group, φ : A → B a homomorphism, and H ≤ K andL ≤ C, whereC = B̂. The following are equivalent: (i) φ : (A, τH) → (B, τL) is s-continuous; (ii) χ ◦ φ is s-continuous on(A, τH) for everys-continuous character χ of (B, τL); (iii) gφ : (A, τgK(H)) → (B, τgC(L)) is continuous. PROOF. Clearly, (i)⇒ (ii), because the composite of a continuous and and an s-continuous map is continuous. The implication (ii) ⇒ (iii) follows from Theorem 2.1(b), because (ii) states that χ ◦ φ ∈ gK(H) for everyχ ∈ gC(L). To show (iii) ⇒ (i), notice thatφ is the composite of (A, τH) → (A, τgK(H)), gφ, and(B, τgC(L)) → (B, τL). Since the three maps are all s-continuous (cf. Theorem 2.1(d)), their composite is also s-continuous. Corollary 2.7. For H1, H2 ≤ K, τH1 andτH2 are s-equivalent if and only if g(H1) = g(H2). In particular, τg(H1) is the finest precompact group topology on A that iss-equivalent toH1. 4 G. Luḱacs / Notes on the g-closure Inspired by the notions of k-group ands-group introduced by Noble [13] & [14], and the term “kk-group” used by Deaconu [7], we say that a topological group G is ansk-group if every s-continuous homomorphism of G into a compact Hausdorff group is continuous. Proposition 2.8. LetH be a subgroup of K. The following are equivalent: (i) everys-continuous homomorphism φ : (A, τH) → (B, τL) into a precompact abelian group (whereB is a discrete group andL ≤ B̂) is continuous; (ii) (A, τH) is ansk-group. (iii) everys-continuous character χ : (A, τH) → T is continuous; PROOF. (i) ⇒ (ii) ⇒ (iii) is obvious, because every compact groups is precompac t, andT is compact. In order to show (iii) ⇒ (i), let φ : (A, τH) → (B, τL) be ans-continuous homomorphism. By Proposition 2.6, χ ◦ φ is s-continuous on(A, τH) for every character χ ∈ L of (B, τL). Applying (iii) to the character χ ◦ φ, one obtains that it is continuous, in other words, χ ◦ φ ∈ H. Therefore,χ ◦ φ ∈ H for everyχ ∈ L, and henceφ is continuous. A combination of Theorem 2.1(c) and Proposition 2.8 yields: Corollary 2.9. A subgroupH ≤ K is g-closed if and only if(A, τH) is ansk-group. In particular, (A, τgK(H)) is ansk-group for everyH ≤ K. Put Prs for the category of precompact groups and their s-continuous homomorphisms, Pr for precompact groups with continuous homomorphisms, and skPr for the full subcategory of Pr consisting of thesk-groups. The next theorem is modeled on the relationship bet ween the category of Hausdorff topological spaces with k-continuous maps, Hausdorff topological spaces with continuous maps, and Hausdorff k-spaces. Theorem 2.10. The assignment g : Prs −→ skPr (A, τH) 7−→ (A, τgK(H)), (K = Â, H ≤ K) is a functor. Furthermore, (a) g : Prs −→ skPr is an equivalence of categories; (b) g : Pr −→ skPr is a coreflection. PROOF. It was explained earlier that every precompact abelian grou p has the form(A, τH), where A is a discrete abelian group and H ≤ Â (A andH are uniquely determined). By Corollary 2.9, g(A, τH) = (A, τgK(H)) ∈ skPr, and by Proposition 2.6, gφ is continuous for everys-continuous homomorphismφ : (A, τH) → (B, τL). Therefore,g is well-defined, and it is a functor. (a) Clearly,g and the inclusionI : skPr −→ Prs are faithful, and by Proposition 2.6, they are also full. Sinceg is idempotent,gI = IdskPr. By Theorem 2.1(d), (A, τH) and (A, τgK(H)) are isomorphic i
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