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
A theorem on the continuity of homomorphisms of locally compact groups
0. Introduction. When f: T1 → T2 is a mapping between two topological spaces we shall say that f is sequentially continuous if and only if in T1 implies that in T2.
Synthesis of quasi-stationary inhomogeneous impedance body of arbitrary shape
Formulas were obtained for the reflection factors of a plane wave from the inhomogeneous impedance plane, which reflects the incident wave in the given direction. The problem of synthesis of impedance inhomogeneous rotation body was solved for an arbitrary shaped generatrix for the given scattering diagram for co- and cross polarizations. Impedance distributions were obtained in explicit form. Restrictions on the class of implementable scat- terplots were formulated.