Bull. London Math. Soc. 2014 Zelenyuk 981 8

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Bull. London Math. Soc. 46 (2014) 981–988 C 2014 London Mathematical Societydoi:10.1112/blms/bdu055

Continuity in GLU C

Yevhen Zelenyuk

Abstract

Given a locally compact group G, GLUC is the largest semigroup compactification of G andG∗ = GLUC \ G. We show that (i) for every locally compact compactly generated Abelian groupG and for every p, q ∈ G∗, the multiplication in GLUC is discontinuous at ( p, q ), (ii) there is alocally compact σ-compact torsion-free Abelian group G for which, assuming Martin’s Axiom,there are p, q ∈ G∗ such that the multiplication in GLUC is continuous at ( p, q ), and (iii) it isconsistent with ZFC that for every locally compact Abelian group G and for every p, q ∈ G∗, theleft translation by p in G∗ is discontinuous at q .

1. Introduction

A semigroup compactification of a topological group G is a semigroup S with a compacttopology together with a continuous homomorphism ψ : G → S such that ψ(G) is dense inS , for each q ∈ S the right translation S x → xq ∈ S is continuous, and for each a ∈ G theleft translation S x → ψ(a)x ∈ S is continuous. All topologies are assumed to be Hausdorff.If the mapping G × S (x, y) → ψ(x)y ∈ S is continuous, then S is said to have the jointcontinuity property. The largest semigroup compactification of G with the joint continuityproperty, in the sense that any other is a natural quotient, is called the LU C -compactificationand denoted by GLUC . The homomorphism ψ : G → GLUC is a topological embedding, sowe identify G with its image, and write G∗ = GLUC \ G. As a topological compactification,GLUC is characterized by the property that a continuous function f : G → [0, 1] extends to acontinuous function f : GLUC → [0, 1] if and only if f is uniformly continuous with respect tothe right uniformity (see [2, Theorem 21.41]). If G is locally compact, then every semigroupcompactification of G has the joint continuity property (see [8, Theorem II.4.3]), so GLUC isthe largest semigroup compactification of G. In the case, where G is discrete, GLUC coincideswith β G, the Stone–Cech compactification.

It is well known that for every locally compact group G and for every p ∈ G∗, the mappingλ p : GLUC x → px ∈ GLUC , the left translation by p in GLUC , is discontinuous [3] (in theAbelian case [7]), and moreover, the mapping λ∗ p : G∗ x → px ∈ G∗, the left translation by

p in G∗, is discontinuous [5] (in the discrete case [9]). It is also known that for every countablediscrete Abelian group G not containing an infinite Boolean subgroup and for every p, q ∈ G∗,

the mapping µ : βG × βG (x, y) → xy ∈ βG, the multiplication in βG, is discontinuous at( p, q ) [4].

On the other hand, assuming additional set-theoretic assumptions, there are nontrivial pointsof continuity.

Recall that a nonprincipal ultrafilter p on ω is a P -point if the intersection of countablymany neighborhoods of p ∈ ω∗ is again a neighborhood of p. Martin’s Axiom (MA) implies theexistence of P -points. However, it is consistent with ZFC, the system of usual axioms of settheory, that there is no P -point (Shelah). A cardinal κ is measurable (Ulam-measurable ),

Received 7 January 2014; revised 3 May 2014; published online 15 July 2014.

2010 Mathematics Subject Classification 22A05, 22A15 (primary), 22D05, 54D35 (secondary).

This research is supported by NRF grant IFR2011033100072.



982 YEVHEN ZELENYUK

if there is a κ-complete (countably complete) nonprincipal ultrafilter on κ. A cardinal isUlam-measurable if and only if it is greater than or equal to the first uncountable measurablecardinal. It is consistent with ZFC that there is no Ulam-measurable cardinal. Moreover, it isconsistent with ZFC that there is neither P -point nor Ulam-measurable cardinal (see [11]).

Here are the examples of points of continuity.

(1) Let G be a countably infinite discrete group and let q ∈ G∗ be a P -point. Then, for every p ∈ G∗, the mapping µ∗ : G∗ × G∗ (x, y) → xy ∈ G∗, the multiplication in G∗, is continuousat ( p, q ) (van Douwen), and if G is Abelian, then λq is continuous at q .

(2) Let G be a discrete group such that κ = |G| is Ulam-measurable and let q ∈ G∗ be acountably complete ultrafilter. Then for any p ∈ G∗ with || p|| = ω, µ is continuous at ( p, q )(Protasov).

(3) Let G be a countably infinite discrete Boolean group. Then, assuming MA, there are p, q ∈ G∗ such that µ is continuous at ( p, q ).

Recently, the following two theorems were proved [11].

(a) If G is a discrete Abelian group containing no infinite Boolean subgroup and |G| is notUlam-measurable, then for every p, q ∈ G∗, µ is discontinuous at ( p, q ).

(b) Assume that there is neither P -point nor Ulam-measurable cardinal. Then for everydiscrete Abelian group G and for every p, q ∈ G∗, λ∗ p is discontinuous at q .

In this paper, we extend examples (1), (2), (3) and theorems (a), (b) to the locally compactAbelian groups. Our results involve the following fact.

Lemma 1.1. Let G be a locally compact Abelian group. Then there is a compact subgroup

K of G such that G/K = Rn × M for some n < ω and a discrete subgroup M of G/K, and

consequently , for the discrete subgroup N = Zn × M of G/K, (G/K )/N = T

n.

Proof. Let G0 be an open compactly generated subgroup of G. By [1, Theorem 9.8],G0 = Rn × Zm × K for some n,m < ω and a compact subgroup K . Then Rn × Zm is an opensubgroup of H = G/K , and so is Rn. Since Rn is divisible, there is a subgroup M of H such thatH = Rn × M algebraically. And since Rn is open, M is discrete and H = Rn × M topologicallyas well.

The number n < ω in Lemma 1.1 is determined uniquely, and the subgroups K and M almost uniquely in the following sense. If (K 1, M 1) and (K 2, M 2) are two such pairs, then foreach i ∈ {1, 2}, F i = K 1K 2/K i is a compact subgroup of H i = R

n × M i, and consequently,a finite subgroup of M i, so there is a third pair (K, M ) with K = K 1K 2 and M = M 1/F 1

or M = M 2/F 2. It follows that |N 1| = |N 2| if G is noncompact, and N 1 contains an infiniteBoolean subgroup if and only if so does N 2. Also note that if G is not compactly generated,then |N | = |G/G0|. Indeed, G/G0 = M/Zm and N = Z

n × M .For every locally compact Abelian group G, choose a compact subgroup K of G and a

discrete subgroup M of G/K such that G/K = Rn × M for some n < ω and define the discrete

subgroup N (G) of G/K by N (G) = Zn × M .

Now, we can state our results.

Theorem 1.2. Let G be a noncompact locally compact Abelian group.

(1) Assume that there is a P -point. Then there are p, q ∈ G∗ such that µ∗ is continuous at

( p, q ) and λ p

is continuous at q .(2) If |N (G)| is Ulam-measurable , then there are p, q ∈ G∗ such that µ is continuous at ( p, q ).



CONTINUITY IN GLUC 983

(3) Assume MA. If N (G) contains an infinite Boolean subgroup , then there are p, q ∈ G∗

such that µ is continuous at ( p, q ).

Statement (3) of Theorem 1.2 implies a surprising fact that has no analog in the discretecase: there is a locally compact σ-compact torsion-free Abelian group G for which, assumingMA, there are p, q ∈ G∗ such that µ is continuous at ( p, q ). Here is an example.

Example 1.3. Let Z(2) denote the additive group of 2-adic integers and let H =ω Z(2).

Define the subgroups K and G of H by K = 2H =ω 2Z(2) and G =

ω Z(2) + K . Define

the group topology on G by taking the natural compact topology on K and declaring thesubgroup K to be open. Then G is a locally compact σ-compact torsion-free Abelian group, K an open compact subgroup of G, and N = G/K =

ω Z2 a countably infinite discrete Boolean

group. Hence, by statement (3) of Theorem 1.2, assuming MA, there are p, q ∈ G∗ such that µis continuous at ( p, q ).

Theorem 1.4. (a) If G is a locally compact Abelian group such that N (G) contains no

infinite Boolean subgroup and |N (G)| is not Ulam-measurable , then for every p, q ∈ G∗, µ is

discontinuous at ( p, q ).

(b) Assume that there is neither P -point nor Ulam-measurable cardinal. Then for every

locally compact Abelian group G and for every p, q ∈ G∗, λ∗ p is discontinuous at q .

As a partial case of statement (a) of Theorem 1.4, we obtain that for every locally compactcompactly generated Abelian group and for every p, q ∈ G∗, µ is discontinuous at ( p, q ). From

statement (b), we obtain that it is consistent with ZFC that for every locally compact Abeliangroup G and for every p, q ∈ G∗, λ∗ p is discontinuous at q .The paper is organized as follows. In Section 2, we study the structure and natural

homomorphisms of GLUC . Then, in Section 3, we prove Theorem 1.2 and Theorem 1.4.

2. Structure and natural homomorphisms of GLUC

Given a topological group G and A ⊆ G, A is the closure of A in GLUC .

Lemma 2.1. Let G be a topological group and let A, B ⊆ G. If there is a neighborhood U of 1 ∈ G such that (U A) ∩ B = ∅, then A ∩ B = ∅.

Proof. Since (U A) ∩ B = ∅, there is a uniformly continuous function f : G → [0, 1] suchthat f (A) = {0} and f (B) = {1} [2, Exercise 21.5.3]. Let f : GLUC → [0, 1] be the continuousextension of f . Then f ( A) = {0} and f ( B) = {1}. Hence, A ∩ B = ∅.

A subset D of a topological group G is uniformly discrete if there is a neighborhood U of 1such that the family {U a : a ∈ D} is disjoint, in which case we say that D is U -discrete . Everydiscrete subgroup is uniformly discrete.

Lemma 2.2 is a version of the Local Structure Theorem (see [2, Lemma 21.42] for part (i)and [6] for the whole result).



984 YEVHEN ZELENYUK

Lemma 2.2. Let G be a topological group and let D be a uniformly discrete subset of G.

Then

(i) D = βD.

Furthermore , if G is locally compact, then

(ii) for every a ∈ G and p ∈ D, the sets U V, where U is a neighborhood of a ∈ G and V a

neighborhood of p ∈ D, form a neighborhood base at ap ∈ GLUC ;(iii) if U is an open neighborhood of 1 ∈ G such that D is U -discrete , then the mapping

U × βD (x, y) → xy ∈ U D is a homeomorphism.

Proof. (i) It suffices to show that any two disjoint subsets A, B of D have disjoint closuresin GLUC . Choose a neighborhood U of 1 ∈ G such that the family {U a : a ∈ D} is disjoint.Then (U A) ∩ B = ∅. Consequently, by Lemma 2.1, A ∩ B = ∅.

(ii) Choose an open neighborhood U 0 of 1 such that a U 0 ⊆ U , U 0 is compact, and D

is U 0-discrete, and choose D0 ⊆ D such that p ∈ D0 ⊆ V . We claim that U 0 D0 is open inGLUC . To see this, let B = G \ (U 0D0). Then U 0D0 ∪ B = GLUC . For every u ∈ U 0, there isa neighborhood W of 1 ∈ G such that W u ⊆ U 0, so (W uD0) ∩ B = ∅, and by Lemma 2.1,(uD0) ∩ B = ∅. Consequently, (U 0D0) ∩ B = ∅. Now let s ∈ U 0D0 \ (U 0D0). Then s = vq forsome v ∈ U 0 \ U 0 and q ∈ D0. Since D0 is U 0-discrete, (vD0) ∩ (U 0D0) = ∅. Consequently,vD0 ⊆ B, and so s ∈ B. Hence, U 0D0 = GLUC \ B.

(iii) Clearly, this mapping is continuous, and by (ii), it is open. We have to check injectivity.For every a ∈ G, the left translation by a in GLUC is a homeomorphism (because λa−1 ◦λa = λa ◦ λa−1 = id), so ap = aq for any distinct p, q ∈ GLUC . Now let u, v ∈ U be distinct.Then (uD) ∩ (vD) = ∅ and (uD) ∪ (vD) is uniformly discrete. To see the latter, choose aneighborhood W of 1 ∈ G such that W u, W v ⊆ U and (W u) ∩ (W v) = ∅, then (uD) ∪ (vD) is

W -discrete. Hence by (i), (u D) ∩ (v D) = ∅, and so up = vq for any p, q ∈ D.

Lemma 2.3. Let G be a topological group and let N be a closed normal subgroup of G.

Then

(i) the natural mapping G → G/N extends to a continuous homomorphism π : GLUC →(G/N )LUC ;

(ii) for every a ∈ G, π−1(π(a)) = a N .

Proof. (i) The natural mapping G → G/N is uniformly continuous, so it is immediatefrom [2, Theorem 21.45].

(ii) Let q ∈ GLUC \ (a N ). Choose a closed neighborhood U of a N not containing q . Usingthe joint continuity property and compactness of N , choose a neighborhood V of 1 ∈ Gsuch that V a N ⊆ U . Let A = aN and B = G \ V A. Then a ∈ A and q ∈ B, so π(a) ∈ π(A)and π(q ) ∈ π(B). But (π(V )π(A)) ∩ π(B) = ∅, consequently, by Lemma 2.1, π(A) ∩ π(B) = ∅.Hence, π(q ) = π(a).

Proposition 2.4. Let G be a locally compact group , let N be a discrete normal subgroup

of G such that G/N is compact, and let π : GLUC → G/N be the natural homomorphism.

Then

(1) N = βN ;(2) for every z ∈ G/N, there is a ∈ G unique modulo N such that π−1(z) = a N ;




(3) GLUC = G N ;(4) for every a ∈ G and p ∈ N , the sets U V, where U is a neighborhood of a ∈ G and V a

neighborhood of p ∈ N , form a neighborhood base at ap ∈ GLUC ;(5) if U is an open neighborhood of 1 such that N is U -discrete , then the mapping U × βN

(x, y) → xy ∈ U N is a homeomorphism.

Proof. (1), (4), and (5) are immediate from Lemma 2.2.(2) For every z ∈ G/N , there is a ∈ G unique modulo N such that π(a) = z, and by

Lemma 2.3, π−1(π(a)) = a N .(3) is immediate from (2).

Proposition 2.5. Let G be a topological group , let N be a compact normal subgroup of

G, and let π : GLUC → (G/N )LUC be the natural homomorphism. Then

(1) for every p ∈ GLUC , π−1(π( p)) = N p;(2) π is open;(3) π(G∗) = (G/N )∗.

Proof. (1) Let q ∈ GLUC \ N p. Choose a closed neighborhood U of N p not containingq . Using the joint continuity property and compactness of N , choose a neighborhood V of 1 ∈ G and a neighborhood W of p ∈ GLUC such that V N W ⊆ U . Let A = N (W ∩ G) and B =G \ V A. Then p ∈ A and q ∈ B, so π( p) ∈ π(A) and π(q ) ∈ π(B). But (π(V )π(A)) ∩ π(B) = ∅,consequently, by Lemma 2.1, π(A) ∩ π(B) = ∅. Hence, π(q ) = π( p).

(2) Let U ⊆ GLUC be open. By (1), π−1(π(U )) = N U . For every a ∈ G, the left translationby a in GLUC is a homeomorphism, so N U =

a∈N aU is open. Then GLUC \ N U is closed,

and consequently, π(GLUC \ N U ) = (G/N )LUC \ π(U ) is closed. Hence, π(U ) is open.(3) Let p ∈ G∗. By (1), π−1(π( p)) = N p. Since N p ⊆ G∗, it follows that

π( p) ∈ (G/N )∗.

3. Proofs of Theorems 1.2 and 1.4

Proof of Theorem 1.2. (1) Suppose first that G is compactly generated. Then G = Rn ×

Zm × K for some n,m < ω and a compact subgroup K , N = Z

n+m is an infinite discretesubgroup of G, and N = βN (Lemma 2.2). Let p, q ∈ N ∗ be such that the multiplication inN ∗ is continuous at ( p, q ) (the left translation by p in N is continuous at q ). We claim that

the multiplication in G∗

is continuous at ( p, q ) (the left translation by p in GLUC

is continuousat q ).

To see that the multiplication in G∗ is continuous at ( p, q ), let U be a neighborhood of 1 ∈ Gand R ∈ pq , so U R∗ is a neighborhood of pq ∈ G∗ (Lemma 2.2). Choose a neighborhood V of 1 ∈ G, P ∈ p, and Q ∈ q such that V 2 ⊆ U and P ∗Q∗ ⊆ R∗. Then V P ∗ is a neighborhood of

p ∈ G∗, V Q∗ a neighborhood of q ∈ G∗, and V P ∗V Q∗ = V 2P ∗Q∗ ⊆ U R∗.To see that the left translation by p in GLUC is continuous at q , let U be a neighborhood of

1 ∈ G and R ∈ pq , so U R is a neighborhood of pq ∈ GLUC . Choose Q ∈ q such that pQ ⊆ R.Then U Q is a neighborhood of q ∈ GLUC and pU Q = U pQ ⊆ U R.

Now suppose that G is not compactly generated. Let G0 be an open compactly generatedsubgroup of G. Then G/G0 is an infinite discrete group. Construct inductively a sequence(x

n)n<ω

in G such that the cosets xn

G0

and xm

xk

G0

, where n < ω and m < k < ω, are pairwisedistinct, so the set D = {xn : n < ω } ∪ {xmxk : m < k < ω} is uniformly discrete. Let q ∈ D∗



986 YEVHEN ZELENYUK

be a P -point such that {xn : n < ω } ∈ q . Then {xn : n < ω}∗q ⊆ D∗. We claim that for every p ∈ {xn : n < ω }∗, the multiplication in G∗ is continuous at ( p, q ), and the left translation byq in GLUC is continuous at q .

To see the first, let U be a neighborhood of 1 ∈ G and R ∈ pq . Choose a neighborhood V

of 1 ∈ G such that V 2 ⊆ U . Choose P ∈ p contained in {xn : n < ω }, and for each xn ∈ P ,Qn ∈ q contained in {xk : n < k < ω} such that

xn∈P

xnQn ⊆ R. Since q is a P -point, thereis Q ∈ q such that Q \ Qn is finite for all n < ω. It follows that P ∗Q∗ ⊆ R∗, and consequently,V P ∗V Q∗ = V 2P ∗Q∗ ⊆ U R∗.

To see the second, let U be a neighborhood of 1 ∈ G and R ∈ qq . Choose P ∈ q containedin {xn : n < ω}, and for each xn ∈ P , Qn ∈ q contained in {xk : n < k < ω} such thatxn∈P

xnQn ⊆ R. Since q is a P -point, there is Q ∈ q such that Q \ Qn is finite for all n < ω .It follows that qQ∗ ⊆ R∗. Choose Q ∈ q in addition so that Qq ⊆ R. Then q Q = qQ ∪ qQ∗ =Qq ∪ qQ∗ ⊆ R, and consequently, qU Q = U q Q ⊆ U R.

(2) Let G0 be an open compactly generated subgroup of G and suppose that κ = |G/G0| isUlam-measurable. Construct inductively a sequence (xα)α<κ in G such that the cosets xnG0

and xmxγ G0, where n < ω and m < γ < κ, are pairwise distinct, so the set D = {xn : n <ω} ∪ {xmxγ : m < γ < κ} is uniformly discrete. Let p, q ∈ D∗ be ultrafilters on D such that{xn : n < ω} ∈ p, {xα : α < κ} ∈ q and q is countably complete. Clearly, pq ∈ D∗. We claimthat the multiplication in GLUC is continuous at ( p, q ).

To see this, let U be a neighborhood of 1 ∈ G and R ∈ pq . Choose a neighborhood V of 1 ∈ Gsuch that V 2 ⊆ U . Choose P ∈ p contained in {xn : n < ω }, and for each xn ∈ P , choose Qn ∈ q contained in {xα : n < α < κ} such that

xn∈P

xnQn ⊆ R. Let Q =n<ω Qn. Then P Q ⊆ R,

and since q is countably complete, Q ∈ q . Hence, V P is a neighborhood of p ∈ GLUC , V Q aneighborhood of q ∈ GLUC , and V P V Q = V 2 P Q ⊆ U R.

(3) Let K be a compact subgroup of G and suppose that G/K contains an infinite discreteBoolean subgroup B. Choose an independent subset {bn : n < ω} of B , for every n < ω , choosexn ∈ G such that xnK = bn, and let D = FP((xn)n<ω). As usual, FP((xn)n<ω) = {

n∈H xn :

H ∈ P f (ω)}, where P f (ω) is the family of finite nonempty subsets of ω. Then D is uniformlydiscrete, so D = βD, T =

m<ω FP((xn)mn<ω) is a closed subsemigroup of D∗ [2, Lemma

5.11], and for every x ∈ D, x2 ∈ K .It follows from the proof of [10, Theorem 10.4] that assuming MA, there is a filter F on D

containing exactly two ultrafilters p, q ∈ T together with a mapping f : D → { p, q } such that

(i) both p and q are idempotents and pq = qp = q ;(ii) f −1( p) ∈ p and f −1(q ) ∈ q ;

(iii) f (i∈I xi) = q if and only if there is i ∈ I such that f (xi) = q ; and

(iv) F has a base consisting of subsets of the form FP((yn)n<ω), where (yn)n<ω is a productsubsystem of (xn)n<ω.

(Recall that (yn)n<ω is a product subsystem of (xn)n<ω if there is a sequence (H n)n<ω inP f (ω) such that for every n < ω , yn =

i∈H n

xi and max H n < min H n+1.) We claim that themultiplication in GLUC is continuous at ( p, q ).

To show this, we first note that the images of p, q under the mapping x → x2 are idempotentultrafilters on {x2 : x ∈ D} ⊆ K , so by [10, Lemma 7.10], they converge to 1. Now let U bea neighborhood of 1 ∈ G and let R ∈ pq = q , so U R is a neighborhood of pq ∈ GLUC . Choosea compact neighborhood V of 1 ∈ G such that V 3 ⊆ U , and choose a product subsystem(yn)n<ω of (xn)n<ω such that FP((yn)n<ω) ∩ f −1(q ) ⊆ R and FP((y2n)n<ω) ⊆ V . Define P ∈ pand Q ∈ q by P = FP((yn)n<ω) ∩ f −1( p) and Q = FP((yn)n<ω) ∩ f −1(q ). Then P Q ⊆ V Q.

Indeed, let x ∈ P and y ∈ Q. Write x =i∈I yi and y =

j∈J yj. For every i ∈ I , f (yi) = p,

and there is j ∈ J such that f (yj) = q . Let u =k∈I ∆J yk and v =

i∈I ∩J y

2i . Then xy = uv,

u ∈ Q, and v ∈ V .




Finally, V P is a neighborhood of p ∈ GLUC , V Q is a neighborhood of q ∈ GLUC , andV P V Q = V 2P Q ⊆ V 2V Q = V 3 Q ⊆ U R.

Theorem 1.4 is a consequence of its discrete case and the following reduction theorem.

Theorem 3.1. Let G be a locally compact group and suppose that there are a compact

normal subgroup K of G and a discrete normal subgroup N of H = G/K such that H/N is

compact. If there are p, q ∈ G∗ such that the multiplication in GLUC is continuous at ( p, q )(the left translation by p in G∗ is continuous at q ), then there are p0, q 0 ∈ N ∗ such that the

multiplication in βN is continuous at ( p0, q 0) (the left translation by p0 in N ∗ is continuous

at q 0).

Proof of Theorem 3.1. We first use Proposition 2.5. Let π : GLUC → H LUC be thenatural homomorphism and let p1 = π( p) and q 1 = π(q ). Then p1, q 1 ∈ H ∗. We claim that the

multiplication in H LUC is continuous at ( p1, q 1) (the left translation by p1 in H ∗ is continuous atq 1). To see this, let U 1 be a neighborhood of p1q 1 ∈ H LUC . Then U = π−1(U 1) is a neighborhoodof pq ∈ GLUC . Choose a neighborhood V of p and a neighborhood W of q such that V W ⊆ U .Since π is open, V 1 = π(V ) is a neighborhood of p1 and W 1 = π(W ) a neighborhood of q 1, andV 1W 1 = π(V )π(W ) = π(V W ) ⊆ π(U ) = U 1. The check for the left translation is similar.

Now we use Proposition 2.4. Write p1 = ap2 and q 1 = bq 0 for some a, b ∈ H and p2, q 0 ∈ βN .Let p0 = b−1 p2b. Then p1q 1 = ap2bq 0 = abb−1 p2bq 0 = abp0q 0. We claim that the multiplicationin βN is continuous at ( p0, q 0) (the left translation by p0 in N ∗ is continuous at q 0). To seethis, let R ∈ p0q 0. Choose a neighborhood U of 1 ∈ H such that N is U -discrete. Then abU R isa neighborhood of p1q 1. Since the multiplication in H LUC is continuous at ( p1, q 1), there are aneighborhood V of 1 ∈ H , P ∈ p2, and Q ∈ q 0 such that (aV P )(bV Q) ⊆ abU R. In particular,

aP bQ = ab(b−1

P b) Q ⊆ abU R, so (b−1

P b) Q ⊆ U R. But then (b−1

P b) Q ⊆ R. The check for theleft translation is similar.

Acknowledgements. The author thanks the referee for a careful reading of the paper anduseful comments.

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988 CONTINUITY IN GLUC

Yevhen Zelenyuk School of Mathematics University of the Witwatersrand (WITS )Private Bag 3

Johannesburg 2050 South Africa

yevhen·zelenyuk@wits·ac·za

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Bull. London Math. Soc. 2014 Zelenyuk 981 8