Compactness and connectedness in topological groups

TOPOLOGY AND ITS APPLICATIONS ELSEVIER

Topology and its Applications 84 (1998) 227-252

Compactness

and connectedness Dikran

Dipartimento

di Matemtira

e Infmmatira,

Dikranjan

in topological groups * ’

Udine University. Via delk Scienze 206 (IocalitLi Rizzi). 33100 l/dine, Italy

Received 7 December 1995: revised 27 December 1996

Abstract We study the dynamic interrelation between compactness and connectedness in topological groups by looking at the scale of various levels of connectedness through the looking glass of compactness and vice versa. More precisely, we are interested in measuring the gap between the connected component c(G) and the quasi-component q(G) of a compact-like group G. Neither local compactness nor countable compactness of G “can distinguish” between the properties: (a) c(G) = 1. (b) q(G) = 1, (c) G is zero-dimensional; in particular, always c(G) = q(G) for such a group G. Pseudocompactness together with minimality “cannot distinguish” between (b) and (c), but pseudocompactness together with total minimality “distinguishes” between (a) and (b). In the opposite direction, connectedness “cannot distinguish” between compactness and {countable compactness plus minimality} for Abelian groups of nonmeasurable size. We also discuss the role of connectedness for the question when topology or algebra alone determine the topological group structure of a compact-like group. 0 1998 Elsevier Science B.V. Keywords: Connected group; Hereditarily disconnected group; Totally disconnected group; Zero-dimensional group; Functorial subgroup: Pseudocompact group; Precompact group; Complete group; Initially a-compact group; Countably compact group: (Totally) minimal group AMS classijicution:

Primary 54F45; 54D0.5; 54D30, Secondary

Dedicated

to Adalberto

22A05; 22D05; 54D2.5

Orsatti on his 60th birthday

Introduction

In this work we consider are Hausdorff.

Following

only Tychonoff

spaces,

[44], we call a topological

*This work has been supported CRG 950347. ’ E-mail: [email protected].

by funds

MURST

in particular, space

60% and by NATO Collaborative

0166~8641/98/$19.00 0 1998 Elsevier Science B.V. All rights reserved. I’ZI 50166.8641(97)00094-l

all topological

groups

X: Research

Grant

D. Dikranjan / Topology and its Applications 84 (1998) 227-252

228

l

hereditarily

l

totaEZy disconnected,

disconnected,

if all connected

if all quasi-components

components

of X are trivial;

of X are trivial (i.e., distinct points of

X can be separated by a partition); l

zero-dimensional,

Obviously,

if X has a base of clopen sets.

zero-dimensional

+ totally disconnected

+ hereditarily

disconnected.

The

reverse implications hereditarily

disconnected

(1,, totally disconnected

&

zero-dimensional

(D)

hold true for some nice spaces:

Theorem (Vedenissov).

A hereditarily

disconnected

locally compact space is zero-di-

mensional. Hence local compactness “cannot distinguish” between hereditary disconnectedness, total disconnectedness and zero-dimensionality. In particular, the connected component and the quasi-component coincide in a locally compact group G. Totally disconnected spaces of positive dimension were constructed first by Sierpidski [76], and Knaster and Kuratowski [.53] in dimension one, and Mazurkiewicz [54] for each finite dimension. The first counterexample for topological groups was given by Erdos [45]-the rational points in e2 form a totally disconnected group which is not zero-dimensional. Later van Mill [55] showed that every locally compact Abelian group G with dim G > n + 1 contains a totally disconnected subgroup H with dim H 3 72 (H can be constructed to be a Bore1 subset of G whenever G is metrizable). Answering a question of Shell ([75], [ 12, Question 3M.31) Ursul [86] found for each natural n a totally disconnected subfield F, of the ring Iw x C’” of dimension n. The aim of this survey is to collect some facts which witness the dynamic interplay between compactness and connectedness in topological groups in the spirit of Vedenissov’s theorem. In other words, we will impose on our groups various compactness-like

condi-

tions in order to ensure the validity of the implications (1) or (2) in the “disconnectedness scale” (D). We consider the following generalizations of compactness for a topological group G, where G will denote the two-sided completion of G: - (two-sided) complete (i.e., G = G), - precompact (i.e., G is compact), - pseudocompact (every continuous function

G + Iw is bounded),

- hereditarily pseudocompact (every closed subgroup of G is pseudocompact), - countably compact (each open countable cover of G admits a finite subcover), - minimal (each continuous isomorphism G 4 H is open [77]), - totally minimal (each Hausdorff quotient of G is minimal [32, 4.31). Compact groups are totally minimal and countably compact (for the converse see Section 2.4), countably compact groups are hereditarily pseudocompact and pseudocompact groups are precompact (Comfort and Ross [20]). The totally minimal groups are precisely the groups G satisfying the open mapping theorem, i.e., each continuous smjective ho-

D. Dikranjan / Topology and its Applications 84 (1998) 227-252

momorphism

G + H is open. Prodanov

229

and Stoyanov proved [65] that Abelian minimal

groups are precompact. Shakhmatov (see Corollary docompactness

proved that the implication

(2) holds for minimal

pseudocompact

1.1.4), while Comfort and van Mill [ 16, Corollary alone does not guarantee

validity of (2):

Theorem. For every natural number n there exists a totally disconnected, pact Abelian group G, such that dim G, = n. and asked if (1) is true for precompact Question

A. Is every precompact

groups

7.71 showed that pseu-

pseudocom-

groups [ 16, Remark 7.81:

hereditarily

disconnected

group totally disconnected?

The answer turned out to be negative even for pseudocompact totally minimal groups (see Section 1.4) which motivated Shakhmatov to ask whether (1) is true for countably compact groups. These two questions

mainly will be the leitmotiv

in Section

1. In Section 2 connect-

edness plays a “dual” role: now relations between various compact-like properties are investigated under appropriate (dis)connectedness conditions. In order to better understand and resolve the main problems in Section 1 we borrow from algebra two tools: preradicals and extensions. Preradicals (i.e., functorial subgroups) give a key to describe connectedness

and measure the gap between the quasi-component

and the connected component of a pseudocompact group (see Section 1.1 and Sections 1.4.2 and 1.4.3). The extension procedure discussed in Section 1.3 permits to construct pseudocompact groups with prescribed properties. This technique permits to answer negatively Question A (Corollary 1.4.2) and to describe the connected component and the quasi-component of a pseudocompact group (Section 1.4.2). In Section 1.2 we see that both (1) and (2) are true for a countably compact group (Theorem 1.2.1). The proof is based on appropriate approximation of pseudocompact groups by simpler ones (Lemma

1.2.3). In Sections

1.4.4 and 1.4.5 some recent results on transfinite

dimensions

and preservation of total disconnectedness under taking quotients are given. The question when the algebraic structure alone of a compact-like group determines its topological

group structure is treated in Section 2.1. For example, if an Abelian group

admits a unique compact group topology then this topology is totally disconnected and these groups can be completely classified (Theorem 2.1.3). In Section 2.2 we check the level of compactness which ensures the validity of Scheinberg’s theorem (homeomorphic compact connected Abelian groups are isomorphic). It turns out that compactness cannot be replaced by initial o-compactness, and in particular by countable compactness. In Section 2.3 we discuss the relation between K-compactness and the known compactness conditions. (A topological group G is K-compact if for each topological group H the projection G x H + H sends closed subgroups of G x H to closed subgroups of H.) Now connectedness “cannot distinguish” between compactness and {local compactness and K-compactness} (Theorem 2.3.5), while discreteness “cannot distinguish” between

D. Dikranjan / Topology and its Applications 84 (1998) 227-252

230

hereditary

total minimality

The question

and K-compactness

when countable

compactness

for countable

plus minimality

groups (Theorem

2.3.6).

imply compactness

is dis-

cussed in Section 2.4. Compactness

and connectedness

are the principal topological properties, so that writing

a complete survey on all recent results on this topic, even if restricted only to topological groups, was out of question.

As a consequence,

I have limited the point of view of this

survey perhaps too close to my personal interest and taste. Nevertheless, of errors and omissions

remains.

Comments

and suggestions

the possibility

are welcome.

We denote by N and P the sets of naturals and primes, respectively, by Z the integers by Q the rationals, by lR the reals, by @ the complex numbers, by T the unit circle group in C, by Z, the p-adic integers (p E P). Let G be a group and A be a subset of G. We denote by 1 the neutral element of G and by (A) the subgroup of G generated by A. The group G is divisible if for every g E G and positive n E N the equation 9 the only divisible subgroup of G.

1. Disconnectedness

= g has a solution in G, G is reduced if 1 is

under the looking glass of compactness

I. 1. How to measure connectedness

There are two typical ways to measure (dis)connectedness. Category theory and algebra suggest to use functorial subgroups, topology offers dimension. 1,l. 1. Functorial

Definition preradical)

whenever

subgroups

1.1.1. Let P be a class of topological

groups. A functorial

subgroup

(or

in P is defined by assigning to each G E P a subgroup r(G) such that f : G --$ H is a continuous homomorphism in P one has f(r(G)) C r(H).

Note that a functorial subgroup is always normal since it must be invariant under all internal automorphisms of the group. A preradical t is a radical if r(G) is closed for each G and r(G/r(G))

= 1. The guiding example of a functorial

subgroup (radical) should be

the connected component c(G) and the quasi-component q(G) of a topological group G. Beyond the radicals c(G) and Q(G) for a topological group G, it is helpful to consider also the bigger radicals z(G) and o(G) defined as follows: _ z(G) is the intersection of all kernels of continuous homomorphisms G + H, where H is a zero-dimensional group, _ o(G) is the intersection of all open subgroups of G. Then c(G) C q(G) C z(G) C o(G) f or every group G. Moreover, z(G) = 1 iff G admits a coarser zero-dimensional group topology, while o(G) = 1 iff G admits a coarser linear topology (i.e., a group topology with a local base at 1 of open normal subgroups). In these terms we have the following:

D. Dikranjan / Topology and its Applications 84 (1998) 227-252

Question 1.1.2 (Arkhangel’skiy).

Does q(G) = 1 always imply z(G)

logical group G (i.e., does a totally disconnected dimensional

231

= 1 for a topo-

group always admit a coarser zero-

group topology)?

This is true for topological

spaces. We see now that the answer is “Yes” for pseudo-

compact groups. Theorem 1.1.3 [25]. For apseudocompactgroup G, q(G) = GIIC(@ = z(G) = o(G). In particular every pseudocompact totally disconnected group admits a coarser linear topology. This theorem gives a new proof of the following 1990 which answers Arkhangel’skii’s Corollary

unpublished

result of Shakhmatov

question for pseudocompact

of

groups:

1.1.4. Let G be a pseudocompact totally disconnected group. Then G admits a

coarser zero-dimensional group topology. In particular, a minimal pseudocompact group G is totally disconnected ifs G is zero-dimensional. Let t be a preradical.

A topological

group G is r-connected (or r-torsion) if r(G) = G,

hereditarily r-disconnected (or r-torsion-free) if r(G) = 1, and r-complete if G > r(G). The following

three examples

of functorial

subgroup are not radicals.

Example 1.1.5. Let G be a topological group, p be a prime number and Q be a cardinal. (i) (a) An element 5 of G is quasi-p-torsion if xpn converges to 1. The set td,(G) of quasi-p-torsion elements This subgroup is functorial.

of the group G forms a subgroup when G is Abelian. The group G is quasi-p-torsion if G consists of quasi-

p-torsion elements, (cf. [78] or [32, Chapter 41). (b) An element x of G is said to be of character < a if the character

of the cyclic

subgroup (z) of G is (Y[33]. The elements of countable character are the metrizable elements introduced

by Wilcox [90] for locally compact groups. Denote by M,(G)

the subset of elements

of G of character less than Q. Clearly Ma(G)

C: MD(G) if

o < /3 and MQ,“)+ (G) = G. If G is precompact, A&(G) coincides with the subset of torsion elements of G. In case G is also Abelian Ma(G) is a subgroup of G, and this defines a functorial

subgroup

in the category of precompact

Abelian

groups

[33]. Following Wilcox, we abbreviate MU, to M and call the elements of M(G) metrizable elements of G. Wilcox [90] proved that for a compact Abelian group G the subgroup M(G) (and so A&(G) for each a > w) is dense and pseudocompact. (c) For a precompact

Abelian

group G set

td(G) := {x E K: x( x ) 1s torsion for every characater This defines a functorial

subgroup

x : K + Ir}.

in the category of precompact

Abelian

groups

1241. For preradicals t and 5 we set r < 5 if for every topological group G r(G) C s(G). We also consider the composition rs of r and EIdefined as m(G) := r@(G)). The preradical

232

D. Dikranjan / Topology and ifs Applications 84 (1998) 227-252

r is said to be idempotent

if rr = r. The preradicals

respect to the meet and join defined for an arbitrarily preradicals

{t,},

by: (f\,

r,)(G)

= n,

r,(G)

form a large complete

lattice with

family (possibly a proper class) of

and (V, ra)(G)

= (U,r-a(G)).

Now we

can define the iterations ra for every ordinal cr by r’ = r, ra+’ = rr” and rO = Aficcu rp for limit Q. For each group G the descending then we set r”(G)

chain {r”(G)}

stabilizes

at some Q and

= t?+’ (G) = roe(G). The smallest a with this property will be called

Ulm r-length of G and denoted by X,(G). In this way we get an idempotent preradical r 03_- /\r”, it is the largest idempotent preradical contained in r. Clearly X,(G) = 1 iff r is idempotent,

so that X,(G) is a measure of the gap between r(G) and r”(G).

In

particular, qoo = c since a group satisfying H = q(H) is connected. It is important to note that the descending ordinal chain {r”} may form a proper class, even if for every group G its values {r”(G)} get stable from X,(G) on (an example to this effect, with r = q, will be given in Theorem 1.4.10). The next example shows that r-completeness with respect to certain preradicals may sometimes lead also to a good measure of compactness. It also provides examples of hereditarily

pseudocompact

non countably

compact Abelian

groups.

Example 1.1.6. Let G be a precompact Abelian group. (a) If G is td-complete, then G is totally minimal (see [32, Exercise 65.1 l(b)]). (b) G is hf-complete iff G is hereditarily pseudocompact and totally minimal. In fact, assume that G is Al-complete. Since td < M, G is also td-complete, hence G is totally minimal by (a). By the final remark of Example 1.1.5(b) G is pseudocompact. Now note that every closed subgroup H of G is still hf-complete since M(g) = Al(G) n k & G n I? = H. Hence H must be pseudocompact by the previous argument. Conversely, if G is hereditarily pseudocompact and totally minimal, then for every closed metrizable subgroup K of G the intersection N n G must be both dense in N and compact (as pseudocompact subgroup of a metrizable Further

group), so that N f? G = N, i.e., N c G.

information

concerning

functorial

subgroups

of topological

found in [24] (see also: [78,32] for td and td,, [24] for the relation c). For preradicals

in pointed categories

groups can be between

td, and

see [38, 5.51.

A general approach to connectedness and disconnectedness, in consonance with that given by Arkhangel’skii and Wiegandt [l] for topological spaces, can be adopted also in the context of topological groups as follows [24]. Let P be a class of topological groups. A touion theory of P is a pair T = (C,D) of subclasses of P such that the following orthogonality properties are satisfied: each continuous homomorphism f : C + D with C E C and D E ZJ is trivial; C (respectively D) contains all groups C (respectively D) of P such that all homomorphisms f : C -+ D with D E D (respectively C E C) are trivial. The groups in C are considered as r-connected or r-torsion and those in D-as hereditarily r-disconnected or T-torsion-free. Moreover, every topological group G has a closed functorial subgroup r(G) (the T-component of G) such that r(G) E C and G/T(G) E V. In particular, T is an idempotent radical. Vice versa, every idempotent

233

D. Dikranjan / Topology and its Applications 84 (1998) 227-252

radical T in P gives rise to a torsion theory of P by setting C = {G E P: r(G) = G} and D = {G E P: r(G) = 1). H ence this approach is less general than the one based on arbitrary preradicals. 1.1.2. Dimension The dimension nectedness.

theory of topological of a topological

All three dimensions

group G when G is locally (TkaEenko

groups

space is usually considered dim G = indG

compact

(Pasynkov

[SO]). (If P is a topological

= IndPG

as a measure of its concoincide

for a topological

[64]) or even locally

property, we call a topological

pseudocompact group G locally

P if there exists an open non-empty subset U of G whose closure in G has P.) Further, zero-dimensionality and strong zero-dimensionality coincide within the class of precompact groups (and more generally, subgroups of locally compact groups, see [68,71]). Therefore, no confusion may arise in discussing dimension throughout this paper. For reference on dimensions for reference

in topological

on transfinite

dimensions

Let N be a closed subgroup dim G = dim G/N

groups see the survey paper of Shakhmatov

[69],

see [43,72].

of a topological

G. The equality

+ dim N

was proved by Yamanoshita

(Y)

[92] for locally compact groups G and every closed subgroup

N of G. The equality dim G = dim G/N

+ dim fi, extending

partially

(Y), was proved

for a locally pseudocompact group G and an arbitrary closed subgroup N of G by Tkai-enko [80]. In particular, Tkacenko’s result implies that (Y) is true if the subgroup N is pseudocompact (in particular, when N splits, i.e., G E G/N x N). Hence (Y) is always true for a hereditarily pseudocompact (in particular, countably compact) group G. Shakhmatov [68] showed that (Y) fails for precompact groups. It turns out that even the following weaker version of (Y) obtained with N = q(G) dim G = dim G/q(G)

+ dim q(G)

may strongly fail for a pseudocompact 1.2. Zero-dimensionality The following (D) coincide

(Y,) group G (see Corollary

of some compact-like

1.4.2).

groups

theorem shows that all three levels of disconnectedness

for countably

considered

in

compact groups.

Theorem 1.2.1 [27]. Let G be a hereditarily disconnected, hereditarily pseudocompact group. Then G is zero-dimensional. In particular; every hereditarily disconnected, countably compact group is zero-dimensional. This theorem substantially enlarges the environment where Vedenissov’s theorem remains valid, since the class of hereditarily pseudocompact groups is as far as possible from that of locally compact groups. In fact, pseudocompact locally compact groups are compact.

D. Dikranjan / Topology and its Applications 84 (1998) 227-252

234

The following

technical

notion will be useful for the proof of Theorem

G is splitting if the connected

component

1.2.1: a group

c(G) splits off in G as a topological

direct

factor, i.e., there exits a closed subgroup N of G such that G 2 c(G) x N. This condition is not quite restrictive:

actually, all pseudocompact

groups appearing in our constructions

in Section 1 are splitting. The proof of Theorem

1.2.1 is based on the following

Lemma 1.2.2. Let G be a splitting connected component

hereditarily

C of the completion

three lemmas.

pseudocompact

G is metrizable.

group such that the

Then C C G.

A nice application of this lemma was found by Itzkowitz and Shakhmatov [52, Example 1.161. The lemma is substantially used also in the proof of the next lemma which is the other important ingredient in the proof of Theorem 1.2.1. Roughly speaking, every pseudocompact group can be approximated by splitting ones (as in Lemma 1.2.2) in the following

sense:

Lemma 1.2.3 (Approximation

Lemma).

Let G be a pseudocompact

group and let C be

the connected component of G. Then for eve_ry closed normal Gs-subgroup

L of C there

exists a closed normal Gg-subgroup N of G, such that the subgroup Gt = NC n G of G satisfies the following conditions: (a) Gi is a closed normal Gs-subgroup of G and G1 = NC; (b) L is a normal subgroup of Gt and the following isomorphism of topological groups holds G,/L

% NL/L

Clearly the subgroup

x C/L.

Gi admits a surjective

continuous

homomorphism

onto C/L.

This is used in the proof of the next lemma and corollary. Lemma 1.2.4. Let G be a pseudocompact group. Then q(G) is dense in c(G) iff G/q(G) is zero-dimensional. This occurs when G is hereditarily pseudocompact (in particular countably compact). For a cardinal Q denote by F, the free group of rank (Y and by Zca) the free Abelian group of rank (Y. For a group G set r(G) 3 a if there exists an embedding F, c-f G, and for an Abelian group H set r(H) 3 o if there exists an embedding Z(“) of H. Then the approximation Lemma 1.2.3 permits to prove also the following: Corollary 1.2.5 [27,37]. A pseudocompact group G with r(G) < 2” and r(G) < 2” is zero-dimensional. In particular a torsion pseudocompact group is zero-dimensional.

1.3. Extension theory of pseudocompact The following properties.

construction

provides

groups splitting pseudocompact

groups with prescribed

235

D. Dikranjan / Topology and its Applications 84 (1998) 227-252

Theorem

1.3.1. Let K be a pseudocompact

Then for each totally disconnected normal subgroup H with T(N/H) G with dimG

group and L be normal subgroup

3 \K/LI

th ere exists a splitting pseudocompact

= dim K, w(G) = max{w(K),wl}

(a) there exists a continuous

of K.

compact group N admitting a dense pseudocompact

open surjective

group

and such that: homomorphism

cp : G -+ K such that

the restriction of cp rq(~) is an embedding of q(G) into L; (b) there exist a continuous

homomorphism II, : G -+ N with dense image and a (dissection s : $J(G) + G, i.e., ker$ splits algebraically as a semidirect

continuous)

factor of G. Moreoven if the group K is connected then the restriction ‘p rn(~) in (a) is an isomorphism q(G) 2 L, ker$ = q(G) in (b) and the following three conditions are equivalent: (I) $ : G -+ $(G) is open; (2) L is dense in K; (3) dimG/q(G)

= 0.

Proof {Sketch). Let H C HI c N such that HI/H is free of rank \K\. Then there exists a surjective (discontinuous) homomorphism f : HI + K with kerf 2 H. Then the graph Pf of f is a dense pseudocompact G =

((1) x

subgroup of N x K and the product

L)rf

has the desired properties.

0

As N one can take (&_ S’,)lKI, where S, is the symmetric group. Now suppose that the group K in Theorem 1.3.1 is connected and metrizable

(in par-

ticular, connected and finite-dimensional). Then we have dim q(G) = dim L; moreover, item (3) gives dim q(G)+dim G/q(G) = dim L if L is dense in K. Since dim G = dim K, we conclude that dim q(G) + dim G/q(G) = dim G holds iff dim L = dim K. This produces

many totally

lary 1.4.2). Constructions

minimal,

pseudocompact

counterexamples

based on the idea of dense pseudocompact

Comfort and Robertson

to (Yp) (see Corol-

graphs were already used by

[ 181.

Theorem 1.3.2. If K and N/H in Theorem 1.3.1 are Abelian with r(N/H) > 2# and h7 is divisible (in particular if K is compact and connected) the homomorphism $ : G 4 N can be chosen surjective, so that G/q(G) is algebraically isomorphic to N. If L is dense in K, then G/q(G) E N is compact. Proof (Sketch).

By [91] the group K is connected,

so that Theorem

1.3.1 applies. The

homomorphism f : HI 4 K considered in the above proof can be extended whole group N since K is divisible. 0 Complete

proofs of these theorems will appear elsewhere.

now to the

D. Dikranjan / Topology and its Applications 84 (1998) 227-252

236

1.4. Applications 1.4.1. High-dimensional

hereditarily

disconnected,

totally

minimal,

pseudocompact

groups The first negative

answer to Question

0 < n < Y there exists a hereditarily

A was given by the author in 1990: for every

disconnected

and non-totally-disconnected

pseu-

docompact group of dimension n [25, Corollary 3.71. Moreover, it was shown that under the assumption of Lusin’s axiom 2 we = 2”, the groups can be chosen to be also totally minimal [25, Theorem 2.31. In 1991 Shakhmatov found a new proof of the existence of hereditarily disconnected and non-totally-disconnected pseudocompact groups of arbitrary dimension. following

An application

theorem (announced

of the extension

technique

from Section

1.3 gives the

in [26, Theorem 71) showing that total minimality

of such

groups can be ensured in ZFC.

Theorem 1.4.1. For every connected compact metrizable Abelian group h’ andfor every totally disconnected compact Abelian group N admitting a dense pseudocompact minimal subgroup H with r(N/H) 3 2” there exists a hereditarily disconnected,

totally totally

minimal, pseudocompact Abelian group G = GK,N such that: (i) dim G = dim K, dim q(G) = dim G/q(G) = 0; (ii) G/q(G) % N is a compact group; (iii) there exists a closed subgroup B of G such that G/B e c(G) g K; (iv) the subgroup q(G) is metrizable and splits algebraically in G.

Proof (Sketch). To get total minimality

of G and dim q(G) = 0 apply Theorem

1.3.2

with a zero-dimensional subgroup L of K such that L x H is totally minimal for every totally minimal Abelian group H. With K = TIT”the n-dimensional torus one can take L = (Q/Z)n the rational points of K, in general L = td(K) works (see Examples 1.1.5(c), 1.1.6 and [32, Theorem 6.1 ,111). By (i) (or (ii)) G is not totally disconnected if K # 0, while q(G) (and hence G) is hereditarily

disconnected

by (i).

0

The next corollary provides the first (negative) answer to Question A since a totally disconnected, minimal, pseudocompact group must be zero-dimensional according to Corollary

1.1.4.

Corollary 1.4.2. For every natural number n or n = w there exists a hereditarily disconnected, totally minimal, pseudocompact Abelian group H, with: (i) dim H, = n, dim q(H,) = dim H,/q(H,) = 0; (ii) H,/q(H,) is a compact totally disconnected group; (iii) H, has a compact connected quotient group of dimension n (isomorphic to c(G)); (iv) the subgroup q(H,) is metrizable and splits algebraically in H,.

Proof (Sketch). For each n E N or n = w apply Theorem 1.4.1 to K, = TIT”,N = n Zg’ and H = n HP, where each HP is a subgroup

of Z,w1 containing

the C-product

CZ,Wl

D. Dikranjan / Topology and its Applications 84 (1998) 227-252

(in the sense of Corson [22]), missing HP n C = 0) and is maximal quotient

Zf;l /HP is torsion-free

[32, Theorem and r(N/H)

a fixed nontrivial

with these properties. with r(Z;l

C of ZF’ (i.e.,

Then HP is pseudocompact

and the

/HP) = 1, hence HP is also totally minimal

4.3.71. Thus H is pseudocompact, = 2”.

cyclic subgroup

231

totally minimal

[32, Theorem

6.2.151

0

For each 7~ there are at least 2” nonisomorphic groups H, as in Corollary 1.4.2 (actually their completions are even pairwise non-homotopically-equivalent, see Theorem 2.2.1). By Theorem 1.2.1 “pseudocompact and totally minimal” cannot be replaced by “hereditarily pseudocompact”. Note that in view of (i) the group H, disproves equality (Y,) from Section

1.1.2.

1.4.2. The quasi-component

and the connected component

We begin with the important

question

of coincidence

of a pseudocompact

of the quasi-component

group and the

connected component in a pseudocompact group (see [88] for coincidence of the quasicomponent and the connected component of the zero in a regular semigroup and [85] for an example of a hereditarily disconnected non-totally-disconnected subgroup of R*). The next theorem shows that the quasi-component and the connected component coincide for countably

compact groups.

Theorem 1.4.3. Let G be a hereditarily pseudocompact G/c(G) is zero-dimensional.

group. Then q(G) = c(G) and

The proof of the equality q(G) = c(G) is b ase d on Lemma follows from Theorem 1.2.1. Let P be a class of topological

1.2.4, the last assertion

groups closed under taking quotient

groups. Then in

P the implication (1) from (D) is equivalent to the coincidence of the quasi-component and the connected component for every group in P. On the other hand, if (2) from (D) and (Y,) hold in P, then dimG Theorem Corollary

= dimq(G)

for every group G E P. In particular,

1.4.3 implies the following 1.4.4. For a hereditarily pseudocompact

group G dim G = dim c(G).

Towards a characterization of the connected component Comfort and van Mill [16] proved in 1989 the following:

of a pseudocompact

group

Theorem 1.4.5. For precompact connected Abelian group L such that L is torsion-free there exists a pseudocompact Abelian group H with c(H) = L. Since every precompact group can be presented as a closed subgroup of a pseudocompact group ([ 151, see also [87] for local precompactness and local pseudocompactness respectively), this result was not completely satisfactory. An application of Theorem 1.4.5 permits to remove the restraint from Theorem 1.45 and shows that the quasi- (respectively

D. Dikranjan / Topology and its Applications 84 (I 998) 227-252

238

connected) necessary

component

of a pseudocompact

condition-precompactness

Corollary

Abelian group is subject to the only obvious

(respectively

1.4.6 [25]. Let L be a precompact

exists a pseudocompact

precompactness

(connected)

Abelian

group H such that q(H) ” L (respectively

Here H can be chosen with any uncountable chosen a direct factor of H. In the non-Abelian case the following some necessary condition

group. Then there c(H) = q(H) g L).

weight 2 W(L) and L 2 q(H)

result announced

on the quasi-component

and connectedness):

in [26, Proposition

can be

91 provides

.of a finite-dimensional

pseudocompact

of a finite-dimensional

pseudocompact

group. Proposition

1.4.7. If L is the quasi-component

group, then L is a precompact completion

metrizable

group which is a normal subgroup

of its

Z.

This result provides a lot of examples of precompact groups which cannot be the quasi-component of a finite-dimensional pseudocompact group (e.g., any dense proper subgroup of a compact simple Lie group). The proof of the proposition goes through the approximation Lemma 1.2.3 and the following consequence of Theorem I .3.1 which characterizes the connected component of splitting pseudocompact

groups.

Lemma 1.4.8. Let L be a connected precompact group. Then L is the connected component of a splitting pseudocompact group iff the normalizer of L in i is Gs-dense in L. This leaves open the following

question in the general case:

Problem 1.4.9. Describe the connected docompact

components

and the quasi-components

of pseu-

groups.

1.4.3. The Ulm q-length of hereditarily

disconnected,

pseudocompact

groups

To measure the gap between the connected component and the quasi-component consider the Ulm q-length X, defined in Section 1.I. Clearly X,(G) = 1 for a group G iff c(G) = q(G), so that X,(G) can be considered as a measure of the gap between c(G) and q(G). Theorem 1.4.10. For every ordinal LY > 0 there exists a pseudocompact disconnected Abelian group H, with X,(G) = cr.

hereditarily

Proof (Sketch). We proceed by transfinite induction. For o = 0 take any nontrivial compact totally disconnected Abelian group Ho. For Q > 0 assume all Hp for ,!? < o are

D. Dikranjan / Topology and its Applications 84 (1998) 227-252

defined. If o is limit set H, Abelian

group H,

This theorem hereditarily

= HP<,

with Q(H,)

provides

disconnected,

of pseudocompact

Hp, if LY= p + 1, there exists a pseudocompact

% Hp by Corollary

again a negative but #(Ha)

groups hereditary

239

answer

1.4.6. This works. to Question

0

A since every H,

is

#

{ 1) f or a > ,/3 3 1; i.e., within the class disconnectedness and total disconnectedness are as

distant as possible. I .4.4. Quotients of totally disconnected

compact-like

groups

Connectedness is preserved under formation of quotients, but this fails for total disconnectedness in general. The first example to this effect can be found in [8, Chapter 3, p. 21, Exercise

211, where R is presented

as a quotient

of a totally disconnected

group. Later Kaplan [94] showed that lR is even quotient of a zero-dimensional metrizable group. Arkhangel’skii [2] showed that every topological group with countable base is a quotient of a zero-dimensional group with countable base. He showed later [3,4] that every topological group G is a quotient of a zero-dimensional hereditarily paracompact topological group H, but w(H) > w(G) in general. Answering Arkhangel’skii’s question [4, Problem 121 TkaCenko [83, Theorem 1.11 proved that every topological group G is a quotient

of a zero-dimensional

topological

group H with ,w(G) = w(H).

On the other hand, for every zero-dimensional locally compact group G and closed subgroup N of G the quotient group G/N is zero-dimensional. This was extended to pseudocompact groups by Tkacenko [80, Corollary 31. One can consider the problem of (non)preservation of zero-dimensionality under quotients in other classes of compactlike topological groups. Following Shakhmatov [71] call a class P of topological groups ind-representable if every G E P is a quotient of a zero-dimensional group from P of the same weight. According

to Tkacenko’s

result the class of all topological

ind-representable. To the list of ind-representable classes announced Theorem 7.21 we add here three new classes (f)-(h): (a) precompact (Abelian) groups [69, Corollary 4.41; (b) metrizable (c) w-bounded (d) (Abelian)

(Abelian)

groups (Bel’nov

groups is

by Shakhmatov

[7 1,

[6] and Coban [23]);

groups ’ [23]; Lindelof

C-groups’

[73];

(e) a-compact (Abelian) groups [73]; (f) totally minimal Abelian groups [29]; (g) torsion-free totally minimal Abelian groups [29]; (h) torsion-free precompact Abelian groups [29]. A simpler proof of the ind-representability of the class of Abelian precompact groups without any recourse to free topological groups is given in [29, Theorem 2.91, while the

’ G is w-bounded if G can be covered by countably many translates of any open nonempty set U & G. 2 A space is a Lindeliif Z-space if it belongs to the smallest class of spaces that contains both compact spaces and metrizable spaces and is closed under taking countable products, closed subspaces and continuous images

WI.

D. Dikranjan / Topology and its Applications 84 (1998) 227-252

240

proof in [70] makes essential use of the free precompact the factorization

technique

The preservation

developed

Abelian

of zero-dimensionality

group and

under quotients in the class !J3 of pseudocom-

pact groups yields that ?J?is not ind-representable.

Nevertheless,

from Section 1.3 gives the following

concerning

nounced in [26, Theorem

topological

by TkaEenko [82,83].

counterpart

151 in the case of connected

the “extension

technique”

total disconnectedness

(an-

group K):

Theorem 1.4.11. Every pseudocompact group K is a quotient of a totally disconnected pseudocompact group H of the same dimension, with w(H) = max{ w(K), WI}. Proof. Take L = 1 in Theorem 3.1. Then the group G will be totally disconnected.

0

This is the best possible result since taking quotients does not increase dimension the class of pseudocompact groups. On the other hand, by Theorem 1.2.1, quotients

in of

hereditarily disconnected hereditarily pseudocompact (in particular, groups are totally disconnected, even zero-dimensional. 1.4.5. Transfinite inductive dimensions

in topological

countably

compact)

groups

Shakhmatov [72] showed that countable compactness is strongly related to the transfinite inductive dimensions of a topological group: a normal homogeneous space X for which the large transfinite inductive dimension trIndX is defined and infinite must be countably compact [72, Theorem for topological groups: Theorem

1.4.12. ZftrInd G is definedfor

This is Theorem

w yields

exists and coincides

Theorem1.4.12 Question

the following

more precise result holds

a normal topological group G, then ind G < w.

4 (plus Note added in proof) of [72], where an application

Corollary 1.4.4 is given. Obviously, indG < trindG

31. Actually,

that

with indG.

the

small

The following

transfinite

inductive

general question

of our

dimension motivated

by

was studied in [72]:

1.4.13. When the existence of the small transfinite inductive

for a compact-like

group G implies indG

dimension

trind G

< w?

This holds true in the following cases: (a) [72, Theorem 61 for G locally compact; (b) [72, Corollary 91 for G connected and locally countably compact; (c) [72, Theorem lo] for G totally minimal and locally countably compact; (d) [72, Theorem 1 l] for G a locally Lindelof C-group. The proof of (b) is based on the following weaker version which is of independent interest: for a locally pseudocompact group G the existence of the small transfinite inductive dimension trind G implies that c( 2) is metrizable [72, Theorem 71. By a suitable modification of the proof given in [72] for (c) (e.g., with a recourse to Lemma 1.2.4

241

D. Dikranjan / Topology and its Applications 84 (1998) 227-252

instead of total minimality), one can replace “totally minimal and locally countably compact” in (c) by “countably compact”. However, it remains unclear whether only “locally countably

compact” can be left in (b) and (c) [72, Question

171.

2. Compactness under the looking glass of (dis)connectedness 2.1. Rigidity of the algebraic structure of a compact-like

group

The interplay between the topology and the algebraic structure of a topological group has two aspects: the impact of the algebraic structure on the topological one and viceversa. Here we discuss the problem: groups are homeomorphic.

Obviously,

when two topological

groups isomorphic

as abstract

one can just take one group with two group topolo-

gies and look at how much they differ. In the case of compact group topologies, the connectedness of the topology is completely determined by the algebraic structure of the group : the group is connected iff it is divisible [57, Corollary 21. Consequently, a compact group is totally disconnected iff it is reduced. Moreover, every divisible pseudocompact group is connected [91], but the converse

need not be true [91, Example

11. For example,

the group Z(2ti) is reduced,

but admits a connected and locally connected pseudocompact group topology [37, Theorem 5.101. Actually, this group has a stronger property due to the following example of Tkacenko [81]:

Example 2.1.1. Under CH the free Abelian group Z(2w) of rank 2” admits a connected and locally connected

countably

compact group topology.

Quite recently Tomita [84] found a weaker assumption, namely MA (a-centered), which gives the same result (it should be noted however, that TkaEenko’s example was in addition hereditarily separable and hereditarily normal). whether such an example can be produced in ZFC.

It is still an open question

Following [ 17, Remark 3.131, we say that a topological group G is a van der Waerden group if G is a compact group on which every homomorphism to a compact group is continuous.

Clearly, the class W of van der Waerden

groups is closed under taking

Hausdorff quotients and finite products and contains all finite groups. It is a theorem of van der Waerden [89], that W contains all compact, connected, semisimple Lie groups (for the special case G = SO(3,lR) see also [19]). A compact group G E W if and only if its topology is the only (hence, the finest) precompact group topology on G (cf. [ 11, p. 1951). Let W* denote the class of precompact groups with this property. By [7, Example 7.13, Remark 7.141 or [17, Remark 3.161 W is properly contained in W*. Those examples show also that W* contains infinite zero-dimensional groups, while our knowledge of W does not permit to decide whether the quotient G/c(G) is always finite for G E W. An infinite van der Waerden group G cannot be Abelian since the weight of any compact group topology on G is always less than IGI, while the finest precompact group

242

D. Dikranjan / Topology and its Applications

84 (1998) 227-252

topology on G has weight 21Gl when G is Abelian. Consequently, the quotient

G/G’

is finite, where G’ is the derived subgroup

for every group G E W of G. This motivates

the

following Definition

2.1.2. A compact Abelian group G is an Orsatti group if G admits a unique

compact group topology. It is well known that the group Z, of p-adic integers is an Orsatti group (see Hulanicki [51, Theorem

8.101 and Orsatti [61,62]). It can be shown similarly that this is true for x FP), where IcP > 0 and is FP a finite Abelian p-group (see every product I&,,@~ [32, Exercise 3.8.161. It turns out that these are the only Orsatti groups: Theorem 2.1.3. Let G be an Orsatti group. Then for every prime p there exist an integer k, 3 0 and a jnite Abelian p-group FP such that G S n,,,(Z? x FP) as topological groups. In particular

G is totally disconnected.

The first step of the proof is establishing

that an Orsatti group G is necessarily

totally

disconnected, i.e., reduced. Then the results from [62] can be applied to conclude that the quasi-p-torsion part of G has the form Bgp x I@, where BP is a compact Abelian p-group. Finally note that if the group BP is infinite, then it has a discontinuous automorphism which could produce a new compact topology on G. The same argument applied to Z$’ shows that /?, must be finite. Detailed proofs and more about W will appear in [301. It is natural to ask about the compact Abelian groups which admit a unique countably compact group topology. Clearly, such groups are Orsatti groups, hence the question is: does an infinite Orsatti group admit a different countably

compact group topology?

In

particular: Question

2.1.4 [40]. Does Z, admit a countably

p-adic one? How many nonhomeomorphic Obviously,

if a topology as in Question

compact

group topology

beyond

the

ones? 2.1.4 exists, then it should be nonmetrizable.

The next theorem shows that such a topology cannot be zero-dimensional. A group containing dense countably compact subgroup is strongly pseudocompact. Obviously, strongly pseudocompact

groups are pseudocompact.

Theorem 2.1.5. Let G be an infinite Orsatti group. Then: (a) G admits no pseudocompact zero-dimensional group topology beyond its natural compact topology; (b) for every cardinal o satisfying w < o 6 22w there exists a connected pseudocompact group topology of weight u on G which may be chosen also to have or not the property “locally connected”; (c) under the assumption of CH G admits a strongly pseudocompact connected and locally connected group topology.

243

D. Dikranjan / Topology and its Applications 84 (1998) 227-252

Proof (Sketch). (a) Follows from the fact that a pseudocompact topology

on G is must be precompact

Corollary

7.61.

zero-dimensional

group

and linear. The proof of (b) is based on [37,

(c) Both in [81] and [84] the topologies on Z(2d) as in Example 2.1.1 have the property that the completion GcTzw.

of Zc2”) is T2W. This permits to embed G into T2W so that Zc2”) C

•I

Theorem 2.1.5(b) shows that the counterpart of Question 2.1.4 concerning pseudocompact topologies has a strongly positive answer. Note that Theorem 2.1.5(c) works under weaker hypothesis,

namely: the assumption of MA (o-centered) instead of CH and 2” < r(G) 6 IGI 6 22d instead of asking G to be an Orsatti group. We do not know whether Theorem 2.1.5(c) works in ZFC. This will depend on the answer to the following Question in ZFC?

2.1.6. Does Z (2w) admit a strongly pseudocompact

2.2. When homeomorphic

compact-like

connected

group topology

groups are isomorphic?

In contrast with Section 2.1 here we ask to what extent the topology of a (compactlike) topological group determines the algebraic structure of the group. For nonconnected groups such a relation may totally fail, for example every compact metrizable totally disconnected group is homeomorphic to (0, I}“, hence this homogeneous topological space admits many (totally different from algebraic point of view) group structures which make it a topological group. Connectedness

helps to get a positive result in this direction:

two compact connected

simple Lie groups are isomorphic if they are homeomorphic (or even if they have the same homotopy type) [5, Theorem 9.31. Here “simple” cannot be omitted: there exist two semisimple

compact connected

Lie groups which are homeomorphic

[5, Theorem 9.41. In the Abelian case we have the following

well-known

but not isomorphic

general result of Scheinberg

[74] (see also [50]): Theorem

2.2.1. Two connected

groups whenever

compact Abelian groups are isomorphic

they are homotopy-equivalent

(in particulal;

as topological

homeomorphic).

Proof (Sketch). If G is a compact connected Abelian group, then the Pontryagin dual G is canonically isomorphic to the cohomology group H’ (G, Z), thus G is isomorphic to the dual of a discrete group which depends only on the topology of the space G. 0 For topological groups G and H set G N H if H embeds homeomorphically into G and G embeds homeomorphically into H. The relation N, unlike homeomorphism, gives very weak impact on the group-structure. Indeed, Cleary and Morris [9, Theorem 2.61 proved the following

244

D. Dikranjan

/ Topology and its Applications

84 (1998)

Theorem 2.2.2. Let G and H be compact groups with w(H) G is connected.

Then H embeds homeomorphically

This theorem implies that for compact connected

227-252

< w(G),

w(G)

> w and

into G. nonmetrizable

groups G and H one

has G N H iff G and H have the same weight. Scheinberg’s

theorem can be extended

to some noncompact

groups. For example,

a

similar assertion was proved to hold true for the special case of the subgroup of metrizable elements

(see Example

for precompact

1.1.5(b)) of compact Abelian

M-complete

and M-connected

groups in [33, Theorem

3.11 (i.e.,

groups G). A slight modification

of that

proof gives the following extension of Scheinberg’s theorem to for pseudocompact G which are functorial subgroups of their completion G [41].

groups

Theorem 2.2.3. Let r be an idempotent preradical in the category of precompact Abelian groups. Then any two connected, groups are isomorphic

r-connected,

as topological

pseudocompact

and r-complete

Abelian

groups whenever they are homeomorphic.

Call a precompact group G rigid if every topological automorphism j of G satisfies j(G) C G. Note that the groups described in the above theorem are rigid. Question

2.2.4. Is it possible

to extend

Scheinberg’s

result to rigid pseudocompact

Abelian groups? Let us note that rigidity cannot be omitted in Question 2.2.4. In fact, we give here a strongly negative answer to the rigidity-free version of Question 2.2.4 within much smaller classes of compact-like groups (in particular, countably showing that even isomorphism as abstract groups is not available

compact groups) by in this case. Let CKbe

an infinite cardinal. A topological space X is initially a-compact, if every open cover of X with ,< (Yelements has a finite subcover. Initially w-compact spaces are the countably compact spaces. Clearly, a space is compact if and only if it is initially

a-compact

for

every infinite cardinal Q. Theorem 2.2.5 [40]. For each a 3 UJ there exists two connected initially a-compact Abelian groups which are homeomorphic as topological spaces but not isomorphic even as abstract groups. Proof (Sketch). Consider the subgroups K = T*. With G = K”’ define GA

= {f E G:

J{PE

a+:

f(P)$

A = ((i, i)) and B = (( 1, -l),

A}/

(-1,1))

of


Every subset of < a elements of G,J is covered by a compact subset of GA, so that G-4 is initially a-compact and hence pseudocompact. Now the connectedness of G = GA yields GA is connected. Define analogously Gg . Then GA and Gg are homeomorphic (so that Gg is connected and initially cr-compact). Finally, every algebraic homomorphism 0 f : Gg --f GA satisfies lcoker fi = IGAI, so that in particular f cannot be surjective.

245

D. Dikranjan / Topology and its Applications 84 (1998) 227-252

None of the groups constructed to countably

above is rigid, so that one can extend Question

2.2.4

compact groups as well.

2.3. Categorical

compactness

The following

versus completeness

theorem of Kuratowski

and minimal@

and Mrowka is well known [56].

Theorem

2.3.1. A topological space X is compact if and only iffor every space Y the

projection

p : X x Y -+ H is a closed map.

This suggests the following: Definition

2.3.2. A topological

group G is Kuratowski compact (briefly, K-compact)

for each topological group H the projection G x H to closed subgroups of H. In the category of linearly topologized

G x H +

H sends closed subgroups

modules 3 the Kuratowski

if of

compact modules (de-

fined similarly) are precisely the linearly compact ones ([39], a linearly topologized module is linearly compact if every filter-base of cosets of closed submodules has nonempty intersection). Definition 2.3.2 is a particular case of categorical compactness with respect to an appropriate notion of “closedness” defined analogously categories by Manes [95]. Categorical compactness is extensively

in the case of general studied in topological

spaces, groups and rings, as well as in abstract categories (see [10,39,41,46-48] and the bibliography there, in most of these papers “closedness” is defined by means of closure operators in the sense of [38]). Obviously, compact groups are K-compact. To establish the converse turned out to be quite hard [41]. Even for Abelian groups one has to make recourse to the deep theorem of precompactness

of Prodanov

and Stoyanov

[65]. More precisely, the following

is known

at present: Theorem

2.3.3 [4 11.

(a) Soluble K-compact (b) K-compactness

groups are compact.

is preserved

under taking continuous homomorphic

images, prod-

ucts and closed subgroups. (c) Continuous

(d) Separable

homomorphic K-compact

images of K-compact

groups are complete.

groups are complete and totally minimal.

A proof of the preservation of K-compactness under taking products was obtained recently by Clementino and Tholen [lo] in the general setting of categorical compactness. It is not clear if one can remove separability in item (d) of the above theorem. In view of (b), it suffices to consider only minimality: 3 Topological

modules having a local base at 0 consisting

of open submodules.

246

Question

D. Dikranjan / Topology and its Applications

2.3.4 [41]. Is every K-compact

84 (1998) 227-252

group minimal?

Now we shall discuss when the inverse of Theorem compact group. We shall be interested local compactness Theorem

K-compactness

2.3.5 [41]. Evep

also to determine

yields compactness.

2.3.3(d) holds true for a locally when under the hypothesis Imposing

connected locally compact K-compact

connectedness

of

gives:

group is compact.

This theorem shows that connectedness cannot help to prove that separable locally compact, totally minimal groups are K-compact (compare with Theorem 2.3.3(d)). In fact, the group SL2(LR) is separable, connected, locally compact, and totally minimal [66], but not K-compact according to Theorem 2.3.5. On the other hand, in the extreme case of disconnectedness we get: Theorem 2.3.6 [41]. A countable subgroup of G is totally minimal. Question

discrete group G is K-compact

2.3.7 [41]. Is every discrete K-compact

if and only if every

group finite?

In 1938 Schmidt asked if there exist quasi-jinite groups, i.e., infinite non-Abelian groups with all proper subgroups finite. Answering this question Ol’shanskii [60, Theorem 28. l] showed that there exist simple quasi-finite groups. Note that by Theorem 2.3.6 a discrete simple quasi-finite group is K-compact iff it is minimal, so that a possible way for a negative answer to Question 2.3.7 is to find a minimal discrete quasi-finite simple group. From the known examples of minimal discrete groups, the one given by Ol’shanskii [59] is neither simple nor quasi-finite and it is not known whether it is totally minimal. The group defined by Shelah [74] is not K-compact since it has nontorsion elements

(see Theorem 2.3.3 (a) and (b)).

2.4. Countable compactness

and minima&

versus compactness

Now we discuss the following: Question

2.4.1. Does countable

compactness

plus minimality

imply compactness?

It is natural to begin with the easier question: does total minimality plus countable compactness imply compactness? The first counterexample to this effect was given by Comfort and Grant [ 131. The group they proposed was zero-dimensional. A connected counterexample was given later by Shakhmatov and the author in [35]. (More precisely, they proved that a connected compact group G admits a proper dense countably compact (necessarily connected) totally minimal subgroup iff the center of G is not a Ggsubset of G.) In both cases the counterexamples were non-Abelian. In fact, Comfort and Soundararajan [21] proved that for Abelian groups total minimality plus countable compactness imply compactness in the zero-dimensional case. Later Dikranjan and

241

D. Dikranjan / Topology and its Applications 84 (1998) 227-252

Shakhmatov

[35] extended

the impression

this result to all Abelian

that (dis)connectedness

it turns out that in the last result “totally case the group is also connected,

topological

has nothing

groups. This may leave

to do with this problem.

minimal”

can be weakened

However,

to “minimal”

in

i.e., Question 2.4.1 has positive answer for connected

Abelian groups. As often happens in general topology, the last word goes to set theory: the result fails for large groups, where large means here “of Ulam-measurable

cardinal-

ity”. (A cardinal

complete

ultrafilter.)

Q: is Ulam-measurable

Let us note that the set-theoretic

weak since the nonexistence Theorem

if Q admits a nonprincipal restraint

of Ulam-measurable

2.4.2. If a connected,

in assertions

cardinals

countably 2.4.2-2.4.4

is equi-consistent

is rather with ZFC.

countably compact, minimal Abelian group is not large,

then it is compact. The proof of this theorem, based on properties of the subgroup of quasi-p-torsion elements of compact Abelian groups (see Example 1.1.5(a)), will appear in [28]. More details and comments (in particular, examples of noncompact large connected, countably compact, minimal Corollary component

Abelian

groups) can be found in [28,31].

2.4.3. Let G be a countably compact, minimal Abelian group. If the connected c(G) is not large, then c(G) is compact.

Proof. Let C = c(G). By Theorem

1.1.3 and Lemma

1.2.4 c(G) = C n G and it is

dense in C. Therefore, c(G) is a dense countably compact minimal subgroup of the connected compact group C (closed subgroup of a minimal Abelian group is minimal [32, Proposition

25.71). By the above theorem c(G) is compact.

In terms of Section

1.I .l we have proved

Abelian group having non-large

connected

that every minimal

component

is essential here due to the existence of noncompact compact group in the non-Abelian

0

is c-complete.

connected

case as mentioned

countably

compact

Note that “Abelian”

(totally) minimal countably

above.

The next theorem reduces, in case the connected component is not large, the study of minimal countably compact Abelian groups to those which are totally disconnected. Theorem component minimal.

2.4.4. Let G be a countably

compact Abelian group such that the connected

c(G) is not large. Then G is minimal iff c(G) is compact

and G/c(G)

is

Proof. Assume G is minimal. Then by Corollary 2.4.3 c(G) is compact. By Lemma 1.2.4 c(G) = G n c(G) is dense in c(G), hence by [32, Exercise 45.151 G/c(G) is minimal. Now assume [42]. 0

that c(G)

is compact

and G/c(G)

is minimal.

Then G is minimal

by

The next theorem shows that in analogy with the compact case, every countably compact minimal totally disconnected Abelian group G decomposes into a direct product

248

D. Dikranjan / Topology and its Applications 84 (1998) 227-252

of its quasi p-torsion

subgroups

compact totally disconnected

G,

= td,(G).

Thus the study of minimal

countably

Abelian groups is reduced to that of quasi-p-torsion

ones.

Theorem 2.4.5 [28]. Let G be a countably compact totally disconnected Abelian group. Then for each p E P there exists a compact subgroup

of the Cartesian product

&,-module

BP such that G is a dense

B = n BP. For each p E P the quasi p-torsion

subgroup G, = G fl BP of G coincides with the projection

of G into BP and G coincides

with nPEP G,. Moreover; G is minimal iff each G, is minimal. Quasi-p-torsion

countably

compact minimal

Abelian

groups exist in profusion

(they

are necessarily totally disconnected). Noncompact examples of such groups were given by Shakhmatov and the author in [36, Corollary 1.61. Under the assumption of MAcountable (a weak version of Martin’s Axiom equivalent to the condition that the real line is not the union of fewer than 2w-many nowhere dense subsets), Hart and van Mill [93] found a countably compact group H such that H* is not countably compact. Tomita [84] constructed under M&ountable a countably compact group HT such that H$ is countably compact and H+ is not countably compact. By an appropriate

modification

of this group one can get the following:

Theorem 2.46 [28]. Under the assumptions of MA, ,,u,&le there exists a minimal, countably compact, zero-dimensional Abelian group H, such that H’ is countably compact, H” is minimal for every c~ and H3 is not countably compact. Note that on one hand H is “close to being compact”,

since H has very strong

minimality properties (see [79,34,36] for the failure of the powers of minimal groups to be minimal). On the other hand, H3 fails to be countably compact.

Acknowledgments It is a pleasure to thank my coauthors A. Berarducci, D. Shakhmatov,

L. Stoyanov,

W. Tholen,

M. Forti, A. Orsatti, N. Rodinb,

A. Tonolo, V. Uspenskij

and S. Watson. My

joint work with them, especially that with D. Shakhmatov and V. Uspenskij, form the backbone of this survey. Thanks are due also to W. Comfort, M. TkaEenko and A. Tomita for helpful discussions or information, and to Professor Krzysztof Ciesielski from the University of West Virginia for his hospitality in October 1995, while the final work on this paper was carried out.

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