1. a pair of countable Fr´echet topological groups whose product is sequential but is not Fr´echet,

151 (1996)

Examples of sequential topological groups under the continuum hypothesis

by

Alexander S h i b a k o v (Auburn, Ala.)

Abstract. Using CH we construct examples of sequential topological groups:

1. a pair of countable Fr´echet topological groups whose product is sequential but is not Fr´echet,

2. a countable Fr´echet and α

topological group which contains no copy of the ratio- nals.

1. Introduction. The classical methods of study of continuity involve consideration of convergent sequences and their images. Although the conti- nuity as it is understood in modern topology cannot be treated only in terms of classical convergent sequences there is a field of topology and the corre- sponding subclass of topological spaces where classical convergence plays an important rˆole. Like general topology itself the field has its origin in metric space theory.

The first natural generalization of metrizability is first-countability. Go- ing further in generalization one can emphasize the following property of the closure operator in a first-countable space: x ∈ A implies existence of a sequence in A converging to x. Spaces having this property are called Fr´echet spaces. The next step is to require only that convergent sequences determine the topology of the space. The corresponding definition is: a space X is sequential if for every A ⊆ X such that A 6= A there exists a sequence in A converging to a point outside A.

Sequentiality and its behaviour in several situations were studied by a number of authors (see [F], [No1], [No2], [NT], [AF]). In the course of their investigation some new convergence properties were introduced and some problems were stressed. Among those problems is the study of products of Fr´echet and, more generally, sequential spaces. It is relatively easy to de-

1991 Mathematics Subject Classification: 54D55, 54A20.

Key words and phrases: topological group, sequential space, Fr´echet space.

[107]

stroy Fr´echetness by product operation (see [vD], [GMT]) so one has to put some restrictions on the factors to see subtle phenomena. The most popular and useful restrictions are compactness and the property to be a topologi- cal group. Several techniques were developed to study products with some or all factors being compact (see [A1], [O], [M2]). It was shown that the product of two Fr´echet compact spaces may be non-Fr´echet (see [Si], [BR], [MS]). Properties α

were introduced to obtain theorems about preservation of Fr´echetness in products (see [A1], [A2]). Since then α

-properties have found several applications in the theory of sequential spaces ([No1], [No2], [N], [NT]). In particular, for topological groups P. Nyikos proved in [N] that a sequential topological group is Fr´echet if and only if it is α

. D. Shakhmatov showed ([Sm]) that one can say no more about Fr´echet topological groups: in a model of ZFC obtained by adding uncountably many Cohen reals there ex- ists a Fr´echet non-α

topological group. As α

-properties play an important rˆole in the study of the product operation and preservation of Fr´echetness, the products of Fr´echet topological groups are also of interest. S. Todorˇcevi´c in [T] constructed (in ZFC) a pair of σ-compact Fr´echet topological groups whose product is non-Fr´echet (even of uncountable tightness).

In Section 2 we construct using CH a pair of countable Fr´echet topolog- ical groups whose product is sequential but not Fr´echet. The sequentiality of the product imposes some restrictions on the factors. For example, at least one group of the pair is a non-α

-space. Indeed, if both groups were α

-spaces then so would be their product by [No1, Theorem 2.2] but being sequential it would be Fr´echet by the result of P. Nyikos cited above.

The technique by which that example was obtained is applied to the construction of a countable Fr´echet and α

topological group containing no copy of the rationals in the conclusion of Section 2. Being necessarily a non-first-countable space, it cannot be obtained without extra set-theoretic assumptions (see [DS]). In fact, slight modification of the technique permits obtaining a topological field with such properties.

The topology of each group is constructed by induction. At each step a pair of topologies is considered and the finer topology is coarsened by adding a new convergent sequence from the usual topology of Q while the coarser one is refined so that the resulting new pair of topologies remains comparable and stays between the usual topology of Q and the discrete topology. The construction is arranged so that the processes of coarsening and refining come together in a single topology. The properties of the topology thus constructed are obtained by considering an appropriate pair of topologies involved in the inductive procedure which “approximate” the topology from above and below.

Let us recall the terminology used in the study of sequential spaces. A

family S = {S

| i ∈ ω} of sequences converging to a common point x ∈ X

is called a sheaf , the point x is called the vertex of the sheaf (see [A1]). A space X is called an α

-space (or X ∈ h1i in the notation of [A1]) if for every sheaf there is a sequence S converging to its vertex such that S

\ S is finite for all i ∈ ω. X is called an α

-space if for every sheaf there is a sequence converging to its vertex which meets infinitely many sequences of the sheaf.

A quotient image of a topological sum of countably many compact spaces is called a k

-space. A product of two k

-spaces is itself a k

-space (see [M2]).

Put ω(n) = {k | k > n} ⊆ ω. A set σ ⊆ ω

will be called thin (resp.

small) if for every n ∈ ω the set σ ∩ {n} × ω is finite (resp. σ ∩ ω(n) × ω is thin for some n ∈ ω). Let σ be a small set and n = min{k | σ ∩ ω(k) × ω is thin}. Then σ ∩ ω(n) × ω = ess(σ) will be called the essential part of σ.

Consider the set S = ω

∪ ω ∪ {ω}. Define a topology on S as follows.

Every point of ω

is isolated, a typical neighborhood of n is {n} ∪ ({n} × ω\

finitely many points), U 3 {ω} is open if (U ∩{n}×ω)∪{n} is a neighborhood of n for every n ∈ U and ω \ U is finite. The resulting space is called Arens’

space S

. Another canonical space S

is obtained from countably many convergent sequences by identifying their limit points.

Let Q be the set of rationals. Let K = {K

}

be an arbitrary family of subsets of Q. Suppose ~a ∈ Q

and ~ K ∈ K

where n ∈ ω \ {0}. Set h~a, ~ Ki = h(a

, . . . , a

), (K

, . . . , K

)i = a

· K

+ . . . + a

· K

⊆ Q, where a

∈ Q. Define Q

= S

n∈ω

Q

and Q

= {0}. If K ⊆ Q and

~a ∈ Q

we set ~ahKi = a

· K + . . . + a

· K. If ~a ∈ Q

then ~ahKi = 0.

Let Q = {b

| i ∈ ω} with b

6= b

for i 6= j, Q(i) = {b

| j ≤ i} and Q

= S

i,j≤k

(Q(i))

. If a ∈ Q \ {0} let n

(a) = n provided a = b

, and n

(0) = ∞ > k for any k ∈ ω. All spaces are assumed to be Hausdorff.

The following simple lemma was proved in [Sh, Lemma 1.1].

Lemma 1.1. A countable nondiscrete sequential topological group con- tains a closed copy of S

provided the group is a k

-space.

2. Examples. Lemmas 2.1–2.6 were proved in [Sh, Lemmas 2.1–2.6].

Lemma 2.1. Let K = {K

}

be an arbitrary family of subsets of Q.

Then there exists a countable family C(K) ⊇ K such that:

(1) {a} ∈ C(K) for all a ∈ Q,

(2) if ~a ∈ Q

and ~ K ∈ C(K)

then h~a, ~ Ki ∈ C(K), (3) if K

∈ C(K), . . . , K

∈ C(K) then S

i≤n

K

∈ C(K), (4) if K ⊆ K

and K

has properties (1)–(3) then C(K) ⊆ K

.

Lemma 2.2. If K = {K

}

is a family of compact subsets of Q then so is C(K).

Lemma 2.3. Let K = {K

}

be an arbitrary family of compact subsets

of Q. Introduce a new topology on Q by declaring U ⊆ Q to be open if and

only if U ∩ F is relatively open for every F ∈ C(K). Denote Q with this topology as G(K). Then:

(5) if ~a ∈ Q

then the mapping p : G(K)

→ G(K), p(~b) = h~a,~bi, is continuous,

(6) G(K) is a k

-space.

Lemma 2.4. If K = S

β<α

K

and K

⊆ K

for β ≤ β

then C(K) = S

β<α

C(K

).

Lemma 2.5. For every family K = {K

}

of compact subsets of Q and every family U = {U

}

of open subsets of G(K) one can fix a topology τ (U, K) on Q such that:

(a) the mapping p : Q

→ Q where p(~a) = h~b,~ai, ~b ∈ Q

, is continuous in τ (U, K),

(b) τ (U, K) is a Hausdorff group topology with a countable base, (c) U

∈ τ (U, K) for all i ∈ ω,

(d) τ (U, K) is stronger than the usual topology of Q and weaker than the topology of G(K), and

(e) if U ⊇ τ

(U

, K

) then τ (U, K) is stronger than τ (U

, K

) where K and K

are countable families of compact subsets of Q and τ

(U, K) is a fixed countable base at 0 ∈ Q in τ (U, K).

Lemma 2.6. C(K ∪ {K}) = { S

i≤k

(~a

hKi + K

) | ~a

∈ Q

, K

∈ C(K), k ∈ ω}.

We need the following technical definition. Let t : ω

→ Q be an injec- tion. We shall call t a correct table in G(K) if the following properties hold with S

= t(ω

) and S

= t({n} × ω):

1(t) t(n, k) → s

as k → ∞ in G(K), 2(t) s

→ 0 as n → ∞ in G(K),

3(t) S

∪ {s

} ⊆ K

and {s

| n ∈ ω} ∪ {0} ⊆ B

where K

∈ C(K) and B

∈ C(K).

Lemma 2.7. Let t and u be correct tables, U = {U

}

be a family of open subsets of G(K), and K = {K

}

be a family of compact subsets of Q. Suppose that for any F ∈ C(K) the set t

(F ) is small. Then there exists an infinite thin subset σ = {σ

| i ≥ 1} ⊆ ω

such that:

(7) u(σ

) → 0 as i → ∞ in τ (U, K) (see Lemma 2.5),

(8) for each ~a ∈ Q

and F ∈ C(K) the set t

(~ahu(σ) ∪ {0}i + F ) is

small.

P r o o f. Let C(K) = {F

}

. Suppose that for some 0 6= a ∈ Q, {b

, . . . , b

} ⊆ Q, k ∈ ω and ~a ∈ Q

the set

(9) u

 [

i≤k

K

∪ B



−  [

i≤k

F

+ ~ah{b

, . . . , b

}i



· a



is not small. Denote the part (. . .) · a

as F

. Then F

∈ C(K). It easily follows from (5) that F

is a compact subspace of G(K) and thus the topology on F

inherited from G(K) coincides with that inherited from Q. Thus F

is a metrizable compact subspace of G(K). Suppose u

(F

) is not small.

Then 1(u) and 2(u) imply that 0 ∈ F

∩ u(ω

). So there exists σ ⊆ ω

such that σ = {σ

| i ≥ 1} is infinite and thin, u(σ) ⊆ F

and u(σ

) → 0 as i → ∞ in G(K). Thus u(σ

) → 0 in τ (U, K) by (d). Now for any ~b ∈ Q

and any F ∈ C(K) we have t

(~bhu(σ) ∪ {0}i + F ) ⊆ t

(~bhF

i + F ) = t

(G) where G ∈ C(K). Thus t

(~bhu(σ) ∪ {0}i + F ) is small by the assumption of the lemma. So σ satisfies both (7) and (8). Thus we may assume without loss of generality that every set of the form (9) is small.

It follows easily from 1(u) and 2(u) that if U 3 0 is open in G(K) then ω

\ u

(U ) is small. Now choose σ

, k ≥ 1, by induction so that:

(10) u(σ

) 6∈ [

n_Q(a)≤k

~a∈Q_k

 [

i≤k

K

∪ B



−  [

i≤k

F

+ ~ahu({σ

| i < k}) ∪ {0}i



· a

,

(11) u(σ

) ∈ \

i≤k

U

, {U

}

= τ

(U, K) (see Lemma 2.5), (12) σ

∈ S

, n

> n

.

The preimage under u of the union on the right hand side of (10) is small by the assumption so using the remark preceding (10) it is easy to choose σ

satisfying (10)–(12). Now by (11), u(σ

) → 0 as k → ∞ so (7) holds.

Consider now the set R = ~ahu(σ) ∪ {0}i + F

where ~a ∈ Q

and n ∈ ω.

We have ~a = (a

, . . . , a

) for some k ∈ ω. So ~a ∈ Q

for some i(~a) ∈ ω. The set A = {h~a,~bi | ~b ∈ {0, 1}

} \ {0} is finite, so r = max{n

(a) | a ∈ A} < ∞.

Put M = max{i(~a), r, n}. Now

(13) R = a

· (u(σ) ∪ {0}) + . . . + a

· (u(σ) ∪ {0}) + F

. Define u(σ

) = p

for i ≥ 1 and p

= 0 and rewrite (13) as

R = [

(i1,...,ik)∈ω^k

a

· p

+ . . . + a

· p

+ F

.

We write i ∈

(i

, . . . , i

) if and only if P

iν=i

a

6= 0 or p

= 0. It is easy to see that if a

· p

+ . . . + a

· p

= b ∈ Q then there are {p

, . . . , p

} ⊆ {p

, . . . , p

}∪{p

} such that a

·p

+. . .+a

·p

= b and j

∈

(j

, . . . , j

) for all m ≤ k. A point (i

, . . . , i

) ∈ ω

is called essential if i

∈

(i

, . . . , i

) for all m ≤ k. Let Ω ⊆ ω

be the set of all the essential points. It is easy to check now, using the properties of essential points discussed above, that

R = [

(i1,...,ik)∈Ω

a

· p

+ . . . + a

· p

+ F

.

Set

L = [

(i₁,...,i_k)∈Ω\{i|i≤M }^k

a

· p

+ . . . + a

· p

+ F

Obviously

~ahu(σ) ∪ {0}i + F

= R = (~ahu({σ

| i ≤ M }) ∪ {0}i + F

) ∪ L.

Let us prove that S

∩ L is finite for m > M . Suppose there is m > M such that S

∩ L is infinite. So we have

(14) a

· p

+ . . . + a

· p

+ f

= t(m, n

)

where n

> n

, f

∈ F

and (i(1, l), . . . , i(k, l)) ∈ Ω \ {i | i ≤ M }

. Suppose that there are s, l ∈ ω such that i(s, l) > m > M . Without loss of generality assume that i(s, l) = max{i(s

, l) | s

≤ k}. Then substituting every occurrence of p

in (14) by p

= 0, leaving the occurrences of other p

untouched and thus obtaining p

we have

 X

i(ν,l)=i(s,l)

a



· p

= t(m, n

) − (f

+ a

· p

+ . . . + a

· p

)

where j(ν, l) < i(s, l) if ν ≤ k and P

i(ν,l)=i(s,l)

a

= a 6= 0 because (i(1, l), . . . , i(k, l)) ∈ Ω; moreover, a ∈ A and thus n

(a) ≤ r ≤ M <

m < i(s, l). It follows that p

∈  [

i≤i(s,l)

K

−  [

i≤i(s,l)

F

+ ~ahu({σ

| i < i(s, l)})i



· a

where n

(a) < i(s, l) and ~a ∈ Q

⊆ Q

, which contradicts (10).

Therefore

a

· p

+ . . . + a

· p

+ f

= t(m, n

)

where n

> n

, i(s, l) ≤ m and (i(1, l), . . . , i(k, l)) ∈ Ω \ {i | i ≤ M }

. The

set [

l∈ω

a

· p

+ . . . + a

· p

⊆ ~ahu({σ

| i ≤ m}) ∪ {0}i

is finite and thus the set F = [

l∈ω

h~a, (p

, . . . , p

)i + F

is compact in G(K). But F ∩ S

is infinite and thus by 1(t) and 3(t) there is a point

a

· p

+ . . . + a

· p

+ f = s

= b

∈ B

, f ∈ F

.

Let j = max{i(j

, l) | j

≤ k}. Note that since (i(1, l), . . . , i(k, l)) 6∈ {i | i ≤ M }

, it follows that j > M . Analogously to the consideration of the previous case we have

 X

i(ν,l)=j

a



· p

= b

− (f + a

· p

+ . . . + a

· p

)

and

p

∈



B

−  [

i≤j

F

+ ~ahu({σ

| i < j})i



· a

where a = P

i(ν,l)=j

a

, n

(a) < j and ~a ∈ Q

, which contradicts (10). Thus S

∩ L is finite for m > M , which implies that t

(L) is small.

Now N = ~ahu({σ

| i ≤ M }) ∪ {0}i + F

∈ C(K) and thus t

(N ) is small. Then R = N ∪ L and t

(N ∪ L) is small. Thus (8) also holds. The lemma is proved.

Let us consider an example of a group G(S). Define S

= {1 | n ∈ N}

∪ {0} and S = {S

}. Consider the topological group G(S). It is obviously nondiscrete and is a k

-space by (6). Then it contains a closed copy of S

by Lemma 1.1. So we can fix an injection t : ω

→ Q such that:

(f) t(n, k) → s

as k → ∞ in G(S), (g) s

→ 0 as n → ∞ in G(S),

(h) 0 6∈ S

= t(ω

) and 0 6= s

6= s

6∈ S

if n 6= k,

(i) if S

= t({n} × ω) then S

∪ {s

} ⊆ K

and {s

| n ∈ ω} ∪ {0} ⊆ B

where K

, B

∈ C(S),

(j) t(ω

) ∪ {s

| i ∈ ω} ∪ {0} is a closed subset of G(S) homeomorphic to S

.

Then properties (f)–(g) and (i) imply 1(t)–3(t) so t is a correct table in G(S). Property (j) implies that t

(F ) is small for all F ∈ C(S). In all further considerations t denotes the injection discussed above.

Assume CH. Let {O

}

be all subsets of Q, and {Z

}

be all subsets of Q

. We assume for convenience that O

= ∅, Z

= ∅ and Z

= {(t(n, k), t(n, k)) | n, k ∈ ω}∪{(s

, s

) | n ∈ ω}∪{(0, 0)}. Let ω

\0 = Λ

∪Λ

and Λ

∩ Λ

= ∅, with Λ

uncountable (ν ∈ {0, 1} here and further on). Let

{u

}

be the family of all the injections u

: ω

→ Q such that every u ∈ {u

}

repeats ω

times in {u

}

as well as in {u

}

.

Lemma 2.8 (CH). For every α < ω

there exist:

• countable families K

of compact subsets of Q,

• countable families U

of subsets of Q,

• compact subsets K

of Q, such that:

(15) K

= S

β<α

K

∪ {K

}, S

∈ K

,

(16) if α ∈ Λ

and u

is a correct table in G(K

) for some β < α then K

⊆ S

∪ {0} and u

(K

) is infinite and thin; otherwise K

= S

,

(17) K

is a nontrivial convergent sequence with limit point 0 in G(K

), (18) if U

∈ U

and β ≤ α then U

is open in G(K

),

(19) U

⊇ S

β<α

τ

(U

, K

),

(20) for every β ≤ α the topology of G(K

) is stronger than τ (U

, K

), (21) if O

is open in G(K

) then O

∈ U

,

(22) Z

is either not closed in G(K

) × G(K

) or closed in τ (U

, K

) × τ (U

, K

),

(23) for every F

∈ C(K

) with β ≤ α the following hold:

(a) t

(F

) is small,

(b) ess(t

(F

)) ∩ ess(t

(F

)) is finite,

(c) t(ess(t

(F

))) is closed and discrete in τ (U

, K

).

P r o o f. Put K

= K

= {S

}, K

= K

= S

and U

= U

= {∅}. Then (15)–(23) are easy to check. Suppose the families K

, U

and the sets K

are already constructed so that they satisfy the conditions (15)–(23) for all α < κ. Put

(24) U

= [

α<κ

τ

(U

, K

) ∪ [

α<κ

U

, K

= [

α<κ

K

.

Suppose that Z

is closed in G(K

) × G(K

). Since Q is countable there exist countable families {L

}

such that for every i ∈ ω, L

is open in G(K

), and for any (a, b) ∈ Q

\ Z

there are i, j ∈ ω such that (a, b) ∈ L

× L

⊆ Q

\ Z

. Put

(25) U

= U

∪ {L

}

. If O

is open in G(K

) then put

(26) U

= U

∪ {O

}.

Otherwise U

= U

. Consider the families {F

}

= C(K

). By (24),

(15) and Lemma 2.4 every F

is in C(K

) for some α < κ. Now consider

the families {θ

}

where θ

= ess(t

(F

)). This definition is correct by induction, (23)(a) and the remark above. It now follows by induction and (23)(c) that for any i ∈ ω the set t(θ

) is closed and discrete in τ (U

, K

) for some β < κ. Thus by (20), t(θ

) is closed and discrete in any G(K

) where β ≤ α < κ and thus by (15) in any G(K

) with α < κ since (15) obviously implies that the topology of G(K

) is stronger than that of G(K

) for γ ≤ α.

Thus by (24), Lemma 2.4 and the definition of G(K

) every t(θ

) is closed and discrete in G(K

). Put W

= (Q \ t(θ

)) ∪ {a} for a ∈ Q. Now every W

is open in G(K

). Put

(27) U

= U

∪ {W

}

.

It follows by induction, (24)–(27)(d), (18)–(20) and the construction of U

that every U

∈ U

is open in G(K

). So we can consider the topology τ (U

, K

).

Assume without loss of generality that κ ∈ Λ

and u

is a correct table in G(K

) for some α < κ. Then obviously u

is a correct table in G(K

).

By induction, (23)(a), (15) and Lemma 2.4, t

(F

) is small for each F

∈ C(K

). Then by Lemma 2.7 choose an infinite and thin σ = {σ

| i ≥ 1} ⊆ ω

such that

(28) u

(σ

) → 0 as i → ∞ in τ (U

, K

) and

(29) for all ~a ∈ Q

and F

∈ C(K

) the set t

(~ahu

(σ) ∪ {0}i + F

) is small.

Put K

= u

(σ) ∪ {0} and K

= S

. Then by (d) and (28), K

is a compact subset of Q. Now put K

= S

α<κ

K

∪ {K

}. Then (15) holds.

Let U

∈ U

. Let us show that U

is open in G(K

). It is enough to prove that U

∩ F

is relatively open (in the usual topology of Q) for every F

∈ C(K

). By Lemma 2.6, (24) and Lemma 2.4 every F

is of the form

(30) F

= [

i≤k

~a

hK

i + F

where ~a

∈ Q

, F

∈ C(K

) for some α < κ, and k ∈ ω. Now K

is compact in τ (U

, K

) and by (a) every ~a

hK

i is compact in τ (U

, K

). Thus F

is compact in τ (U

, K

) and thus has the topology induced from Q by (d). But U

∈ τ (U

, K

) by (c) so U

∩ F

is relatively open.

Now let us show that every set of the form

(31) t(ess(t

(F

))), F

∈ C(K

),

is closed and discrete in G(K

). Note that ess(t

(F

)) exists due to (29),

Lemma 2.6 and the construction of K

. First we have C(K

) = C(K

).

So if F

∈ C(K

) then F

∈ C(K

) for some α < κ and thus F

= F

and ess(t

(F

)) = θ

. Then for any point a ∈ Q there is a neighborhood a ∈ (Q \ t(θ

)) ∪ {a} = W

∈ U

open in G(K

) by what we have proved above. Thus t(θ

) is closed and discrete in G(K

). So F

∩ t(θ

) is finite. Then t(ess(t

(F

))) ∩ t(ess(t

(F

))) is finite for all F

∈ C(K

).

So t(ess(t

(F

))) ∩ F

is finite for all F

∈ C(K

). Thus t(ess(t

(F

))) is closed and discrete in G(K

).

Consider the family {V

}

where V

= (Q\t(ess(t

(H

))))∪{a}

and {H

}

= C(K

). By what we have proved above every V

is open in G(K

). Put

(32) U

= U

, U

= U

∪ {V

}

.

Let U

∈ U

with α ≤ κ. If α < κ we have already proved that U

is open in G(K

). If α = κ then if U

∈ U

we have proved before that U

is open in G(K

). Now it follows from (32) that (18) holds. Then (20) is obvious because if β < κ then by (24) and (e), τ (U

, K

) is stronger than τ (U

, K

) and the topology of G(K

) is stronger than τ (U

, K

) by (d). If O

is open in G(K

) then it is open in G(K

) and thus O

∈ U

⊆ U

by (26)–(27).

So (21) holds.

If Z

is closed in G(K

) × G(K

) then it is closed in G(K

) × G(K

).

Then the construction of L

and (25) give that Z

is closed in τ (U

, K

) × τ (U

, K

). Thus (22) holds. Now (16), (17) and (19) are obvious. Let now F

∈ C(K

) with α ≤ κ. Then in fact F

∈ C(K

) for some β < κ. So by induction and (23)(a), t

(F

) is small. If F

∈ C(K

) with α ≤ κ then by Lemma 2.6, Lemma 2.4 and (15),

F

= [

i≤k

~a

hK

i + F

, where F

∈ C(K

).

Now by (29) each t

(~a

hK

i + F

) = t

(~a

hu

(σ) ∪ {0}i + F

) is small so (23)(a) holds. By the choice of {V

}

and the fact that every V

is open in τ (U

, K

) every set of the form t(ess(t

(F

))) where F

∈ C(K

) is closed and discrete in τ (U

, K

). It follows that t(ess(t

(F

))) ∩ F

is finite for all F

∈ C(K

). Thus ess(t

(F

)) ∩ ess(t

(F

)) is finite for all F

∈ C(K

). So (23)(b) holds. To prove (23)(c) it remains to show that for every F

∈ C(K

) the set t(ess(t

(F

))) is closed and discrete in τ (U

, K

).

This can be proved using the properties of W

.

Let us now construct a pair of countable Fr´echet topological groups whose product is sequential but is not Fr´echet.

Example 2.9 (CH). Let K

= S

α<ω1

C(K

) where the families K

are

constructed in Lemma 2.8. Let τ

be the topology on Q defined as follows.

U ⊆ Q is open in τ

if and only if U ∩F

is relatively open for each F

∈ K

. The following fact follows easily from (20) and the definition of G(K

):

Fact. For every α < ω

the topology τ

is stronger than τ (U

, K

).

Consider now an arbitrary O ∈ τ

. Then O = O

for some α < ω

and O

is open in the topology of G(K

), which is stronger than τ

. Thus by (21) and (c), O

is open in τ (U

, K

). It follows from the Fact and the above considerations that τ

is a common refinement for the family {τ (U

, K

) | α < ω

}. So τ

is a group topology.

Let Z ⊆ (Q, τ

) × (Q, τ

) be an arbitrary subset. Then Z = Z

for some α < ω

. Let Z be a nonclosed subset of G(K

) × G(K

). Then since G(K

) has a k

-topology it follows that G(K

) × G(K

) is sequential and thus there is a sequence in Z converging to a point outside Z in the topology of G(K

) × G(K

) and thus in the weaker topology τ

× τ

. If Z is closed in G(K

) × G(K

) then Z is closed in τ (U

, K

) × τ (U

, K

) by (22) and thus Z is closed in the stronger topology τ

× τ

. So τ

× τ

is sequential.

Suppose τ

is not Fr´echet. Then there exists an injection u : ω

→ Q such that u(n, k) → s

as k → ∞ in (Q, τ

) and s

→ 0 as n → ∞ in (Q, τ

) and there is no sequence in u(ω

) converging to 0 in (Q, τ

).

Using the definition of τ

we may assume without loss of generality that u({n} × ω) ∪ {s

} ⊆ K

, {s

| n ∈ ω} ∪ {0} ⊆ B

where K

∈ C(K

) and B

∈ C(K

). Let γ = sup({α

| n ∈ ω}∪{α}). Obviously u is a correct table in G(K

). By the choice of u

there exists β ∈ Λ

with β > γ + 1 such that u = u

. Now by (16) and (17), K

⊆ S

∪{0} and K

is homeomorphic to a nontrivial convergent sequence with limit point 0 in G(K

). So K

is a convergent sequence in the weaker topology τ

. A contradiction. So τ

is Fr´echet.

Obviously (t(n, k), t(n, k)) → (s

, s

) as k → ∞ in τ

×τ

and (s

, s

) → (0, 0) as n → ∞ in τ

× τ

. Suppose (t(n

, k

), t(n

, k

)) → (0, 0) as i → ∞ in τ

× τ

. Then we may assume without loss of generality that {t(n

, k

) | i ∈ ω} ⊆ F

∈ K

and {t(n

, k

) | i ∈ ω} ⊆ F

∈ K

for some F

, F

. Also, F

∈ G(K

) and F

∈ G(K

) for some α < ω

. The set {(n

, k

) | i ∈ ω}

is infinite and thin. Then ess(t

(F

)) ∩ ess(t

(F

)) ⊇ {(n

, k

) | i ∈ ω}, which contradicts (23)(b). So τ

× τ

is not Fr´echet. The argument above also shows that the set Z

= {(t(m, n), t(m, n)) | m, n ∈ ω} ∪ {(s

, s

) | n ∈ ω}∪{(0, 0)} is homeomorphic to S

in the topology induced by (Q

, τ

×τ

) and the proof of Lemma 2.8 shows that Z

is closed in (Q

, τ

× τ

).

In the next example we construct two countable Fr´echet topological groups whose product is not sequential.

Example 2.10 (CH). Let τ

be the topologies constructed in the pre-

vious example. Put G

= (Q, τ

), G

= (Q, τ

) and G

= G

× Q. Then G

can be embedded into G

× [0, 1] and since G

is an α

-space by the result of [N], it follows from [A3, Corollary 5.26] that G

is Fr´echet. The product G

× G

contains a closed copy of S

as was shown in Example 2.9. Now S

× Q is a closed subset of the product G

× G

. Since S

× Q is not a k-space (see [M2]), neither is G

× G

.

Let {v

}

be the family of all mappings v

: ω

→ Q, and {P

}

be the family of all compact subsets of Q.

The following lemma may be proved by an argument similar to that of Lemma 2.8 (see [EKN] for a discussion of spaces containing a copy of the rationals).

Lemma 2.11 (CH). For every α < ω

there is a convergent sequence K

⊆ Q, a countable family K

of compact subsets of Q, a subset D

of Q and a countable family U

of subsets of Q such that:

(33) K

= S

β<α

K

∪ {K

} and S

∈ K

,

(34) if for all i ∈ ω, v

(i, j) → 0 as j → ∞ in some G(K

) with β < α then K

⊆ v

(ω

) ∪ {0} and {v

(i, j) | j ∈ ω} \ K

is finite for all i ∈ ω,

(35) if O

is open in G(K

) then O

∈ U

, (36) U

⊇ S

β<α

τ

(U

, K

),

(37) if U ∈ U

then U is open in G(K

),

(38) the topology of G(K

) is stronger than τ (U

, K

) for β ≤ α, (39) if there is no finite κ ⊆ K

such that P

⊆ S

κ then D

is an infinite closed and discrete subset of P

in τ (U

, K

).

Let us now indicate briefly how to construct an α

and Fr´echet countable topological group which contains no copy of the rationals. Let us recall the definition of a well known topological invariant. For a topological space K let K

= K\isolated points of K, K

= K

\isolated points of K

and K

= T

β<α

K

for limit α. Let sc(K) = min{α | K

= ∅}. It is well known that sc(K) is well defined for every countable compact space and that if sc(K

) and sc(K

) are finite for K

, K

⊆ Q then sc(K

∪ K

) and sc(K

+ K

) are finite. So in the construction of Lemma 2.2 it can be shown that sc(K) is finite for all K ⊆ K

.

Example 2.12 (CH). Let K = S

α<ω1

C(K

) where K

were constructed in Lemma 2.11. Define a topology on Q as in Example 2.9. We obtain a topological group G. The conditions (34), (35) and (38) easily give that G is α

and sequential and hence Fr´echet. Now it follows from (39) that for each compact P ⊆ G there is a finite κ ⊆ K

for some α < ω

such that P ⊆ S

κ. This implies that sc(P ) is finite. But if G contained a copy of the

rationals it would contain a compact P such that sc(P ) = ω + 1.

The author wishes to thank Professor T. Nogura for many useful sugges- tions, advice and moral support, and the referee for important suggestions and corrections.

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Department of Mathematics Auburn University

Auburn, Alabama 36849 U.S.A.

E-mail: shobaay@mallard.duc.auburn.edu Web: http://www.auburn.edu/˜shobaay

Received 3 May 1995;

in revised form 31 May 1996