����������� ������ 58 (2006), 131–138

UDK 515.122

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METRIZABLE GROUPS AND STRICT o-BOUNDEDNESS

Liljana Babinkostova

Abstract. We show that for metrizable topological groups being a strictly o-bounded group is equivalent to being a Hurewicz group. In [5] Hernandez, Robbie and Tkachenko ask if there are strictly o-bounded groups G and H for which G × H is not strictly o-bounded. We show that for metrizable strictly o-bounded groups the answer is no. In the same paper the authors also ask if the product of an o-bounded group with a strictly o-bounded group is again an o-bounded group. We show that if the strictly o-bounded group is metrizable, then the answer is yes.

1. Definitions and notation

Let H and G be topological spaces with G a subspace of H. We shall use the notations:

• O_{H}: The collection of open covers of H.

• O_{HG}: The collection of covers of G by sets open in H.

An open cover U of a topological space H is said to be

• an ω-cover if H is not a member of U, but for each finite subset F of H there is a U ∈ U such that F ⊂ U [3]. The symbol ω denotes the collection of ω-covers of H.

• groupable if there is a partition U = ⋃_{n∈N} U_{n}, where each U_{n} is finite, and for each x ∈ H the set {n : x ∉ ⋃ U_{n}} is finite [9]. The symbol O^{gp} denotes the collection of groupable open covers of the space.

• γ-cover if it is infinite, and each infinite subset of it is still an open cover of the space. The symbol γ denotes the collection of γ-covers of the space.

• large if each element of the space is contained in infinitely many elements of the cover. The symbol Λ denotes the collection of large covers of the space. Now let (G, ∗) be a topological group with identity element e. We will assume

that G is not compact. For a neighborhood U of e, and for a finite subset F of G the set F ∗U is a neighborhood of the finite set F. Thus, the set {F ∗U : F ⊂ G finite} is an ω-cover of G, which is denoted by the symbol Ω(U). The set

Ω_{nbd} = {Ω(U) : U a neighborhood of e}

is the set of all such ω-covers of G.

AMS Subject Classification: 54H11 Keywords and phrases: Metrizable topological group, strict o-boundedness, product groups.

The set O(U) = {x ∗ U : x ∈ G} is an open cover of G. The symbol

O_{nbd} = {O(U) : U a neighborhood of e}

denotes the collection of all such open covers of G. Now we describe the selection principles relevant to this topic. Let S be an infinite set, and let A and B be collections of families of subsets of S.

The symbol S_{1}(A, B) denotes the statement that there is for each sequence (O_{n} : n ∈ N) of elements of A a sequence (T_{n} : n ∈ N) such that for each n T_{n} ∈ O_{n}, and {T_{n} : n ∈ N} ∈ B. The earliest example of this sort of selection principle was introduced in [14] by Rothberger, and is S_{1}(O, O) in our notation.

The symbol S_{fin}(A, B) is defined as follows: For each sequence (O_{n} : n ∈ N) from A there is a sequence (T_{n} : n ∈ N) of finite sets such that for each n T_{n} ⊂ O_{n}, and ⋃_{n∈N} T_{n} ∈ B. The earliest example of this selection principle was introduced by Hurewicz in [7], and in our notation is S_{fin}(O, O). Because of its equivalence in metric spaces with a basis property of Menger, S_{fin}(O, O) is also called the Menger property. In [7] Hurewicz introduced also a second selection principle, now called the Hurewicz property, and characterized in [9] as the statement S_{fin}(Λ, O^{gp}).

Each selection principle has a corresponding game. The game that we will use in this paper is the game corresponding to S_{1}(A, B). The game is denoted G_{1}(A, B) and is played as follows: In the n-th inning ONE chooses an O_{n} ∈ A, and TWO responds with a T_{n} ∈ O_{n}. They play an inning for each natural number n. A play

O_{1}, T_{1}, ⋯, O_{n}, T_{n}, ⋯

is won by TWO if {T_{n} : n ∈ N} is in B. Otherwise, ONE wins.

2. Hurewicz-bounded groups

The notion of Hurewicz-bounded group was introduced by Kočinac in 1998 in one of his unpublished notes. A topological group (G, ∗) is said to be a Hurewiczbounded group if it satisfies the selection principle S_{1}(Ω_{nbd}(G), O_{G}^{gp}). Let (G, ∗) be a subgroup of the group (H, ∗). Then G is Hurewicz-bounded H if the selection principle S_{1}(Ω_{nbd}(H), O_{HG}^{gp}) holds.

Theorem 1. For a subgroup (G, ∗) of an infinite topological group (H, ∗) the following are equivalent:

1. S_{1}(Ω_{nbd}(H), O_{HG}^{gp}).

2. S_{1}(Ω_{nbd}(H), Γ_{HG}).

Metrizable groups and strict o-boundedness 133

Proof. We need to prove only that 1 ⇒ 2. For each n let U_{n} be an element of Ω_{nbd}(H), and choose for each n an open neighborhood U_{n} of e such that U_{n} = Ω(U_{n}). For each n put V_{n} = ∩_{j≤n}U_{j}. For each n put V_{n} = Ω(V_{n}). Apply S_{1}(Ω_{nbd}, O^{gp}) to (V_{n} : n ∈ N). For each n choose W_{n} ∈ V_{n} such that {W_{n} : n ∈ N} is a groupable open cover of G. Choose a sequence m_{1} < m_{2} < ⋯ < m_{n} < ⋯ such that for each x ∈ G, for all but finitely many n, x ∈ ⋃_{m_{n}≤j<m_{n+1}} W_{j}. For each n also choose a finite set F_{n} ⊂ H with W_{n} = F_{n} ∗ V_{n}. Now define, for each k, the finite set G_{k} by

G_{k} = \{ \begin{matrix}⋃_{i≤m_{1}} F_{i}, & if k ≤ m_{1} \\ ⋃_{m_{n}<i≤m_{n+1}} F_{i}, & if m_{n} < k ≤ m_{n+1} \end{matrix}

For each n put A_{n} = G_{n} ∗ U_{n}, an element of Ω(U_{n}). We claim that {A_{n} : n ∈ N} is a γ-cover of G. For consider g ∈ G. Choose M ∈ N so that for all n ≥ M we have g ∈ ⋃_{m_{n}<i≤m_{n+1}} W_{i}. But for m_{n} < i ≤ m_{n+1} we have W_{i} = F_{i} ∗V_{i} ⊂ A_{k} = G_{k} ∗U_{k} for m_{n} < k ≤ m_{n+1}. Thus, for all k > m_{M} we have g ∈ A_{k}. It follows that {A_{k} : k ∈ N} is a γ-cover of G.

Theorem 2. Let (G, ∗) be a subgroup of the topological group (H, ∗). Then the following are equivalent:

1. S_{1}(Ω_{nbd}(H), O_{HG}^{gp}).

2. S_{1}(Ω_{nbd}(G), O_{G}^{gp}).

Proof. The implication (2) ⇒ (1) is evidently true. We must show that (1) ⇒ (2). Thus, let (Ω(U_{n}) : n ∈ N) be a sequence in Ω_{nbd}(G). Since each U_{n} is a neighborhood (in G) of the group identity we may choose for each n a neighborhood T_{n} in H of the identity, such that U_{n} = T_{n} ∩ G. Next, choose for each n a neighborhood S_{n} in H of the identity, such that S_{n}^{−1} ∗ S_{n} ⊆ T_{n}.

Apply (2) to the sequence (Ω(S_{n}) : n ∈ N) we get for each n a finite set F_{n} ⊂ H such that for each element x of G there is an N such that for all n ≥ N we have x ∈ F_{n} ∗ S_{n}. For each n, and for each f ∈ F_{n}, choose a g_{f} ∈ G as follows:

g_{f}\{ \begin{matrix}∈ G ∩ f ∗ S_{n}, & if nonempty \\ = e, the group identity, & otherwise \end{matrix}

Then put G_{n} = {g_{f} : f ∈ F_{n}}, a finite subset of G. For each n we have G_{n} ∗ U_{n} ∈ Ω(U_{n}) ∈ Ω_{nbd}(G). We must show that for each x ∈ G there is an N such that for each n ≥ N, x ∈ G_{n} ∗ U_{n}.

Let x ∈ G be given. Choose N so that for all n ≥ N we have x ∈ F_{n} ∗ S_{n}. Fix n ≥ N and choose f_{x} ∈ F_{n} so that x ∈ f_{x} ∗ S_{n}. Then evidently G ∩ f_{x} ∗ S_{n} is nonempty, and so g_{f_{x}} ∈ G is defined as an element of this intersection. Since g_{f_{x}} ∈ f_{x} ∗ S_{n}, we have f_{x} ∈ g_{f_{x}} ∗ S_{n}^{−1}, and so x ∈ g_{f_{x}} ∗ S_{n}^{−1} ∗ S_{n} ⊆ g_{f_{x}} ∗ T_{n}. Now g_{f_{x}}^{−1} ∗ x ∈ G ∩ T_{n} = U_{n}, and so we have x ∈ g_{f_{x}} ∗ U_{n} ⊂ G_{n} ∗ U_{n}. This completes the proof.

This result implies the following:

Corollary 3. If (H, ∗) has property S_{1}(Ω_{nbd}(H), Γ_{H}), then for each infinite subgroup G of H, S_{1}(Ω_{nbd}(G), Γ_{G}) holds.

3. A characterization of strict o-boundedness for metrizable groups

According to Hausdorff [4, 25] a metric space (X, d) is totally bounded if there is for each δ > 0 a partition of X into finitely many sets, each of diameter less than δ. The metric space is σ-totally bounded if it is a union of countably many sets, each totally bounded.

Measure-like properties of metrizable spaces can be equivalent to selection properties. Similarly, for metrizable groups measure-like properties with respect to left-invariant metrics can be equivalent to selection properties. Let A be a collection of sets, each a set of subsets of H. As in [1] we say: Metric space (H, d) has Ameasure zero if for each sequence (ϵ_{n} : n ∈ N) of positive real numbers, there is a sequence (F_{n} : n ∈ N) where:

1. For each n, F_{n} is a finite set of subsets of H,

2. Each element of F_{n} has diameter less than ϵ_{n}, and

3. ⋃_{n∈N} F_{n} ∈ A.

The following theorem of Kakutani will be used below:

Theorem 4. [6] Let (U_{k} : k < ∞) be a sequence of subsets of the topological group (H, ∗) where {U_{k} : k < ∞} is a neighborhood basis of the identity element e and each U_{k} is symmetric^{1}, and for each k also U_{k+1}^{2} ⊆ U_{k}. Then there is a left-invariant metric d on H such that

1. d is uniformly continuous in the left uniform structure on H × H.

2. If y^{−1} ∗ x ∈ U_{k} then d(x, y) ≤ (\frac{1}{2})^{k−2}.

3. If d(x, y) < (\frac{1}{2})^{k} then y^{−1} ∗ x ∈ U_{k}.

The equivalence of the first two statements of the following theorem apparently has been independently obtained also by H. Michalewski [13].

Theorem 5. For a subgroup (G, ∗) of a metrizable group (H, ∗) the following are equivalent:

1. TWO has a winning strategy in the game G_{1}(Ω_{nbd}(H), O_{HG}).

2. (G, ∗) is σ-totally bounded in all left-invariant metrics generating the topology of H.

3. S_{1}(Ω_{nbd}(H), Γ_{HG}) holds.

4. H has the O_{HG}^{gp}-measure zero property in all left invariant metrics on H.

Proof. 1 ⇒ 2: Since (H, ∗)is a metrizable group, it is first countable. Let d be a left-invariant metric of H and let (U_{n} : n ∈ N) be a neighborhood basis of the identity element e of H such that for each n, U_{n} ⊃ U_{n+1} and diam_{d}(U_{n}) < \frac{1}{2^{n}}. Let σ be TWO’s winning strategy. Define:

G_{∅} = ⋂_{n∈N} σ(Ω(U_{n})).

1U_{k} is symmetric if U_{k} = U_{k}^{−1}

Then, for n_{1}, ⋯, n_{k} < ∞ given, define

G_{n_{1},··· ,n_{k}} = ⋂_{n∈N} σ(Ω(U_{n_{1}}), ⋯, Ω(U_{n_{k}}), Ω(U_{n})).

First, we show that

G ⊆ ⋃_{τ∈^{<ω}N} G_{τ} .  (1)

For suppose on the contrary that (1) is false. Choose an x ∈ G \ ⋃_{τ∈^{<ω}N} G_{τ}. Since x is not in G_{∅}, choose n_{1} so that x ∉ σ(Ω(U_{n_{1}})). Since x is not in G_{n_{1}}, choose n_{2} so that x ∉ σ(Ω(U_{n_{1}}), Ω(U_{n_{2}}), and so on. In this way we obtain a σ-play

Ω(U_{n_{1}}), σ(Ω(U_{n_{1}})), Ω(U_{n_{2}}), σ(Ω(U_{n_{1}}), Ω(U_{n_{2}})), ⋯

which is lost by TWO, since x is never covered by TWO. This contradicts the fact that σ is a winning strategy for TWO.

Next, we observe that each G_{τ} is totally bounded. Fix τ = (n_{1} ⋯, n_{k}). Let an ϵ > 0 be given. Choose n so large that \frac{1}{2^{n}} < ϵ. Now G_{τ} is a subset of σ(Ω(U_{n_{1}}), ⋯, Ω(U_{n_{k}}), Ω(U_{n})), which is of the form F ∗ U_{n} for some finite subset F of G. But then {f ∗ U_{n} : f ∈ F } is a finite family of sets, each of diameter less than \frac{1}{2^{n}} < ϵ, and covers G_{τ}.

This completes the proof that G is σ-totally bounded.

2 ⇒ 1: Fix a left-invariant metric d of H and assume that G is σ-totally bounded. Write G = ⋃_{n∈N} G_{n}, where each G_{n} is totally bounded. We define a winning strategy σ for TWO as follows:

When player ONE plays Ω(U_{1}), put ϵ_{1} = diam(U_{1}), and δ_{1} = \frac{ϵ_{1}}{4} . Since G_{1} is totally bounded, choose a finite family F of open sets, each of diameter less than δ_{1}, so that G_{1} ⊆ ⋃ F. For each F ∈ F, choose a point x_{F} ∈ F. Then F ⊆ x_{F} ∗ U (by diameter considerations), and so, setting S_{1} = {x_{F} : F ∈ F}, TWO responds with σ(Ω(U_{1})) = S_{1} ∗ U_{1} ∈ Ω(U_{1}).

Suppose it is the n-th inning, and ONE has played Ω(U_{1}), ⋯, Ω(U_{n}) so far. Put ϵ_{n} = min\{diam(U_{j}) : j ≤ n\}, and put δ_{n} = \frac{ϵ_{n}}{4} . Since G_{1} ∪ ⋯ ∪ G_{n} is totally bounded, choose a finite set F of open subsets of G, each of diameter less than δ_{n}, such that G_{1} ∪ ⋯ ∪ G_{n} ⊆ ∪F. For each F ∈ F choose an x_{F} ∈ F and put S_{n} = {x_{F} : F ∈ F}. Then by diameter considerations, ⋃ F ⊂ S_{n} ∗ U_{n}. Now TWO plays σ(Ω(U_{1}), ⋯, Ω(U_{n})) = S_{n} ∗ U_{n} ∈ Ω(U_{n}).

It is evident that σ is a winning strategy for TWO.

2 ⇒ 3: The winning strategy σ described for TWO above has the effect of choosing for a sequence (Ω(U_{n}) : n ∈ N) a corresponding sequence (T_{n} : n ∈ N) such that for each n we have T_{n} ∈ Ω(U_{n}), and G_{1} ∪ ⋯ ∪ G_{n} ⊆ T_{n}. This witnesses S_{1}(Ω_{nbd}(H), Γ_{HG}).

3 ⇒ 2: Since H is metrizable it is first countable and has a left-invariant metric. Choose a left invariant metric d for H and a sequence (U_{n} : n ∈ N) of neighborhoods of the identity e, which forms a neighborhood basis at e, and such that lim_{n→∞} diam_{d}(U_{n}) = 0. Apply S_{1}(Ω_{nbd}(H), Γ_{HG}) to the sequence (Ω(U_{n}) : n ∈ N): For each n we find a T_{n} = S_{n} ∗ U_{n} ∈ Ω(U_{n}) where S_{n} is a finite subset of G, such that {T_{n} : n ∈ N} is a γ-cover of G. For each n ∈ N define G_{n} = ∩_{k≥n}T_{k}. Then G ⊆ ⋃_{n∈N} G_{n}. Each G_{n} is totally bounded: For let an n be given and let ϵ > 0 be given. Choose an N > n so large that diam(U_{N}) < ϵ. Then T_{N} is a union of finitely many sets of the form x ∗ U_{N}, and G_{n} ⊂ T_{N}.

3 ⇒ 4: Let d be a left-invariant metric on H and let (ϵ_{n} : n ∈ N) be a sequence of positive real numbers. Choose for each n a neighborhood U_{n} of the identity element e of H such that diam_{d}(U_{n}) < ϵ_{n}. Then apply S_{1}(Ω_{nbd}(H), O_{HG}^{gp}) to the sequence (Ω_{H}(U_{n}) : n ∈ N). For each n we find F_{n} of H such that each element of G is in all but finitely many of the F_{n} ∗ U_{n}’s. Since d is left-invariant, each element of V_{n} = {x ∗ U_{n} : x ∈ F_{n}} has diameter less than ϵ_{n}.

4 ⇒ 3: Let (U_{n} : n ∈ N) be a sequence of neighborhoods of e, the identity element of H. Choose for each n a symmetric neighborhood V_{n} such that V_{n+1}^{2} ⊆ V_{n} ⊂ U_{n}. By Kakutani’s theorem choose a left-invariant metric d as in Theorem 5 corresponding to the sequence(V_{n} : n ∈ N). For each n put ϵ_{n} = \frac{1}{2^{n}}. We may assume that G is not totally bounded in d. Choose δ ≥ 0 witnessing this. Choose N large enough such that ϵ_{N} ≥ δ, and apply 4 to (ϵ_{n} : n ≥ N). For each n ≥ N choose a finite set H_{n} of subsets of H such that each set in H_{n} has d-diameter less that ϵ. For each S ∈ H_{n} choose T_{s} ∈ O_{H}(V_{n}) with S ∈ T_{s}. Choose W_{s} ∈ O_{H}(U_{n}) with S ⊆ W_{s}. For each n, put G = {W_{s} : S ∈ H_{n}}, and put G_{n} = ⋃ G_{n} ∈ Ω(U_{n}). Then, {G_{n} : n ≥ N} is a γ cover of G.

Problem 4.1 of [5] asks if there are strictly o-bounded groups G and H for which G × H is not strictly o-bounded. We show that for metrizable strictly obounded groups the answer is no. Problem 4.2 asks if the product of an o-bounded group with a strictly o-bounded group is again an o-bounded group. We show that if the strictly o-bounded group is metrizable, then the answer is yes. Both of these answers are results of the following theorem.

Theorem 6. Let (G, ∗) be a group satisfying S_{1}(Ω_{nbd}, Γ). Let A be one of O, Ω, Γ. If (H, ∗) is a group with property S_{1}(Ω_{nbd}, A), then (G × H, ∗) also has this property.

Proof. Consider a sequence (Ω(U_{n}) : n ∈ N) in Ω_{nbd}(G × H). For each n choose neighborhoods V_{n} of e_{G} and W_{n} of e_{H} such that V_{n} × W_{n} ⊂ U_{n}. Consider the sequences (Ω(V_{n}) : n ∈ N) and (Ω(W_{n}) : n ∈ N). For each n choose a finite set G_{n} ⊂ G and a finite set H_{n} ⊂ H such that (G_{n} ∗ V_{n} : n ∈ N) is in Γ, and (H_{n} ∗ W_{n} : n ∈ N) is in A. For each n put F_{n} = G_{n} × H_{n}. We claim that (F_{n} ∗ U_{n} : n ∈ N) has the required properties:

1. A = O: By the Remark following Theorem 1 of [2], we may assume that there is for each y ∈ H infinitely many n so that y ∈ H_{n} ∗ W_{n}. Consider any (x, y) ∈ G × H. Fix an N so that for all n ≥ N we have x ∈ G_{n} ∗ V_{n}. Then choose an n > N so that also y ∈ H_{n} ∗ W_{n}. Then (x, y) ∈ F_{n} ∗ U_{n}.

2. A = Ω: Observe that for each finite subset K of G there is an N such that for all n ≥ N, K ⊆ G_{n} ∗V_{n}. Since for each finite subset L of H there are infinitely many n with L ⊆ H_{n} ∗ W_{n}, apply the argument from before.

3. A = Γ: The argument is similar for this case.

Corollary 7. If (G_{j}, ∗_{j}), j ≤ n are metrizable strictly o-bounded groups, then so is their product.

Proof. By Theorem 5, metrizable strictly o-bounded groups are characterized by S_{1}(Ω_{nbd}, Γ). Apply Theorem 6.

Corollary 8. If (G, ∗) is an o-bounded group and (H, ∗) is a metrizable strictly o-bounded group, then G × H is an o-bounded group.

Proof. Apply Theorems 5 and 6.

We characterize for metrizable groups when TWO has a winning strategy in the game G_{1}(O_{nbd}(H), O_{HG}):

Theorem 9. Let (H, ∗) be a metrizable group with a subgroup (G, ∗). The following are equivalent:

1. TWO has a winning strategy in G_{1}(O_{nbd}(H), O_{HG}).

2. G is a countable set.

Proof. We only need to prove 1 ⇒ 2: Let σ be a winning strategy for TWO. Since (H, ∗) is metrizable, let d be a left-invariant metric for it, and let (U_{n} : n ∈ N) be a neighborhood basis for e in H such that lim_{n→∞} diam_{d}(U_{n}) = 0. Define G_{∅} = ⋂_{n∈N} σ(O(U_{n})). For (n_{1}, ⋯, n_{k}) a given finite sequence, define G_{(n_{1},··· ,n_{k})} = ⋂_{n∈N} σ(O(U_{n_{1}}), ⋯, O(U_{n_{k}}), O(U_{n})). We claim that

G ⊆ ⋃_{τ∈^{<ω}N} G_{τ} .  (2)

For suppose on the contrary x is not in ⋃_{τ∈^{<ω}N} G_{τ}. Choose an n_{1} with x ∉ σ(O(U_{n_{1}})), and then choose n_{2} so that x ∉ σ(O(U_{n_{1}}), O(U_{n_{2}})), and so on. In this way we obtain a sequence n_{1}, n_{2}, ⋯, n_{k}, ⋯ so that for each k, x ∉ σ(O(U_{n_{1}}), ⋯, O(U_{n_{k}})). But then we obtain a σ-play lost by TWO, a contradiction.

Next, we observe that each G_{τ} has at most one element: This is because d is left-invariant, so that diam(σ(O(U_{n_{1}}), ⋯, O(U_{n_{k}}), O(U_{n})) = diam(U_{n}), and since these U_{n}’s form a neighborhood base for e, the diameter of H_{n_{1},··· ,n_{k}} is 0.

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(received 12.11.2005, in revised form 22.01.2006)