- Email: [email protected]
Discrete subsets and convergent sequences in topological groups
Topology and its Applications 191 (2015) 137–142
Contents lists available at ScienceDirect
Topology and its Applications www.elsevier.com/locate/topol
Discrete subsets and convergent sequences in topological groups Valentin Keyantuo a , Yevhen Zelenyuk b,∗,1 a
Department of Mathematics, University of Puerto Rico, PO Box 70377, San Juan, PR 00936-8377, USA b School of Mathematics, University of the Witwatersrand, Private Bag 3, Wits 2050, Johannesburg, South Africa
a r t i c l e
i n f o
Article history: Received 28 October 2014 Received in revised form 29 May 2015 Accepted 31 May 2015 Available online xxxx MSC: primary 03E35, 54H11 secondary 22A05, 54G05
a b s t r a c t We show that (1) assuming there is no rapid ultrafilter, every countable nondiscrete maximally almost periodic topological group contains a discrete nonclosed subset which is a convergent sequence in some weaker totally bounded group topology, (2) every infinite Abelian totally bounded topological group contains such a discrete subset (in ZFC), and (3) if an extremally disconnected topological group contains such a discrete subset, then there is a selective ultrafilter. © 2015 Elsevier B.V. All rights reserved.
Keywords: Topological group Maximally almost periodic Totally bounded Extremally disconnected Discrete subset Convergent sequence Rapid ultrafilter P -point Selective ultafilter
1. Introduction In [8], O. Sipacheva obtained partial solutions to Arhangel’ski˘ı’s question of whether there exists in ZFC a countable nondiscrete extremally disconnected topological group [1] and to Protasov’s question of whether there exists in ZFC a countable nondiscrete topological group in which every discrete subset is closed [6]. It had been known previously that the existence of an extremally disconnected topological group containing a countable discrete nonclosed subset implies the existence of a P -point, and similar result holds for a nondiscrete group topology T on ω Z2 which is finer than the direct sum topology, has no discrete subset * Corresponding author. 1
E-mail addresses: [email protected] (V. Keyantuo), [email protected] (Y. Zelenyuk). The second author was supported by NRF grant IFR2011033100072.
http://dx.doi.org/10.1016/j.topol.2015.05.089 0166-8641/© 2015 Elsevier B.V. All rights reserved.
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with a unique accumulation point, and satisfies one more condition [10]. (The direct sum topology is the one induced by the product topology.) Sipacheva showed that the existence of a nondiscrete extremally disconnected group topology on ω Z2 finer than the direct sum topology implies the existence of a rapid ultrafilter, and similar result holds for a group topology on ω Z2 which is finer than the direct sum topology and has no discrete subset with a unique accumulation point. In this paper we show that (1) assuming there is no rapid ultrafilter, every countable nondiscrete maximally almost periodic topological group contains a discrete nonclosed subset which is a convergent sequence in some weaker totally bounded group topology, (2) every infinite Abelian totally bounded topological group contains such a discrete subset (in ZFC), and (3) the existence of an extremally disconnected topological group containing such a discrete subset implies the existence of a selective ultrafilter. Notice that such a subset has a unique accumulation point and converges to that point in that weaker topology. Recall that a nonprincipal ultrafilter p on ω is (i) a P -point if for every partition {An : n < ω} of ω with An ∈ / p, there is A ∈ p such that |A ∩ An | < ω for all n, (ii) selective if for every partition {An : n < ω} of ω with An ∈ / p, there is A ∈ p such that |A ∩ An | ≤ 1 for all n, and (iii) rapid if for every partition {An : n < ω} of ω with finite An , there is A ∈ p such that |A ∩ An | ≤ n for all n. Clearly, every selective ultrafilter is rapid. MA implies the existence of P -points and rapid ultrafilters. However, it is consistent with ZFC that there is no P -point [7, VI, §4], and it is consistent with ZFC that there is no rapid ultrafilter [5]. A topological group is totally bounded (maximally almost periodic) if it can be topologically and algebraically (continuously and algebraically) embedded into a compact group. All topologies are assumed to be Hausdorff. 2. Rapid ultrafilters Every group topology on a countable group can be weakened to a metrizable (= first countable) group topology [2]. Consequently, every maximally almost periodic group topology on a countable group can be weakened to a metrizable totally bounded group topology. In this section we prove the following result. Theorem 2.1. Assume there is no rapid ultrafilter. Let (G, T ) be a countable nondiscrete maximally almost periodic topological group and let T0 be a metrizable totally bounded group topology on G weaker than T . Then (G, T ) contains a discrete nonclosed subset which is a convergent sequence in T0 . As a consequence we obtain that Corollary 2.2. It is consistent with ZFC that every countable nondiscrete maximally almost periodic topological group contains a discrete nonclosed subset which is a convergent sequence in some weaker totally bounded group topology. The proof of Theorem 2.1 involves the following result [11, Theorem 3.1]. Theorem 2.3. Let G be a countably infinite metrizable totally bounded topological group. Then there exist a sequence (mn )n<ω of integers ≥ 2 and a homeomorphism h : G → n<ω Zmn with h(1) = 0 such that
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(a) h(xy) = h(x)+h(y) whenever φ(x)+2 ≤ θ(y), where φ(x) = max supp(h(x)) and θ(y) = min supp(h(y)), (b) φ(xi y j ) ≤ max{φ(x), φ(y)} + 1 for all i, j ∈ {1, −1}, and if |φ(x) − φ(y)| ≥ 2, then φ(xi y j ) ≥ max{φ(x), φ(y)} − 1, and (c) if m < ω and h(x)(n) = h(y)(n) for each n ≤ m, then θ(x−1 y), θ(xy −1 ) ≥ m. Here, n<ω Zmn is endowed with the direct sum topology, and the functions φ and θ are not defined at x = 1. Notice that in [11, Theorem 3.1] property (c) reads as θ(x−1 y), θ(xy −1 ) ≥ m − 1, but the proof shows that in fact θ(x−1 y), θ(xy −1 ) ≥ m holds (although this small improvement plays no role in our result). The next theorem is a more precise statement of Theorem 2.1. Theorem 2.4. Let (G, T ) be a countable nondiscrete maximally almost periodic topological group and let T0 be a metrizable totally bounded group topology on G weaker than T . Suppose (G, T ) contains no discrete nonclosed subset which is a convergent sequence in T0 . Let h : G → n<ω Zmn be a mapping guaranteed by Theorem 2.3. Then for every ultrafilter p on G converging to 1 in T , the ultrafilter φ(p) on ω is rapid. Here, φ(p) is the ultrafilter on ω with a base consisting of subsets φ(A), where A ∈ p. In the proof of Theorem 2.4 we use the following characterization of rapid ultrafilters [5]: An ultrafilter p on ω is rapid if and only if there is a function h : ω → ω with the property that, whenever f : ω → ω, there exists A ∈ p such that for every n < ω, |A ∩ f (n)| ≤ h(n), where A ∩ f (n) = A ∩ {0, 1, . . . , f (n) − 1}. Proof. Let f : ω → ω be given. Without loss of generality one may suppose that f is increasing. Define U ⊆ G by U = {1} ∪ {x ∈ G : f (θ(x)) < φ(x)}. It then follows that U is a neighborhood of 1 in T . Indeed, assume the contrary. Then 1 is an accumulation point of G \ U = {x ∈ G : f (θ(x)) ≥ φ(x)}. But for every n < ω, there are only finitely many x ∈ G \ U with θ(x) = n, because φ(x) ≤ f (θ(x)). Consequently, G \ U is a sequence converging to 1 in T0 , a contradiction. Pick a neighborhood V of 1 in T such that V V −1 ⊆ U . We claim that for every n < ω, |φ(V ) ∩ f (n)| ≤
mi .
i≤n
To see this, assume the contrary. Then there exist distinct x, y ∈ V \ {1} such that φ(x), φ(y) < f (n) and h(x)(i) = h(y)(i) for each i ≤ n. Let z = xy −1 . Then 1 = z ∈ U , θ(z) ≥ n, and φ(z) ≤ f (n). Hence, f (n) ≤ f (θ(z)) < φ(z) ≤ f (n), a contradiction. 2 3. Totally bounded topological groups In [6] it was shown that every infinite totally bounded topological group contains a countable discrete nonclosed subset. In this section we derive from Theorem 2.4 the following result.
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Theorem 3.1. Let (G, T ) be a countably infinite totally bounded topological group and let T0 be a metrizable group topology on G weaker than T . Then (G, T ) contains a discrete nonclosed subset which is a convergent sequence in T0 . Given a group G and a left invariant topology T on G, Ult(T ) ⊆ βG consists of all nonprincipal ultrafilters on G converging to 1 in T and is called the ultrafilter semigroup of T (see [9, Section 7.1]). Here, βG is the Stone–Čech compactification of G. The points of βG are the ultrafilters on G and the topology is generated by taking as a base the subsets A = {p ∈ βG : A ∈ p}, where A ⊆ G. For any p, q ∈ βG, the ultrafilter pq ∈ βG has a base consisting of subsets x∈A xBx , where A ∈ p and Bx ∈ q for every x ∈ A. See [4] and [9] for more information about βG. Lemma 3.2. Let (G, T ) be a countably infinite metrizable totally bounded topological group, let h : (G, T ) → ∗ n<ω Zmn be a mapping guaranteed by Theorem 2.1, and let S = Ult(T ). Then φ(S) = ω , and for every u ∈ ω ∗ , φ−1 (u) ∩ S is a closed left ideal of S. Proof. For every n < ω, pick xn ∈ G with supp(h(x)) = {n}. The sequence (xn )n<ω converges to 1 in T . It then follows that φ(S) = ω ∗ . Now let p ∈ S and q ∈ φ−1 (u) ∩ S. Then pq ∈ S, and we claim that φ(pq) = u. To see this, let A ∈ u. Put P = G \ {1}, and for every x ∈ P , pick Qx ∈ q such that φ(Qx ) ⊆ A and φ(x) + 2 ≤ θ(y) for all y ∈ Qx , so φ(xy) = φ(y). Let R = x∈P xQx . Then R ∈ pq and φ(R) ⊆ A. 2 Proof of Theorem 3.1. Let h : (G, T0 ) → n<ω Zmn be a mapping guaranteed by Theorem 2.1 and let S = Ult(T ) and S0 = Ult(T0 ). By Lemma 3.2, φ(S0 ) = ω ∗ , and for every u ∈ ω ∗ , φ−1 (u) ∩ S0 is a closed left ideal of S0 , so φ−1 (u) ∩ S0 contains an idempotent (see [9, Theorem 6.12]). Since S contains all the idempotents of G∗ [9, Lemma 7.10], it follows that φ(S) = ω ∗ . Consequently, there are ultrafilters in φ(S) which are not rapid. Hence by Theorem 2.3, (G, T ) contains a discrete nonclosed subset which is a convergent sequence in T0 . 2 Now we are going to extend Theorem 3.1 to all infinite Abelian totally bounded topological groups. Lemma 3.3. Let (G, T ) be an Abelian totally bounded topological group, let H be a subgroup of G, and let T0 be a group topology on H weaker than T |H . Then there is a group topology T1 on G weaker than T such that T1 |H = T0 . Proof. Let Γ denote the set of all characters of (G, T ) (= continuous homomorphisms ϕ : (G, T ) → T) and Γ0 the set of all characters of (H, T0 ) and let Γ1 = {ϕ ∈ Γ : ϕ|H ∈ Γ0 }. Then {ϕ|H : ϕ ∈ Γ1 } = Γ0 , and for every a ∈ G \ H, there is ϕ ∈ Γ1 such that ϕ(a) = 1 ([3, Corollary 24.12]). Let T1 be the coarsest topology on G in which each character from Γ1 is continuous. 2 Theorem 3.4. Every infinite Abelian totally bounded topological group contains a discrete nonclosed subset which is a convergent sequence in some weaker group topology. Proof. Let (G, T ) be an infinite Abelian totally bounded topological group. Pick a countably infinite subgroup H of G and a metrizable group topology T0 on H weaker than T |H . By Lemma 3.3, there is a group topology T1 on G weaker than T such that T1 |H = T0 . By Theorem 3.1, there is a sequence (xn )n<ω in H having 1 as an accumulation point in T |H and converging to 1 in T0 . But then (xn )n<ω converges to 1 in T1 , and consequently, 1 is the unique accumulation point of (xn )n<ω in T . 2 Using the same argument as in the proof of Theorem 3.4, we obtain the following result.
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Theorem 3.5. Assume there is no rapid ultrafilter. Then every nondiscrete Abelian maximally almost periodic topological group containing a countable nondiscrete subset has a discrete nonclosed subset which is a convergent sequence in some weaker totally bounded group topology. Corollary 3.6. It is consistent with ZFC that every nondiscrete Abelian maximally almost periodic topological group containing a countable nondiscrete subset has a discrete nonclosed subset which is a convergent sequence in some weaker totally bounded group topology. 4. P -points and selective ultrafilters Let X be a space, let D be a discrete subset of X, and let x ∈ X be an accumulation point of D. We say that D is locally maximal with respect to x if there is an open U ⊆ X containing D such that, whenever C ⊆ U \ D and D ∪ C ⊆ U is discrete, x is not an accumulation point of C. Notice that if X is extremally disconnected and D is countable, then D is locally maximal with respect to x and there is only one ultrafilter on X containing D and converging to x (see [9, Lemma 5.13]). Theorem 4.1. Let G be a topological group and let D be a countable discrete subset of G with 1 being an accumulation point of D. Suppose that D is locally maximal with respect to 1 and there is only one ultrafilter p on G containing D and converging to 1. Then there is a mapping f : G → ω such that f (p) is a P -point. Furthermore, if D is a convergent sequence in some weaker group topology on G, then f can be chosen so that f (p) is a selective ultrafilter. Proof. Let U be an open neighborhood of D ⊆ X witnessing that D is locally maximal with respect to 1. Enumerate D as {xn : n < ω}. Construct inductively a decreasing sequence (Un )n<ω of open neighborhoods of 1 with U0 = G such that xn ∈ / Un+1 and the subsets xn Un+1 , where n < ω, are pairwise disjoint and contained in U . Define f : D → ω by f (x) = n if x ∈ Un \ Un+1 , and let q = f (p). Since n<ω Un ∩ D = ∅, q is a nonprincipal ultrafilter on ω. We prove that q is a P -point. / q and let Dm = f −1 (Am ). Then {Dm : m < ω} is a Let {Am : m < ω} be a partition of ω with Am ∈ partition of D with Dm ∈ / p, so 1 is not an accumulation point of Dm . Every element xn ∈ D belongs to some subset Dmn of the partition. Put C = n<ω xn (Un+1 ∩ Dmn ). Then C is disjoint from D and D ∪ C is a discrete subset of U . Since D is locally maximal, it follows that G \ C is a neighborhood of 1. Pick a neighborhood V of 1 such that V 2 ⊆ G \ C. Put E = D ∩ V and A = f (E). We claim that for every m < ω, A ∩ Am is finite. To see this, assume the contrary. Then there exist xn ∈ E ∩ Dm and xk ∈ Un+1 ∩ E ∩ Dm . On the one hand, xn xk ∈ E 2 ⊆ V 2 , and on the other hand, xn xk ∈ xn (Un+1 ∩ Dm ) ⊆ C, a contradiction. Notice that we would have |A ∩ Am | ≤ 1, and so f (p) would be a selective ultrafilter, if (Un )n<ω were a decreasing sequence of open neighborhoods of 1 with n<ω Un ∩ D = ∅ such that the sets yn Un+1 , where n < ω and yn ∈ (Un \ Un+1 ) ∩ D, are pairwise disjoint. Indeed, otherwise there exist n < k < ω, yn ∈ (Un \ Un+1 ) ∩ Dm ∩ E and yk ∈ (Uk \ Uk+1 ) ∩ Dm ∩ E, and then, on the one hand, yn yk ∈ E 2 ⊆ V 2 , and on the other hand, yn yk ∈ yn (Un+1 ∩Dm ) ⊆ C, a contradiction. It remains to show how to arrive at this situation supposing that D is a convergent sequence is some weaker group topology T0 on G. Construct inductively a decreasing sequence (Un )n<ω of open neighborhoods of 1 in T0 such that n<ω Un ∩ D = ∅ and the sets yn Un+2 , where n < ω and yn ∈ (Un \ Un+1 ) ∩ D, are pairwise disjoint and contained in U . We can do this because D is a convergent sequence in T0 , and so (Un \ Un+1 ) ∩ D is finite. Let D = n<ω (U2n \ U2n+1 ) ∩ D and D = n<ω (U2n+1 \ U2n+2 ) ∩ D. The sets D and D form a partition of D, so one of them is a member of p. Without loss of generality one may suppose that D ∈ p. Let Un = U2n . Then the sets yn Un+1 , where n < ω and yn ∈ (Un \ Un+1 ) ∩ D , are pairwise disjoint and contained in U . 2
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Corollary 4.2. Let G be an extremally disconnected topological group and let D be a countable discrete nonclosed subset of G. Then there is a mapping f : G → ω such that f (p) is a P -point. Furthermore, if D is a convergent sequence in some weaker group topology on G, then f can be chosen so that f (p) is a selective ultrafilter. The first part of Corollary 4.2 is the result from [10]. From the second part of Corollary 4.2 and Theorem 2.1 we obtain the result from [8]. Corollary 4.3. Assume there is no rapid ultrafilter. Then there is no countable nondiscrete extremally disconnected topological group that is maximally almost periodic. Proof. Let G be a countable nondiscrete extremally disconnected maximally almost periodic topological group. If G contains no discrete nonclosed subset being a convergent sequence in some weaker totally bounded group topology, then by Theorem 2.1, there is a rapid ultrafilter. Otherwise by Corollary 4.2, there is a selective ultrafilter. 2 References [1] A. Arhangel’ski˘ı, Every extremally disconnected bicompactum of weight c is inhomogeneous, Sov. Math. Dokl. 8 (1967) 897–900. [2] A. Arhangel’ski˘ı, Cardinal invariants of topological groups. Embeddings and condensations, Sov. Math. Dokl. 20 (1979) 783–787. [3] E. Hewitt, K. Ross, Abstract Harmonic Analysis, I, Springer-Verlag, 1979. [4] N. Hindman, D. Strauss, Algebra in the Stone–Čech Compactification, De Gruyter, Berlin, 1998. [5] A. Miller, There are no Q-points in Laver’s model for the Borel conjecture, Proc. Am. Math. Soc. 78 (1980) 103–106. [6] I. Protasov, Discrete subsets of topological groups, Math. Notes 55 (1994) 101–102. [7] S. Shelah, Proper and Improper Forcing, Springer, Berlin, 1998. [8] O. Sipacheva, Nonexistence of countable extremally disconnected groups with many open subgroups, Topol. Appl. 179 (2015) 193–199. [9] Y. Zelenyuk, Ultrafilters and Topologies on Groups, De Gruyter, Berlin, 2011. [10] Y. Zelenyuk, On extremally disconnected topological groups, Topol. Appl. 153 (2006) 177–181. [11] Y. Zelenyuk, Local homomorphisms of topological groups, J. Aust. Math. Soc. 83 (2007) 135–148.
Contents lists available at ScienceDirect
Topology and its Applications www.elsevier.com/locate/topol
Discrete subsets and convergent sequences in topological groups Valentin Keyantuo a , Yevhen Zelenyuk b,∗,1 a
Department of Mathematics, University of Puerto Rico, PO Box 70377, San Juan, PR 00936-8377, USA b School of Mathematics, University of the Witwatersrand, Private Bag 3, Wits 2050, Johannesburg, South Africa
a r t i c l e
i n f o
Article history: Received 28 October 2014 Received in revised form 29 May 2015 Accepted 31 May 2015 Available online xxxx MSC: primary 03E35, 54H11 secondary 22A05, 54G05
a b s t r a c t We show that (1) assuming there is no rapid ultrafilter, every countable nondiscrete maximally almost periodic topological group contains a discrete nonclosed subset which is a convergent sequence in some weaker totally bounded group topology, (2) every infinite Abelian totally bounded topological group contains such a discrete subset (in ZFC), and (3) if an extremally disconnected topological group contains such a discrete subset, then there is a selective ultrafilter. © 2015 Elsevier B.V. All rights reserved.
Keywords: Topological group Maximally almost periodic Totally bounded Extremally disconnected Discrete subset Convergent sequence Rapid ultrafilter P -point Selective ultafilter
1. Introduction In [8], O. Sipacheva obtained partial solutions to Arhangel’ski˘ı’s question of whether there exists in ZFC a countable nondiscrete extremally disconnected topological group [1] and to Protasov’s question of whether there exists in ZFC a countable nondiscrete topological group in which every discrete subset is closed [6]. It had been known previously that the existence of an extremally disconnected topological group containing a countable discrete nonclosed subset implies the existence of a P -point, and similar result holds for a nondiscrete group topology T on ω Z2 which is finer than the direct sum topology, has no discrete subset * Corresponding author. 1
E-mail addresses: [email protected] (V. Keyantuo), [email protected] (Y. Zelenyuk). The second author was supported by NRF grant IFR2011033100072.
http://dx.doi.org/10.1016/j.topol.2015.05.089 0166-8641/© 2015 Elsevier B.V. All rights reserved.
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V. Keyantuo, Y. Zelenyuk / Topology and its Applications 191 (2015) 137–142
with a unique accumulation point, and satisfies one more condition [10]. (The direct sum topology is the one induced by the product topology.) Sipacheva showed that the existence of a nondiscrete extremally disconnected group topology on ω Z2 finer than the direct sum topology implies the existence of a rapid ultrafilter, and similar result holds for a group topology on ω Z2 which is finer than the direct sum topology and has no discrete subset with a unique accumulation point. In this paper we show that (1) assuming there is no rapid ultrafilter, every countable nondiscrete maximally almost periodic topological group contains a discrete nonclosed subset which is a convergent sequence in some weaker totally bounded group topology, (2) every infinite Abelian totally bounded topological group contains such a discrete subset (in ZFC), and (3) the existence of an extremally disconnected topological group containing such a discrete subset implies the existence of a selective ultrafilter. Notice that such a subset has a unique accumulation point and converges to that point in that weaker topology. Recall that a nonprincipal ultrafilter p on ω is (i) a P -point if for every partition {An : n < ω} of ω with An ∈ / p, there is A ∈ p such that |A ∩ An | < ω for all n, (ii) selective if for every partition {An : n < ω} of ω with An ∈ / p, there is A ∈ p such that |A ∩ An | ≤ 1 for all n, and (iii) rapid if for every partition {An : n < ω} of ω with finite An , there is A ∈ p such that |A ∩ An | ≤ n for all n. Clearly, every selective ultrafilter is rapid. MA implies the existence of P -points and rapid ultrafilters. However, it is consistent with ZFC that there is no P -point [7, VI, §4], and it is consistent with ZFC that there is no rapid ultrafilter [5]. A topological group is totally bounded (maximally almost periodic) if it can be topologically and algebraically (continuously and algebraically) embedded into a compact group. All topologies are assumed to be Hausdorff. 2. Rapid ultrafilters Every group topology on a countable group can be weakened to a metrizable (= first countable) group topology [2]. Consequently, every maximally almost periodic group topology on a countable group can be weakened to a metrizable totally bounded group topology. In this section we prove the following result. Theorem 2.1. Assume there is no rapid ultrafilter. Let (G, T ) be a countable nondiscrete maximally almost periodic topological group and let T0 be a metrizable totally bounded group topology on G weaker than T . Then (G, T ) contains a discrete nonclosed subset which is a convergent sequence in T0 . As a consequence we obtain that Corollary 2.2. It is consistent with ZFC that every countable nondiscrete maximally almost periodic topological group contains a discrete nonclosed subset which is a convergent sequence in some weaker totally bounded group topology. The proof of Theorem 2.1 involves the following result [11, Theorem 3.1]. Theorem 2.3. Let G be a countably infinite metrizable totally bounded topological group. Then there exist a sequence (mn )n<ω of integers ≥ 2 and a homeomorphism h : G → n<ω Zmn with h(1) = 0 such that
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(a) h(xy) = h(x)+h(y) whenever φ(x)+2 ≤ θ(y), where φ(x) = max supp(h(x)) and θ(y) = min supp(h(y)), (b) φ(xi y j ) ≤ max{φ(x), φ(y)} + 1 for all i, j ∈ {1, −1}, and if |φ(x) − φ(y)| ≥ 2, then φ(xi y j ) ≥ max{φ(x), φ(y)} − 1, and (c) if m < ω and h(x)(n) = h(y)(n) for each n ≤ m, then θ(x−1 y), θ(xy −1 ) ≥ m. Here, n<ω Zmn is endowed with the direct sum topology, and the functions φ and θ are not defined at x = 1. Notice that in [11, Theorem 3.1] property (c) reads as θ(x−1 y), θ(xy −1 ) ≥ m − 1, but the proof shows that in fact θ(x−1 y), θ(xy −1 ) ≥ m holds (although this small improvement plays no role in our result). The next theorem is a more precise statement of Theorem 2.1. Theorem 2.4. Let (G, T ) be a countable nondiscrete maximally almost periodic topological group and let T0 be a metrizable totally bounded group topology on G weaker than T . Suppose (G, T ) contains no discrete nonclosed subset which is a convergent sequence in T0 . Let h : G → n<ω Zmn be a mapping guaranteed by Theorem 2.3. Then for every ultrafilter p on G converging to 1 in T , the ultrafilter φ(p) on ω is rapid. Here, φ(p) is the ultrafilter on ω with a base consisting of subsets φ(A), where A ∈ p. In the proof of Theorem 2.4 we use the following characterization of rapid ultrafilters [5]: An ultrafilter p on ω is rapid if and only if there is a function h : ω → ω with the property that, whenever f : ω → ω, there exists A ∈ p such that for every n < ω, |A ∩ f (n)| ≤ h(n), where A ∩ f (n) = A ∩ {0, 1, . . . , f (n) − 1}. Proof. Let f : ω → ω be given. Without loss of generality one may suppose that f is increasing. Define U ⊆ G by U = {1} ∪ {x ∈ G : f (θ(x)) < φ(x)}. It then follows that U is a neighborhood of 1 in T . Indeed, assume the contrary. Then 1 is an accumulation point of G \ U = {x ∈ G : f (θ(x)) ≥ φ(x)}. But for every n < ω, there are only finitely many x ∈ G \ U with θ(x) = n, because φ(x) ≤ f (θ(x)). Consequently, G \ U is a sequence converging to 1 in T0 , a contradiction. Pick a neighborhood V of 1 in T such that V V −1 ⊆ U . We claim that for every n < ω, |φ(V ) ∩ f (n)| ≤
mi .
i≤n
To see this, assume the contrary. Then there exist distinct x, y ∈ V \ {1} such that φ(x), φ(y) < f (n) and h(x)(i) = h(y)(i) for each i ≤ n. Let z = xy −1 . Then 1 = z ∈ U , θ(z) ≥ n, and φ(z) ≤ f (n). Hence, f (n) ≤ f (θ(z)) < φ(z) ≤ f (n), a contradiction. 2 3. Totally bounded topological groups In [6] it was shown that every infinite totally bounded topological group contains a countable discrete nonclosed subset. In this section we derive from Theorem 2.4 the following result.
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Theorem 3.1. Let (G, T ) be a countably infinite totally bounded topological group and let T0 be a metrizable group topology on G weaker than T . Then (G, T ) contains a discrete nonclosed subset which is a convergent sequence in T0 . Given a group G and a left invariant topology T on G, Ult(T ) ⊆ βG consists of all nonprincipal ultrafilters on G converging to 1 in T and is called the ultrafilter semigroup of T (see [9, Section 7.1]). Here, βG is the Stone–Čech compactification of G. The points of βG are the ultrafilters on G and the topology is generated by taking as a base the subsets A = {p ∈ βG : A ∈ p}, where A ⊆ G. For any p, q ∈ βG, the ultrafilter pq ∈ βG has a base consisting of subsets x∈A xBx , where A ∈ p and Bx ∈ q for every x ∈ A. See [4] and [9] for more information about βG. Lemma 3.2. Let (G, T ) be a countably infinite metrizable totally bounded topological group, let h : (G, T ) → ∗ n<ω Zmn be a mapping guaranteed by Theorem 2.1, and let S = Ult(T ). Then φ(S) = ω , and for every u ∈ ω ∗ , φ−1 (u) ∩ S is a closed left ideal of S. Proof. For every n < ω, pick xn ∈ G with supp(h(x)) = {n}. The sequence (xn )n<ω converges to 1 in T . It then follows that φ(S) = ω ∗ . Now let p ∈ S and q ∈ φ−1 (u) ∩ S. Then pq ∈ S, and we claim that φ(pq) = u. To see this, let A ∈ u. Put P = G \ {1}, and for every x ∈ P , pick Qx ∈ q such that φ(Qx ) ⊆ A and φ(x) + 2 ≤ θ(y) for all y ∈ Qx , so φ(xy) = φ(y). Let R = x∈P xQx . Then R ∈ pq and φ(R) ⊆ A. 2 Proof of Theorem 3.1. Let h : (G, T0 ) → n<ω Zmn be a mapping guaranteed by Theorem 2.1 and let S = Ult(T ) and S0 = Ult(T0 ). By Lemma 3.2, φ(S0 ) = ω ∗ , and for every u ∈ ω ∗ , φ−1 (u) ∩ S0 is a closed left ideal of S0 , so φ−1 (u) ∩ S0 contains an idempotent (see [9, Theorem 6.12]). Since S contains all the idempotents of G∗ [9, Lemma 7.10], it follows that φ(S) = ω ∗ . Consequently, there are ultrafilters in φ(S) which are not rapid. Hence by Theorem 2.3, (G, T ) contains a discrete nonclosed subset which is a convergent sequence in T0 . 2 Now we are going to extend Theorem 3.1 to all infinite Abelian totally bounded topological groups. Lemma 3.3. Let (G, T ) be an Abelian totally bounded topological group, let H be a subgroup of G, and let T0 be a group topology on H weaker than T |H . Then there is a group topology T1 on G weaker than T such that T1 |H = T0 . Proof. Let Γ denote the set of all characters of (G, T ) (= continuous homomorphisms ϕ : (G, T ) → T) and Γ0 the set of all characters of (H, T0 ) and let Γ1 = {ϕ ∈ Γ : ϕ|H ∈ Γ0 }. Then {ϕ|H : ϕ ∈ Γ1 } = Γ0 , and for every a ∈ G \ H, there is ϕ ∈ Γ1 such that ϕ(a) = 1 ([3, Corollary 24.12]). Let T1 be the coarsest topology on G in which each character from Γ1 is continuous. 2 Theorem 3.4. Every infinite Abelian totally bounded topological group contains a discrete nonclosed subset which is a convergent sequence in some weaker group topology. Proof. Let (G, T ) be an infinite Abelian totally bounded topological group. Pick a countably infinite subgroup H of G and a metrizable group topology T0 on H weaker than T |H . By Lemma 3.3, there is a group topology T1 on G weaker than T such that T1 |H = T0 . By Theorem 3.1, there is a sequence (xn )n<ω in H having 1 as an accumulation point in T |H and converging to 1 in T0 . But then (xn )n<ω converges to 1 in T1 , and consequently, 1 is the unique accumulation point of (xn )n<ω in T . 2 Using the same argument as in the proof of Theorem 3.4, we obtain the following result.
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Theorem 3.5. Assume there is no rapid ultrafilter. Then every nondiscrete Abelian maximally almost periodic topological group containing a countable nondiscrete subset has a discrete nonclosed subset which is a convergent sequence in some weaker totally bounded group topology. Corollary 3.6. It is consistent with ZFC that every nondiscrete Abelian maximally almost periodic topological group containing a countable nondiscrete subset has a discrete nonclosed subset which is a convergent sequence in some weaker totally bounded group topology. 4. P -points and selective ultrafilters Let X be a space, let D be a discrete subset of X, and let x ∈ X be an accumulation point of D. We say that D is locally maximal with respect to x if there is an open U ⊆ X containing D such that, whenever C ⊆ U \ D and D ∪ C ⊆ U is discrete, x is not an accumulation point of C. Notice that if X is extremally disconnected and D is countable, then D is locally maximal with respect to x and there is only one ultrafilter on X containing D and converging to x (see [9, Lemma 5.13]). Theorem 4.1. Let G be a topological group and let D be a countable discrete subset of G with 1 being an accumulation point of D. Suppose that D is locally maximal with respect to 1 and there is only one ultrafilter p on G containing D and converging to 1. Then there is a mapping f : G → ω such that f (p) is a P -point. Furthermore, if D is a convergent sequence in some weaker group topology on G, then f can be chosen so that f (p) is a selective ultrafilter. Proof. Let U be an open neighborhood of D ⊆ X witnessing that D is locally maximal with respect to 1. Enumerate D as {xn : n < ω}. Construct inductively a decreasing sequence (Un )n<ω of open neighborhoods of 1 with U0 = G such that xn ∈ / Un+1 and the subsets xn Un+1 , where n < ω, are pairwise disjoint and contained in U . Define f : D → ω by f (x) = n if x ∈ Un \ Un+1 , and let q = f (p). Since n<ω Un ∩ D = ∅, q is a nonprincipal ultrafilter on ω. We prove that q is a P -point. / q and let Dm = f −1 (Am ). Then {Dm : m < ω} is a Let {Am : m < ω} be a partition of ω with Am ∈ partition of D with Dm ∈ / p, so 1 is not an accumulation point of Dm . Every element xn ∈ D belongs to some subset Dmn of the partition. Put C = n<ω xn (Un+1 ∩ Dmn ). Then C is disjoint from D and D ∪ C is a discrete subset of U . Since D is locally maximal, it follows that G \ C is a neighborhood of 1. Pick a neighborhood V of 1 such that V 2 ⊆ G \ C. Put E = D ∩ V and A = f (E). We claim that for every m < ω, A ∩ Am is finite. To see this, assume the contrary. Then there exist xn ∈ E ∩ Dm and xk ∈ Un+1 ∩ E ∩ Dm . On the one hand, xn xk ∈ E 2 ⊆ V 2 , and on the other hand, xn xk ∈ xn (Un+1 ∩ Dm ) ⊆ C, a contradiction. Notice that we would have |A ∩ Am | ≤ 1, and so f (p) would be a selective ultrafilter, if (Un )n<ω were a decreasing sequence of open neighborhoods of 1 with n<ω Un ∩ D = ∅ such that the sets yn Un+1 , where n < ω and yn ∈ (Un \ Un+1 ) ∩ D, are pairwise disjoint. Indeed, otherwise there exist n < k < ω, yn ∈ (Un \ Un+1 ) ∩ Dm ∩ E and yk ∈ (Uk \ Uk+1 ) ∩ Dm ∩ E, and then, on the one hand, yn yk ∈ E 2 ⊆ V 2 , and on the other hand, yn yk ∈ yn (Un+1 ∩Dm ) ⊆ C, a contradiction. It remains to show how to arrive at this situation supposing that D is a convergent sequence is some weaker group topology T0 on G. Construct inductively a decreasing sequence (Un )n<ω of open neighborhoods of 1 in T0 such that n<ω Un ∩ D = ∅ and the sets yn Un+2 , where n < ω and yn ∈ (Un \ Un+1 ) ∩ D, are pairwise disjoint and contained in U . We can do this because D is a convergent sequence in T0 , and so (Un \ Un+1 ) ∩ D is finite. Let D = n<ω (U2n \ U2n+1 ) ∩ D and D = n<ω (U2n+1 \ U2n+2 ) ∩ D. The sets D and D form a partition of D, so one of them is a member of p. Without loss of generality one may suppose that D ∈ p. Let Un = U2n . Then the sets yn Un+1 , where n < ω and yn ∈ (Un \ Un+1 ) ∩ D , are pairwise disjoint and contained in U . 2
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Corollary 4.2. Let G be an extremally disconnected topological group and let D be a countable discrete nonclosed subset of G. Then there is a mapping f : G → ω such that f (p) is a P -point. Furthermore, if D is a convergent sequence in some weaker group topology on G, then f can be chosen so that f (p) is a selective ultrafilter. The first part of Corollary 4.2 is the result from [10]. From the second part of Corollary 4.2 and Theorem 2.1 we obtain the result from [8]. Corollary 4.3. Assume there is no rapid ultrafilter. Then there is no countable nondiscrete extremally disconnected topological group that is maximally almost periodic. Proof. Let G be a countable nondiscrete extremally disconnected maximally almost periodic topological group. If G contains no discrete nonclosed subset being a convergent sequence in some weaker totally bounded group topology, then by Theorem 2.1, there is a rapid ultrafilter. Otherwise by Corollary 4.2, there is a selective ultrafilter. 2 References [1] A. Arhangel’ski˘ı, Every extremally disconnected bicompactum of weight c is inhomogeneous, Sov. Math. Dokl. 8 (1967) 897–900. [2] A. Arhangel’ski˘ı, Cardinal invariants of topological groups. Embeddings and condensations, Sov. Math. Dokl. 20 (1979) 783–787. [3] E. Hewitt, K. Ross, Abstract Harmonic Analysis, I, Springer-Verlag, 1979. [4] N. Hindman, D. Strauss, Algebra in the Stone–Čech Compactification, De Gruyter, Berlin, 1998. [5] A. Miller, There are no Q-points in Laver’s model for the Borel conjecture, Proc. Am. Math. Soc. 78 (1980) 103–106. [6] I. Protasov, Discrete subsets of topological groups, Math. Notes 55 (1994) 101–102. [7] S. Shelah, Proper and Improper Forcing, Springer, Berlin, 1998. [8] O. Sipacheva, Nonexistence of countable extremally disconnected groups with many open subgroups, Topol. Appl. 179 (2015) 193–199. [9] Y. Zelenyuk, Ultrafilters and Topologies on Groups, De Gruyter, Berlin, 2011. [10] Y. Zelenyuk, On extremally disconnected topological groups, Topol. Appl. 153 (2006) 177–181. [11] Y. Zelenyuk, Local homomorphisms of topological groups, J. Aust. Math. Soc. 83 (2007) 135–148.