Dense partitions of groups

DefinitionProtasov's question on absolutely dense partitions (Kourovka 13.47)#✓fully proved

The proposition that every countable abelian group \(G\) with only finitely many elements of order \(2\) can be partitioned into two disjoint sets \(A_1\) and \(A_2\) with \(A_1 \cup A_2 = G\) such that \(A_1\) and \(A_2\) are both dense in every non-discrete Hausdorff group topology on \(G\).

def countableAbsolutelyDensePartition : Prop :=
  ∀ (G : Type) [CommGroup G], Countable G → {g : G | g ≠ 1 ∧ g ^ 2 = 1}.Finite →
    ∃ A₁ A₂ : Set G, A₁ ∪ A₂ = Set.univ ∧ Disjoint A₁ A₂ ∧
      ∀ τ : GroupTopology G, @T2Space G τ.toTopologicalSpace → τ.toTopologicalSpace ≠ ⊥ →
        @Dense G τ.toTopologicalSpace A₁ ∧ @Dense G τ.toTopologicalSpace A₂

Used by 1

Used byKourovka Problem 13.47 has a positive answer

TheoremKourovka Problem 13.47 has a positive answer#●ready to prove

Every countable abelian group \(G\) with only finitely many elements of order \(2\) can be partitioned into two disjoint sets \(A_1\) and \(A_2\) with \(A_1 \cup A_2 = G\) such that \(A_1\) and \(A_2\) are both dense in every non-discrete Hausdorff group topology on \(G\).

The Kourovka Archive answers 13.47 "Yes, it can", citing Zelenyuk.

public theorem_wanted countableAbsolutelyDensePartition_holds : countableAbsolutelyDensePartition

Sources

• E. I. Khukhro and V. D. Mazurov (eds.), Unsolved Problems in Group Theory: The Kourovka Notebook, 21st issue, arXiv:1401.0300v49 (2026), Archive of Solved Problems, Problem 13.47 (I. V. Protasov); resolution: E. G. Zelenyuk, Ukrain. Math. J. 51 (1999), no. 1, 44-50, Kourovka Archive, Problem 13.47 (answer: Yes, it can).

Uses 1

Statement usesProtasov's question on absolutely dense partitions (Kourovka 13.47)

DefinitionExpansive sequence#✓fully proved

In a topological space \(G\), a sequence \((F_n)_{n \in \mathbb{N}}\) of finite subsets of \(G\) is expansive if the sets \(F_n\) are pairwise disjoint and for every nonempty open set \(U \subseteq G\) there is \(m \in \mathbb{N}\) such that \(F_n\) meets \(U\) for all \(n > m\).

def IsExpansiveSequence {G : Type*} [TopologicalSpace G] (F : ℕ → Finset G) : Prop :=
  Pairwise (fun i j => Disjoint (F i) (F j)) ∧
    ∀ U : Set G, IsOpen U → U.Nonempty → ∃ m : ℕ, ∀ n, m < n → ∃ x ∈ F n, x ∈ U

Used by 1

Used byKourovka Problem 15.80: dense partitions and expansive sequences

DefinitionKourovka Problem 15.80: dense partitions and expansive sequences#✓fully proved

Every topological group \(G\) admitting a sequence \((P_i)_{i \in \mathbb{N}}\) of pairwise disjoint dense subsets with union \(G\) has an expansive sequence.

def countableDensePartitionHasExpansiveSequence : Prop :=
  ∀ (G : Type) [Group G] [TopologicalSpace G] [IsTopologicalGroup G],
    (∃ P : ℕ → Set G, Pairwise (fun i j => Disjoint (P i) (P j)) ∧ (⋃ i, P i) = Set.univ ∧
      ∀ i, Dense (P i)) → ∃ F : ℕ → Finset G, IsExpansiveSequence F

Uses 1 · Used by 1

Statement usesExpansive sequence

Used byKourovka Problem 15.80 has a negative answer

TheoremKourovka Problem 15.80 has a negative answer#●ready to prove

It is not true that every topological group \(G\) admitting a sequence \((P_i)_{i \in \mathbb{N}}\) of pairwise disjoint dense subsets with union \(G\) has an expansive sequence, that is, a sequence \((F_n)_{n \in \mathbb{N}}\) of pairwise disjoint finite subsets of \(G\) such that every nonempty open set \(U \subseteq G\) meets \(F_n\) for all \(n\) larger than some \(m\).

S. Corson's counterexample: \(\prod_{\mathbb{N}} \mathbb{R}\) with the box topology has no countable dense subset, hence no expansive sequence (the union of one is countable and dense), but its base of \(2^{\aleph_0}\) open sets of size \(2^{\aleph_0}\) lets a transfinite induction partition it into countably many dense subsets.

public theorem_wanted countableDensePartitionHasExpansiveSequence_refuted :
    ¬ countableDensePartitionHasExpansiveSequence

Sources

• E. I. Khukhro and V. D. Mazurov (eds.), Unsolved Problems in Group Theory: The Kourovka Notebook, 21st issue, arXiv:1401.0300v49 (2026), Problem 15.80 (I. V. Protasov); resolution: S. Corson, letter of 19 May 2025, as recorded in the Kourovka Notebook (21st issue), Problem 15.80, G = ∏_ℕ ℝ with the box topology: not separable, hence without expansive sequences, but partitionable into countably many dense subsets by transfinite induction; the Kourovka Notebook (21st issue) marks 15.80 solved (answer: Not necessarily), https://arxiv.org/abs/1401.0300v49.

Uses 1

Statement usesKourovka Problem 15.80: dense partitions and expansive sequences

3 fully proved2 ready to prove

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