Open research questions in Advanced Topology and Set Theory

What the literature leaves open

• The authors suggest exploring the relationship between ultrafilters and the Galvin property further. The paper proposes studying the Galvin property for ultrafilters on Pκ(λ) in more detail.

  SUPERCOMPACT MEASURES AND THE GALVIN PROPERTY · 2026 · DOI

• Future research could focus on extending the results of the paper to other classes of monoids. The study of the properties of strongly sofic monoids and their actions on compact spaces could be continued.

  Strongly sofic monoids, sofic topological entropy, and surjunctivity · 2026 · DOI

• There is a gap in the understanding of the properties of strongly sofic monoids, particularly with regards to their surjunctivity and topological entropy. The paper fills this gap by defining and investigating sofic topological entropy for actions of strongly sofic monoids on compact spaces.

  Strongly sofic monoids, sofic topological entropy, and surjunctivity · 2026 · DOI

• To obtain an exact formula for the cardinality of Kakeya sets. To extend the method of multiplicities to analyze other types of sets in finite fields. To study the geometric properties of Kakeya sets in finite fields.

  On an analogue of BRK-type sets in finite fields · 2026 · DOI

• The problem of finding a lower bound on the cardinality of Kakeya sets in finite fields was previously unsolved. The method of multiplicities had not been extended to analyze (n, d)-BRK-type sets.

  On an analogue of BRK-type sets in finite fields · 2026 · DOI

• The paper identifies a gap in the prior work by Bartoszyński and Shelah, which only partially solved the problem. The paper aims to provide a complete characterization of small sets and Fσ measure zero sets.

  Cardinal characteristics associated with small subsets of reals · 2026 · DOI

• The paper identifies a gap in understanding linear homeomorphisms between function spaces for spaces with one non-isolated point. The authors note that prior work has not fully addressed this gap. The paper aims to fill this gap by investigating these homeomorphisms using free filters on the countable discrete space ω.

  Function spaces and ultrafilters · 2026 · DOI

• The paper identifies a gap in the study of local characterisation of Sidon sets, showing that such a characterisation fails. The paper also identifies a gap in the study of B k,ℓ -sets, showing that the implication of a local characterisation fails already for δ = 1/4.

  Ramsey-type problems for generalised Sidon sets · 2026 · DOI

• The gap identified is the lack of generalization of earlier results to an unstable context. Specifically, prior work did not address infinite dimensional groups.

  On groups definable in geometric fields with generic derivations · 2026 · DOI

• The paper identifies a gap in current research on the Galvin property for ultrafilters on Pκ(λ). The authors note that the Galvin property is not well understood for ultrafilters on Pκ(λ).

  SUPERCOMPACT MEASURES AND THE GALVIN PROPERTY · 2026 · DOI

• The classification of the epimorphism relation among countable groups was unknown. The epimorphism relation on pointed reflexive graphs was not previously classified.

  The epimorphism relation among countable groups is a complete analytic quasi‐order · 2026 · DOI

• Problem 1. Is it consistent that there are exactly ℵ0-many Q-points? Note that a positive answer would require all but finitely many of these Q-points to be non-Ramsey, since for any countable set {Uk : k ∈ ω} of pairwise non-isomorphic Ramsey ultrafilters, the ultrafilter U0- (cid:88) k>0 Uk := {X ⊆ ω × ω : {k > 0 : {i ∈ ω : ⟨k, i⟩ ∈ X} ∈ Uk} ∈ U0} on ω ×ω is a non-Ramsey Q-point (see [6, Section 2.1]). Hence, by partitioning {Uk : k > 0} into an almost disjoint family of size c, each U0-indexed sum over one of the c-many pieces yields a Q-point, and these will be pairwise non-isomorphic by an old result due to Rudin ([20]). This suggests the following second question: Problem 2. Is it consistent that there is a unique Q-point, and it is non-Ramsey?

  There may be exactly n Q-points · 2026 · DOI

• The paper suggests that future research should focus on further developing the theory of equi-families of functions. It recommends investigating the properties of equi-families in different topological and metric spaces. The study proposes exploring the applications of equi-families in various fields, including mathematics, physics, and engineering.

  Borel 1 type mappings and the respective equi-families · 2026 · DOI

• The paper identifies a gap in the current understanding of equi-families of functions. It notes that the study of equi-families is an underdeveloped area of research. The paper aims to address this gap by investigating the properties of equi-families and their closure relative to the topology of pointwise convergence.

  Borel 1 type mappings and the respective equi-families · 2026 · DOI

• The paper identifies a gap in the understanding of the Bousfield-Kan R-completion and its applications. The paper leaves one case open in the non-orientable case.

  Bousfield-Kan completion and very large groups · 2026 · DOI

• The paper suggests that future research should focus on further understanding the properties of strong ergodicity. The paper suggests that future research should explore the applications of the results to other areas of mathematics and science.

  Coamenability and strong ergodicity · 2026

• The paper identifies a gap in the understanding of strong ergodicity for coamenable inclusions of ergodic, probability measure-preserving relations. The paper identifies a need for a characterization of strong ergodicity for such inclusions.

  Coamenability and strong ergodicity · 2026

• The paper identifies the gap in understanding ultrapowers in von Neumann algebras. The authors note that it is a delicate and important question how much of the *-algebra structure can be preserved by the lifted sequences.

  Elementary embeddings into ultrapower $\mathrm{II}_1$ factors without a ucp lift · 2026 · DOI

• To resolve the question of whether MA implies the failure of the Lipschitz axioms. To fully explore the implications between the different Lipschitz axioms.

  VARIANTS OF BAUMGARTNER’S AXIOM FOR LIPSCHITZ FUNCTIONS ON BAIRE AND CANTOR SPACE · 2026 · DOI

• We finish the paper by collecting some open questions which have appeared throughout the text. The first of these is whether there are more implications between the four axioms we have been discussing. Question 1. Are there further implications between BALip(ωω), BALip(2ω), BALip(ωω), and BALip(2ω) than those given in Theorem 3.4? In particular are they all equivalent? Do the weak and strong versions equate? Do the Baire and Cantor versions equate? We can also ask whether the implications of the stronger Lipschitz axioms follow from the weaker ones. Question 2. Do either BALip(ωω) or BALip(2ω) imply 2ℵ0 = 2ℵ1? Do either of BALip(ωω) or BALip(2ω) imply add(N ) > ℵ1? Similarly we note that we still have not resolved the analogue of the question asked in [15] regarding p. Question 3. Do any of the Lipschitz axioms imply p > ℵ1? We would also like to know whether MA itself, and not just the large fragment considered in the previous section, does not imply the Lipschitz axioms. Question 4. Does MA imply BALip(2ω)? Finally floating in the background is the question of the relation between BA and the axioms considered in this paper. Question 5. Are there provable relations between BA and the Lipschitz variations?

  VARIANTS OF BAUMGARTNER’S AXIOM FOR LIPSCHITZ FUNCTIONS ON BAIRE AND CANTOR SPACE · 2026 · DOI

• The need to extend Proposition 2.4 from group actions to groupoids. The lack of understanding of measurewise amenability for locally compact groupoids.

  Appendix to “Amenable actions of real and <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>p</mml:mi> </mml:math> -adic algebraic groups” by Alain Valette · 2026 · DOI

• The gap is the question of whether a proper dense subgroup of the symmetric group can have a transitive action on certain collections of models. The paper identifies this gap in the context of universal algebra and model theory.

  A permutation group acting transitively on certain collections of models · 2026 · DOI

• Investigating the connection between the part-whole principle and the quantity ded κ further. Exploring the implications of Corollary 9 in different situations.

  A note on generalized probability functions and the part-whole principle · 2026 · DOI

• The extent to which the existence of regular total generalized probabilities can be characterized in terms of the cardinalities of the associated sets Ω and V. The connection between the part-whole principle and the quantity ded κ.

  A note on generalized probability functions and the part-whole principle · 2026 · DOI

• Whether a tall Borel ideal can have the Ramsey property is an open question of Hrušák, Meza-Alcántara, Thümmel and Uzcátegui; a coanalytic example exists in ZFC, so a negative answer must use definability essentially.

  Reductions and necessary conditions for tall Borel Ramsey ideals · 2026

Most-cited papers in Advanced Topology and Set Theory

• What is Cantor's Continuum Problem? · American Mathematical Monthly · 1947 · 161 citations
• Partial justification of the Borda count · Public Choice · 1976 · 106 citations
• Preference structures I: Distances between transitive preference relations† · Journal of Mathematical Sociology · 1973 · 99 citations
• The Copeland choice function an axiomatic characterization · Social Choice and Welfare · 1985 · 60 citations
• The Staudt-Clausen Theorem · Mathematics Magazine · 1961 · 29 citations
• On the relation provable equivalence and on partitions in effectively inseparable sets · Studia Logica · 1981 · 24 citations
• The completeness of S · Studia Logica · 1979 · 21 citations
• Wide Sets, ZFCU, and the Iterative Conception · The Journal of Philosophy · 2014 · 16 citations
• Heidegger on intersubjectivity · Human Studies · 1980 · 11 citations
• Individualizing the Multiproblem Family · Social Casework · 1963 · 8 citations

Most recent work

• Appendix to “Amenable actions of real and <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>p</mml:mi> </mml:math> -adic algebraic groups” by Alain Valette · Annales de la faculté des sciences de Toulouse Mathématiques · 2026
• Genuinely nonlocal sets with smallest cardinality · Physical Review A · 2026
• The epimorphism relation among countable groups is a complete analytic quasi‐order · Bulletin of the London Mathematical Society · 2026
• There may be exactly n Q-points · Archive for Mathematical Logic · 2026
• Borel 1 type mappings and the respective equi-families · Topology and its Applications · 2026
• SUPERCOMPACT MEASURES AND THE GALVIN PROPERTY · The Journal of Symbolic Logic · 2026
• Bousfield-Kan completion and very large groups · Proceedings of the American Mathematical Society · 2026
• Elementary embeddings into ultrapower $\mathrm{II}_1$ factors without a ucp lift · Journal of Logic and Analysis · 2026
• Amorphous sets and dual Dedekind finiteness · Annals of Pure and Applied Logic · 2026
• VARIANTS OF BAUMGARTNER’S AXIOM FOR LIPSCHITZ FUNCTIONS ON BAIRE AND CANTOR SPACE · Journal of Symbolic Logic · 2026

Related topics in Mathematics