Science AI Journal
Mathematics · Research topic

Open research questions in Advanced Topology and Set Theory

92 gap statements mined from Advanced Topology and Set Theory papers in our 4.5M-paper local library, which holds 725 papers on the topic — drawn mostly from each paper's own stated research gap, future-work, challenge and limitation notes, and its abstract. The ones listed below are a selection still marked open; each names the study that raised it, with a DOI link where the paper has one.

What the literature leaves open

  • The need to determine the complexity of index sets for certain classes of structures. The lack of a criterion for almost primeness for linear orders elementarily equivalent to the ordinal omega n+1. The need to generalize Proposition 1 to characterize linear orders elementarily equivalent to omega n+1.

    Index Sets of Decidably Categorical Models that are Elementarily Equivalent to the Powers of Omega · 2026 · DOI

  • There is a lack of understanding of the homogeneity of certain ideals. The relationships between different ideals are not well understood. Prior work has not fully characterized the comparability of ideals in various orders.

    Homogeneity and orders between ideals · 2026 · DOI

  • It is an open question whether compositional truth with the principle of propositional soundness: “All arithmetical sentences which are propositional tautologies are true” is conservative over Peano Arithmetic.

    Saturation properties for compositional truth with propositional correctness · 2024 · 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

  • 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

  • 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 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 need for a characterization of strong ergodicity for such inclusions.

    Coamenability and strong ergodicity · 2026

  • 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?

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

  • The lack of a convenient analogue of 'nice names' for subsets of κ in the setting of nonstationary-support products. The need to code each element X ∈ (P(κ)) V [G] by a sequence ⃗ τ ∈ (V κ κ) V. The need to develop an 'interpretation procedure' that retrieves X from ⃗ τ.

    A note on the Ketonen order and Lipschitz reducibility between ultrafilters · 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

  • 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

  • The remainder of this section explores three consequences of this synthesis: a continuum-theoretic reading of Γ 3 via embeddings into standard continua, a measure-theoretic construction of canonical self-similar measures supported on Γ 3(α), and an outlook tying the framework to the open problems of Section ??.

    Descriptions of Cantor Sets: A Set‑Theoretic Survey and Open Problems · 2026 · DOI

  • The paper notes that prior definitions of OP, IP, and SOP involve ¬ϕ, while TP, TP1, and TP2 are positively definable.

    RESULTS ON COLORED TREE PROPERTIES · 2026 · DOI

  • 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

  • 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

  • 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

  • Explore the option of taking the index structure to be the free superposition of the tree structure with a Ramsey index structure I, - Study of a more general form of c-TP, - Investigate whether c-TP 1 is a strictly weaker notion than TP…

    RESULTS ON COLORED TREE PROPERTIES · 2026 · DOI

  • What is the possible complexity of the index set I(K dc) for a finitely axiomatizable class…

    Index Sets of Decidably Categorical Models that are Elementarily Equivalent to the Powers of Omega · 2026 · DOI

  • Along the way, we give a largely algebraic characterization of the back-and-forth relations on reduced Abelian p -groups, making progress on an open problem of Ash and Knight's from.

    Scott complexity of reduced Abelian p-groups · 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

  • Investigating the relationship between the Ketonen and Lipschitz orders in other models - Exploring the implications of the Weak Ultrapower Axiom on the set-theoretic…

    A note on the Ketonen order and Lipschitz reducibility between ultrafilters · 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

  • Chaos and Fractals By distilling these gaps into a concise list of open problems, the paper sets a clear agenda for future work in descriptive set theory and fractal analysis on the real line.

    Descriptions of Cantor Sets: A Set‑Theoretic Survey and Open Problems · 2026 · DOI

Most-cited papers in Advanced Topology and Set Theory

Most recent work

Find a gap in your own Advanced Topology and Set Theory sub-topic

This page shows what the Advanced Topology and Set Theory literature already flags as unresolved. To narrow it to your specific question, search the Research Gap Finder: the search is free with a free account and lists the papers closest to your topic first. Unlocking that topic (50 credits, charged once) fills the comparison table from our 4.5M-paper local library and writes the gaps from its rows.

Open the Research Gap Finder →

Related topics in Mathematics

Cognitive and developmental aspects of mathematical skills
817 open questions
Advanced Causal Inference Techniques
390 open questions
COVID-19 epidemiological studies
369 open questions
Navier-Stokes equation solutions
368 open questions
Nonlinear Partial Differential Equations
361 open questions
Geometric Analysis and Curvature Flows
358 open questions
Mathematics Education and Pedagogy
357 open questions
Statistics Education and Methodologies
348 open questions
Open questions raised by papers listed on this page include: (1) The lack of a criterion for almost primeness for linear orders elementarily equivalent to the ordinal omega n+1 (“Index Sets of Decidably Categorical Models that are Elementarily Equivalent to…”, 2026); (2) There is a lack of understanding of the homogeneity of certain ideals (“Homogeneity and orders between ideals”, 2026); (3) The paper suggests that future research should focus on further developing the theory of equi-families of functions (“Borel 1 type mappings and the respective equi-families”, 2026); (4) The lack of a convenient analogue of 'nice names' for subsets of κ in the setting of nonstationary-support products (“A note on the Ketonen order and Lipschitz reducibility between ultrafilters”, 2026); (5) The lack of understanding of measurewise amenability for locally compact groupoids (“Appendix to “Amenable actions of real and <mml:math…”, 2026). 92 gap statements have been mined from Advanced Topology and Set Theory papers in our 4.5M-paper local library, which holds 725 papers on the topic; the gaps come from whichever of those papers state one. They are mostly the research gaps the authors state and the papers' abstracts, plus future-work, limitations and challenges passages. The ones listed below are a selection still marked open; each names the study that raised it, with a DOI link where the paper has one, so you can read the original claim in context.

Tools for your next paper

Gaps & Literature SearchFind open questions in any field — 120,000+ open gaps mined from our own library.Pre-CheckIs your paper ready? Acceptance odds for top, mid and open/emerging journals, in seconds.Journal FinderA ranked shortlist of target journals from a 17,500-venue index.Duplicate Publication CheckerCheck for prior publication across 7 sources and our local library.Citation GeneratorFreeA DOI or title → APA, MLA, Vancouver, Chicago, BibTeX, RIS. No signup.Graphical Abstract MakerFreeTurn your findings into a colorblind-safe graphical abstract.

How Pre-Check worksHow AI Review works

Compare the category — Honest roundups of the AI research tools, ours listed alongside the alternatives.

Command palette

Jump anywhere, run any action.