Datasets:
Proposed collection: Open problems from Santi Spadaro and collaborators
This discussion proposes a curated collection of 139 research problems extracted from papers by Santi Spadaro and collaborators, mainly in general topology and set-theoretic topology.
The goal is to contribute these problems to the UnsolvedMath dataset in a form that is mathematically clear, bibliographically traceable, and suitable for further verification by the community. For each item, I attempted to identify the precise statement, original source, attribution, year, exact location in the paper, relevant literature, and current status.
The collection is intentionally conservative with respect to status. Problems are marked as open, partially_solved, possibly_solved, or unknown depending on the strength of the available evidence. In particular, the appearance of a question in an older paper is not taken by itself as evidence that it remains open today.
I also tried to distinguish carefully between:
- problems originally proposed by the authors;
- questions attributed to other mathematicians;
- classical problems merely mentioned in a paper;
- repeated or reformulated versions of earlier questions;
- explicit Questions, Problems, and Conjectures versus editorially reconstructed formulations.
Possible duplicates and cases requiring additional verification are flagged rather than silently removed.
The records were prepared from the original academic sources whenever possible, with links and bibliographic information included for traceability. I am happy to adapt the contribution to the exact schema, conventions, or submission format preferred by the UnsolvedMath maintainers.
This discussion proposes a curated collection of 139 research problems extracted from papers by Santi Spadaro and collaborators, mainly in general topology and set-theoretic topology.
The goal is to contribute these problems to the UnsolvedMath dataset in a form that is mathematically clear, bibliographically traceable, and suitable for further verification by the community. For each item, I attempted to identify the precise statement, original source, attribution, year, exact location in the paper, relevant literature, and current status.
The collection is intentionally conservative with respect to status. Problems are marked as open, partially_solved, possibly_solved, or unknown depending on the strength of the available evidence. In particular, the appearance of a question in an older paper is not taken by itself as evidence that it remains open today.
I also tried to distinguish carefully between:
- problems originally proposed by the authors;
- questions attributed to other mathematicians;
- classical problems merely mentioned in a paper;
- repeated or reformulated versions of earlier questions;
- explicit Questions, Problems, and Conjectures versus editorially reconstructed formulations.
Possible duplicates and cases requiring additional verification are flagged rather than silently removed.
The records were prepared from the original academic sources whenever possible, with links and bibliographic information included for traceability. I am happy to adapt the contribution to the exact schema, conventions, or submission format preferred by the UnsolvedMath maintainers.
UNSOLVEDMATH CONTRIBUTION — SANTI SPADARO AND COLLABORATORS
Introductory note for the discussion
PROPOSED COLLECTION: PROBLEMS FROM SANTI SPADARO AND COLLABORATORS
Plain-text submission file for UnsolvedMath
Prepared: 2026-10-20
Total records: 139
EDITORIAL NOTE
This file contains a curated audit of research questions extracted from papers by Santi Spadaro and collaborators. A historical question is not automatically labelled open. Status values record the strength of the evidence found during the audit: open, partially_solved, possibly_solved, or unknown. The value unknown means that the current status was not certified and must not be interpreted as evidence that the problem remains open. Attribution fields are null when the original proposer could not be verified with sufficient confidence.
STATUS SUMMARY
open: 8
partially_solved: 5
possibly_solved: 9
unknown: 117
SUBMISSION-READINESS SUMMARY
do_not_submit_yet: 14
needs_verification: 121
ready: 4
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UNSOLVEDMATH SPADARO COLLECTION — COMMENT 1/46 — PROBLEMS 1–3
PROBLEM 1
Title: Normal functionally countable spaces and exponential separability
Statement: Let $X$ be a normal and functionally countable space. Does it follow that $X$ is exponentially separable?
Original statement: Let $X$ be a normal and functionally countable space. Does it follow that $X$ is exponentially separable?
Background: The source paper, “Functional countability and exponential separability of product spaces and subspaces,” poses this item as Question 3 in its study of general/set-theoretic topology. The August 2026 version of the source explicitly states that Question 3 remains open. The cleaned statement below is an editorially normalized version of the source formulation; the transcription retained in original_statement is kept separately for provenance.
Status: open
Proposed by: Rodrigo Hernández-Gutiérrez; Santi Spadaro
Proposed year: 2026
Category: Topology
Tags: general-topology; selection-principles; exponential-separability
Source URL: https://arxiv.org/abs/2603.02018
Source citation: Rodrigo Hernández-Gutiérrez, Santi Spadaro, “Functional countability and exponential separability of product spaces and subspaces,” arXiv:2603.02018 (2026); Question 3.
Exact location: {"label": "Question 3", "page": null}
arXiv: 2603.02018
References: https://arxiv.org/abs/2603.02018
Literature sources: Rodrigo Hernández-Gutiérrez and Santi Spadaro, “Functional countability and exponential separability of product spaces and subspaces,” arXiv:2603.02018v2 (2026).
Literature assessment: The August 2026 version of the source explicitly states that Question 3 remains open.
Literature checked at: 2026-10-20
Possible duplicate: null
Notes: Exact problem label verified from the prepared source audit; exact PDF page has not been independently verified in this pass unless explicitly stated otherwise.
Submission readiness: ready
Quality control: {"attribution_verified": "YES", "ready_to_submit": "YES", "source_verified": "YES", "status_verified": "YES"}
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PROBLEM 2
Title: Whether can $X$ be an increasing union of closed Menger subspaces if $\mathsf{U_{fin}}(\Delta_X,\mathcal O_X)$ holds but $\mathsf{U_{fin}}(\mathcal…
Statement: Can $X$ be an increasing union of closed Menger subspaces if $\mathsf{U_{fin}}(\Delta_X,\mathcal O_X)$ holds but $\mathsf{U_{fin}}(\mathcal O_X,\mathcal O_X)$ does not? Is it interesting to know the answer?
Original statement: Can $X$ be an increasing union of closed Menger subspaces if $\mathsf{U_{fin}}(\Delta_X,\mathcal O_X)$ holds but $\mathsf{U_{fin}}(\mathcal O_X,\mathcal O_X)$ does not? Is it interesting to know the answer?
Background: The source paper, “On (non-Menger) spaces whose closed nowhere dense subsets are Menger,” poses this item as Question 2.8 in its study of general/set-theoretic topology. The exact source item is identifiable, but this audit did not locate sufficiently strong later evidence to certify its status as of 20 October 2026; it is therefore marked unknown rather than assumed to remain open. The cleaned statement below is an editorially normalized version of the source formulation; the transcription retained in original_statement is kept separately for provenance.
Status: unknown
Proposed by: null
Proposed year: null
Category: Topology
Tags: general-topology; covering-properties; selection-principles
Source URL: https://arxiv.org/abs/2501.13220
Source citation: Mathieu Baillif, Santi Spadaro, “On (non-Menger) spaces whose closed nowhere dense subsets are Menger,” arXiv:2501.13220 (2025); Question 2.8.
Exact location: {"label": "Question 2.8", "page": null}
arXiv: 2501.13220
References: https://arxiv.org/abs/2501.13220
Literature sources: []
Literature assessment: null
Literature checked at: 2026-10-20
Possible duplicate: null
Notes: Exact problem label verified from the prepared source audit; exact PDF page has not been independently verified in this pass unless explicitly stated otherwise. Original proposer attribution has not been established with enough confidence; proposed_by is therefore null unless independently verified. unknown means that the current status was not certified; it must not be interpreted as evidence that the problem is open. This source item is programmatic/rhetorical or asks for a characterization rather than a single sharply testable proposition; editorial decomposition is recommended before catalog submission.
Submission readiness: do_not_submit_yet
Quality control: {"attribution_verified": "NO", "ready_to_submit": "NO", "source_verified": "YES", "status_verified": "NO"}
================================================================================================
PROBLEM 3
Title: Question on What happens (to the Menger-like properties) after a countable number of steps
Statement: Is it useful to iterate (`a la Cantor) the procedure $X_{\mathfrak b}$, $(X_{\mathfrak b}){\mathfrak b}$, $((X_{\mathfrak b}){\mathfrak b}){\mathfrak b},\ldots$? What happens (to the Menger-like properties) after a countable number of steps?
Original statement: Is it useful to iterate (`a la Cantor) the procedure $X{\mathfrak b}$, $(X_{\mathfrak b}){\mathfrak b}$, $((X{\mathfrak b}){\mathfrak b}){\mathfrak b},\ldots$? What happens (to the Menger-like properties) after a countable number of steps?
Background: The source paper, “On (non-Menger) spaces whose closed nowhere dense subsets are Menger,” poses this item as Question 2.14 in its study of general/set-theoretic topology. The exact source item is identifiable, but this audit did not locate sufficiently strong later evidence to certify its status as of 20 October 2026; it is therefore marked unknown rather than assumed to remain open. The cleaned statement below is an editorially normalized version of the source formulation; the transcription retained in original_statement is kept separately for provenance.
Status: unknown
Proposed by: null
Proposed year: null
Category: Topology
Tags: general-topology; covering-properties; selection-principles
Source URL: https://arxiv.org/abs/2501.13220
Source citation: Mathieu Baillif, Santi Spadaro, “On (non-Menger) spaces whose closed nowhere dense subsets are Menger,” arXiv:2501.13220 (2025); Question 2.14.
Exact location: {"label": "Question 2.14", "page": null}
arXiv: 2501.13220
References: https://arxiv.org/abs/2501.13220
Literature sources: []
Literature assessment: null
Literature checked at: 2026-10-20
Possible duplicate: null
Notes: Exact problem label verified from the prepared source audit; exact PDF page has not been independently verified in this pass unless explicitly stated otherwise. Original proposer attribution has not been established with enough confidence; proposed_by is therefore null unless independently verified. unknown means that the current status was not certified; it must not be interpreted as evidence that the problem is open. This source item is programmatic/rhetorical or asks for a characterization rather than a single sharply testable proposition; editorial decomposition is recommended before catalog submission.
Submission readiness: do_not_submit_yet
Quality control: {"attribution_verified": "NO", "ready_to_submit": "NO", "source_verified": "YES", "status_verified": "NO"}
================================================================================================
UNSOLVEDMATH SPADARO COLLECTION — COMMENT 2/46 — PROBLEMS 4–6
PROBLEM 4
Title: ZFC existence of a Lindelöf od-Menger non-Menger space
Statement: Is there a Lindelöf od-Menger non-Menger space in ZFC?
Original statement: Is there a Lindelöf od-Menger non-Menger space in ZFC?
Background: The source paper, “On (non-Menger) spaces whose closed nowhere dense subsets are Menger,” poses this item as Question 3.6 in its study of general/set-theoretic topology. The exact source item is identifiable, but this audit did not locate sufficiently strong later evidence to certify its status as of 20 October 2026; it is therefore marked unknown rather than assumed to remain open. The cleaned statement below is an editorially normalized version of the source formulation; the transcription retained in original_statement is kept separately for provenance.
Status: unknown
Proposed by: null
Proposed year: null
Category: Topology
Tags: general-topology; set-theoretic-topology; covering-properties; selection-principles
Source URL: https://arxiv.org/abs/2501.13220
Source citation: Mathieu Baillif, Santi Spadaro, “On (non-Menger) spaces whose closed nowhere dense subsets are Menger,” arXiv:2501.13220 (2025); Question 3.6.
Exact location: {"label": "Question 3.6", "page": null}
arXiv: 2501.13220
References: https://arxiv.org/abs/2501.13220
Literature sources: []
Literature assessment: null
Literature checked at: 2026-10-20
Possible duplicate: null
Notes: Exact problem label verified from the prepared source audit; exact PDF page has not been independently verified in this pass unless explicitly stated otherwise. Original proposer attribution has not been established with enough confidence; proposed_by is therefore null unless independently verified. unknown means that the current status was not certified; it must not be interpreted as evidence that the problem is open.
Submission readiness: needs_verification
Quality control: {"attribution_verified": "NO", "ready_to_submit": "PARTIAL", "source_verified": "YES", "status_verified": "NO"}
================================================================================================
PROBLEM 5
Title: Existence of ZFC example of a non-trivial M-nw-selective (resp. R-nw-selective, H-nw-selective) space $X$? What about countable spaces
Statement: Is there a ZFC example of a non-trivial M-nw-selective (resp. R-nw-selective, H-nw-selective) space $X$? What about countable spaces?
Original statement: Is there a ZFC example of a non-trivial M-nw-selective (resp. R-nw-selective, H-nw-selective) space $X$? What about countable spaces?
Background: The source paper, “On some recent selective properties involving networks,” poses this item as Question 2.13(1) in its study of general/set-theoretic topology. The exact source item is identifiable, but this audit did not locate sufficiently strong later evidence to certify its status as of 20 October 2026; it is therefore marked unknown rather than assumed to remain open. The cleaned statement below is an editorially normalized version of the source formulation; the transcription retained in original_statement is kept separately for provenance.
Status: unknown
Proposed by: null
Proposed year: null
Category: Topology
Tags: general-topology; set-theoretic-topology; selection-principles; networks
Source URL: https://arxiv.org/abs/2407.18713
Source citation: Maddalena Bonanzinga, Davide Giacopello, Santi Spadaro, Lyubomyr Zdomskyy, “On some recent selective properties involving networks,” arXiv:2407.18713 (2024); Question 2.13(1).
Exact location: {"label": "Question 2.13(1)", "page": null}
arXiv: 2407.18713
References: https://arxiv.org/abs/2407.18713
Literature sources: []
Literature assessment: null
Literature checked at: 2026-10-20
Possible duplicate: null
Notes: Exact problem label verified from the prepared source audit; exact PDF page has not been independently verified in this pass unless explicitly stated otherwise. Original proposer attribution has not been established with enough confidence; proposed_by is therefore null unless independently verified. unknown means that the current status was not certified; it must not be interpreted as evidence that the problem is open.
Submission readiness: needs_verification
Quality control: {"attribution_verified": "NO", "ready_to_submit": "PARTIAL", "source_verified": "YES", "status_verified": "NO"}
================================================================================================
PROBLEM 6
Title: Question on the existence of a non-trivial uncountable M-nw-selective (resp. R-nw-selective, H-nw-selective) space $X$ consistent with ZFC
Statement: Is the existence of a non-trivial uncountable M-nw-selective (resp. R-nw-selective, H-nw-selective) space $X$ consistent with ZFC?
Original statement: Is the existence of a non-trivial uncountable M-nw-selective (resp. R-nw-selective, H-nw-selective) space $X$ consistent with ZFC?
Background: The source paper, “On some recent selective properties involving networks,” poses this item as Question 2.13(2) in its study of general/set-theoretic topology. The exact source item is identifiable, but this audit did not locate sufficiently strong later evidence to certify its status as of 20 October 2026; it is therefore marked unknown rather than assumed to remain open. The cleaned statement below is an editorially normalized version of the source formulation; the transcription retained in original_statement is kept separately for provenance.
Status: unknown
Proposed by: null
Proposed year: null
Category: Topology
Tags: general-topology; set-theoretic-topology; selection-principles; networks
Source URL: https://arxiv.org/abs/2407.18713
Source citation: Maddalena Bonanzinga, Davide Giacopello, Santi Spadaro, Lyubomyr Zdomskyy, “On some recent selective properties involving networks,” arXiv:2407.18713 (2024); Question 2.13(2).
Exact location: {"label": "Question 2.13(2)", "page": null}
arXiv: 2407.18713
References: https://arxiv.org/abs/2407.18713
Literature sources: []
Literature assessment: null
Literature checked at: 2026-10-20
Possible duplicate: null
Notes: Exact problem label verified from the prepared source audit; exact PDF page has not been independently verified in this pass unless explicitly stated otherwise. Original proposer attribution has not been established with enough confidence; proposed_by is therefore null unless independently verified. unknown means that the current status was not certified; it must not be interpreted as evidence that the problem is open.
Submission readiness: needs_verification
Quality control: {"attribution_verified": "NO", "ready_to_submit": "PARTIAL", "source_verified": "YES", "status_verified": "NO"}
================================================================================================