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On families of copulas and quasi-copulas homeomorphic to the Hilbert cube and related results
Abstract
Working with tools from infinite-dimensional topology and using measure-theoretic ideas we analyze numerous well-known fami\-lies of copu\-las and quasi-copulas from a topological perspective, prove that most of them are homeomorphic to the Hilbert cube ℋ and hence have the Fixed Point as well as the Disjoint Cells property. Working with so-called Z-sets, we then study how large one set is within another. Doing so, in order to show that a set A is small/topologically negligible within a compact metric space (Ω,m), we construct a homeomorphism h: [0,1] × ℋ → Ω mapping {0} × ℋ to A. The obtained results are to a certain extent surprising since various supposedly small families are topologically large.
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BibTeX
@misc{durante2026families,
title = {{On families of copulas and quasi-copulas homeomorphic to the Hilbert cube and related results}},
author = {Fabrizio Durante and Juan Fernández Sánchez and Wolfgang Trutschnig},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.35912},
eprint = {2610.35912},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.35912},
}
RIS
TY - UNPB
TI - On families of copulas and quasi-copulas homeomorphic to the Hilbert cube and related results
AU - Fabrizio Durante
AU - Juan Fernández Sánchez
AU - Wolfgang Trutschnig
PY - 2026
DA - 2026/10/07
PB - arXiv
DO - 10.48550/arxiv.2610.35912
UR - https://arxiv.org/abs/2610.35912
AB - Working with tools from infinite-dimensional topology and using measure-theoretic ideas we analyze numerous well-known fami\-lies of copu\-las and quasi-copulas from a topological perspective, prove that most of them are homeomorphic to the Hilbert cube ℋ and hence have the Fixed Point as well as the Disjoint Cells property. Working with so-called Z-sets, we then study how large one set is within another. Doing so, in order to show that a set A is small/topologically negligible within a compact metric space (Ω,m), we construct a homeomorphism h: [0,1] × ℋ → Ω mapping {0} × ℋ to A. The obtained results are to a certain extent surprising since various supposedly small families are topologically large.
ER -