Search arXiv⌕ Search
Home/Archive/arXiv

arXiv · 2611.00857

Continuous, locally Lipschitz and Lipschitz selections for intuitionistic fuzzy sets, with applications to expert voting

Giuseppe Failla·Gianmarco La Rosa·Marco Elio Tabacchi
View original source ↗Read source document ↗Export metadata ↓

Abstract

This paper studies continuous selections in the framework of intuitionistic fuzzy sets. To each intuitionistic fuzzy set $A=(μ,ν)$ on a topological space $X$ we associate the interval-valued map $F_{A}(x)=[μ(x),1-ν(x)]$, whose width is the hesitation degree. Hence, in this setting, a selection of $F_{A}$ is an ordinary fuzzy membership function compatible with the intuitionistic information, and every selection is the membership function of Atanassov's operator $D_{\vartheta}$ for a suitable pointwise parameter $\vartheta$. When $X$ is normal and $μ$, $ν$ are upper semi-continuous, a continuous selection exists by the Katětov--Tong insertion theorem. We show that upper semi-continuity arises naturally in a voting model, in which a panel of experts divides a metric universe into closed regions of acceptance and rejection, and for this model we construct an explicit continuous selection, the consensus selection, illustrated by a non-compact example in the plane. On metric spaces we characterise the intuitionistic fuzzy sets that admit an $L$-Lipschitz selection through a non-local version of the condition $μ+ν\leq 1$, and we give the largest and the smallest such selections in closed form. Finally, we prove that the set of continuous selections depends in a $1$-Lipschitz way on the intuitionistic fuzzy set, with respect to the Hausdorff-type distance of Grzegorzewski. We also characterise the intuitionistic fuzzy sets admitting locally Lipschitz selections.

Explore related subjects

math.GNmath.FA

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

Giuseppe Failla, Gianmarco La Rosa, Marco Elio Tabacchi. 2026-10-31. Continuous, locally Lipschitz and Lipschitz selections for intuitionistic fuzzy sets, with applications to expert voting. https://arxiv.org/abs/2611.00857

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

Borelness of Moduli Spaces of Metrics Implies Separability

Let $X$ be a metrizable space, and let $\mathrm{Met}(X)$ denote the space of metrics compatible with the topology of $X$, regarded as a subspace of the space of continuous pseudometrics with the topology induced by the supremum metric. We first prove that if $Z$ is a discrete space whose cardinality is the first uncountable cardinal, then $\mathrm{Met}(Z)$ is not Borel. As a consequence, if $\mathrm{Met}(X)$ is Borel, then $X$ is separable. Combined with a theorem of Koshino, our result yields that the space of bounded compatible metrics on a metrizable space $X$ is completely metrizable if and only if $X$ is $σ$-compact. We also establish non-Archimedean analogues for spaces of ultrametrics. We further prove componentwise versions of these results, where two metrics belong to the same component if their distance is finite.

math.GN↗
arXiv2026

Four problems about the irreducible determined $T_0$-topologies

We answer four questions on irreducible determined $T_0$-topologies and a related question on monotone determined topologies posed by Xu, Yang and Luo (Topology Appl.\ 382 (2026), 109762). For every $T_0$ topology $τ$, the family $τ_r$ used to define its irreducible-derived topology is already a topology and preserves all nonempty irreducible subsets and their closures. Consequently, the least irreducible determined refinement is obtained in one step: $τ_r=\rd(τ)=\Rd(τ)$. For an arbitrary poset $P$, we describe the weakest order-compatible irreducible determined topology $ρ(P)$ using Dedekind--MacNeille cuts and a finite-cover condition involving principal lower sets. The Scott space of the binary-prefix domain provides a common counterexample to saturated heredity for irreducible determined spaces and monotone determined spaces: its maximal-point subspace is Cantor space. We also describe $ρ(P)$-closure by iterating an irreducible-cut operator, characterize Scott continuity of this operator, and give an example requiring two applications of the cut operator. A finite-decomposition condition extends the topology and closure-preservation results to general subset systems. For the Rudin and well-filtered determined systems, we exhibit a common $T_1$ example whose first and second derived topologies are the co-countable and discrete topologies, respectively. Thus the first derived topology need not be determined by the same subset system.

math.GN↗
arXiv2026

Separately continuous maps into regular spaces and continuity of operations in separable semitopological groups

A space is feebly compact if every locally finite family of nonempty open subsets of it is finite; a Tychonoff space is feebly compact if and only if it is pseudocompact. M. Tkachenko asked whether a regular feebly compact semitopological group is a topological group provided that it is separable, has countable tightness, or is a $k$-space; for Tychonoff groups the answer is known to be positive. We answer this question in the separable case: every regular separable feebly compact semitopological group is a topological group. The proof is based on the following result: if $X$ is a separable Baire space, $Y$ is a feebly compact space and $Z$ is a regular space, then every separately continuous map $X\times Y\to Z$ is strongly quasicontinuous. Unlike the known approaches for Tychonoff spaces, the proof does not use real-valued functions. More generally, every regular separable semitopological group which is $Γ^{BM}_o$-nonmeager, a Baire-type property defined by a modification of the Banach-Mazur game, is a topological group. This class of groups contains all regular separable Baire groups which are feebly $p$-spaces; feebly $p$-spaces form a wider class of spaces, which contains all feebly compact spaces and all $p$-spaces. The results remain valid for spaces satisfying the assumptions locally. In particular, every regular locally separable locally feebly compact semitopological group is a topological group.

math.GN↗