Group-Equivariant Fuzzy Topological Vector Spaces: A New Framework via Symmetric Group Actions

The theory of fuzzy topological vector spaces, initiated by Katsaras and Liu, requires algebraic operations to be fully fuzzy continuous but does not account for any underlying symmetry structure present in many mathematical and physical contexts. This paper introduces Group-Equivariant Fuzzy Topological Vector Spaces, a new algebraic-topological framework that systematically incorporates the action of a symmetry group G into the structure of fuzzy topological vector spaces by equipping a vector space X with a Lowen fuzzy topology T and a linear group action ϕ: G → GL(X), requiring both the fuzzy topology and the algebraic operations to be G-invariant and G-equivariant respectively, with classical results extended using invariant neighbourhood filters, Baire category arguments, and categorical methods. We establish that every group-equivariant fuzzy topological vector space admits a G-invariant fuzzy neighbourhood base at the origin, the fixed-point subspace Xᴳ inherits a classical fuzzy topological vector space structure, a G-equivariant fuzzy open mapping theorem holds under completeness assumptions, the orbit space X/G carries a natural quotient structure, the category of G-equivariant fuzzy continuous linear maps is complete and cocomplete, and new symmetry-induced separation axioms G-T₀, G-T₁, G-T₂, strictly stronger than classical fuzzy separation axioms, are introduced and characterised. Examples on finite-dimensional Euclidean spaces with orthogonal group actions, the real line with two-element group symmetry, and bounded sequence spaces with permutation symmetry illustrate the theory, while counter-examples confirm that G-equivariance is a strictly non-trivial condition. The proposed framework strictly generalises classical fuzzy topological vector spaces, recovered as the special case G = {e}, opening new directions in symmetric fuzzy functional analysis, equivariant spectral theory, and invariant approximation theory.