preimagea

Preimagea is a hypothetical extension of the classical inverse image operation in mathematics. Given a function f: X → Y and an index set A, the preimagea of a subset S ⊆ Y with respect to a ∈ A is denoted P_a(S) and defined as the set of all x ∈ X for which f(x) lies in a neighborhood N_a(S) of S in Y. The neighborhood N_a(S) is chosen from a family of neighborhoods {N_a} indexed by A, often determined by a metric d on Y and a tolerance t(a) ≥ 0, with N_a(S) = { y ∈ Y | ∃ s ∈ S with d(y, s) ≤ t(a) }.

When the tolerance t(a) is zero for a particular a0, so N_{a0}(S) = S, the preimagea P_{a0}(S) coincides with the ordinary preimage f^−1(S).

Properties of preimagea include monotonicity in S, since S ⊆ T implies P_a(S) ⊆ P_a(T). If a fixed a is used, P_a distributes over unions: P_a(∪_i S_i) = ∪_i P_a(S_i) under standard set operations and the chosen neighborhood definitions. For continuous f and a metric Y, P_a respects limit processes in the sense that smaller tolerances yield smaller preimagea sets.

Example: let X = R, Y = R, f(x) = x^2, S = [1,4], and A = {a0, a1} with t(a0)=0 and t(a1)=0.5. Then P_{a0}(S) = { x | x^2 ∈ [1,4] } = [-2,-1] ∪ [1,2], while P_{a1}(S) = { x | x^2 ∈ [0.5,4.5] } = [-√4.5, -√0.5] ∪ [√0.5, √4.5].

Applications include approximate inverse problems, robust set estimation, and parameterized data analysis. The term is used primarily in theoretical expositions and has not achieved broad consensus in standard references. See also inverse image, neighborhood, and set-valued analysis.