Hutchinson’s Theorem in Semimetric Spaces

The paper proves Hutchinson’s theorem in semimetric spaces by working on the hyperspace (K(X), D) built from the Hausdorff–Pompeiu distance D, shows that D inherits the triangle function Φ and that (K(X), D) is complete, then applies a fixed point theorem for ϕ‑contractions to obtain a unique fractal; see Theorem 7 (Φ is a triangle function for D), Theorems 8–9 (completeness of F(X) and K(X)), and Theorem 14 (existence/uniqueness) . The candidate solution replaces D with h(A,B)=max{e(A,B),e(B,A)} and proceeds to re‑prove semimetric and completeness facts on (K(X), h). Key steps are incorrect or rely on unstated assumptions: (i) it uses compactness to extract finite ball covers and sequential compactness—neither is valid in general semimetric spaces where balls need not be open and compact need not imply sequential compactness; the paper avoids this by working with D and explicit covering arguments (Theorem 7) . (ii) it builds finite ε‑nets in K(X) from “compactness,” implicitly assuming total boundedness, which the paper proves separately under appropriate hypotheses (Theorems 4, 8–9) . (iii) it infers “T maps compacts to compacts” from sequential continuity; the paper correctly deduces continuity from a Lipschitz bound implied by ϕ‑contractivity (Lemma 3) . Consequently, the paper’s argument is sound and complete within its framework, whereas the model’s proof has gaps and uses an inappropriate hyperspace distance.