$$AC(\sigma)$$ spaces for polygonally inscribed curves

For certain families of compact subsets the plane, isomorphism class algebra absolutely continuous functions on a set is completely determined by homeomorphism set. This analogous to Gelfand–Kolmogorov theorem for C(K) spaces. In this paper, we define family sets comprising finite unions convex curves and show that has ‘Gelfand–Kolmogorov’ property.