For each n ≥ 2, we investigate a family of iterated function systems which is parameterized by a common contraction ratio s ∈ Dx ≡ {s ∈ C : 0 < |s| < 1} and possesses a rotational symmetry of order n. Let Mn be the locus of contraction ratio s for which the corresponding self-similar set is connected. The purpose of this paper is to show that Mn is regular-closed, i.e. int Mn = Mn holds for n ≥ 4. This gives a new result for n =4 and a simple geometric proof of the previously known result by [BanHu] for n ≥5.