Let \(E=E(f)\) be a compact totally disconnected set in the Riemann sphere \(\overline{\mathbb{C}}=\mathbb{C}\cup\{\infty\}\) and let \(f\) be a meromorphic function in \(E^c=\overline{\mathbb{C}}-E\) such that \(C(f,E^c,z_0)=\overline{\mathbb{C}}\) for \(z_0\in E\), where \[ C(f,E^c,z_0)=\{w\in \overline{\mathbb{C}}\mid w=\lim_{n\to\infty}f(z_n),z_n\in E^c \rightarrow z_0\}. \] Let \(\text{sing}(f)\) be the set consisting of both singularities of the inverse function of \(f\) and the limit values of these singularities. Set \[ P(f)=\bigcup_{p=1}^\infty\bigcup_{k=0}^pf^k(\text{sing}(f)-E(f^k)), \] where \(f^k\) denotes the \(k\)th iterate of \(f\). The authors prove that the Julia set \(J(f)\) of \(f\) is uniformly perfect if \(d(J(f),\overline{P(f)}-J(f))>0\), where \(d\) is the Euclidean distance of two sets.