We study open topological dynamical systems with countably many shrinking holes. Specifically, let f be a transitive homeomorphism on a compact metric space , and take \(\{p_i\}_{i\in \mathbb {N}}\) to be a sequence of distinct points in disjoint from its derived set. To each point, associate a collection \(\{B_i^n\}_{n\in \mathbb {N}}\) of shrinking open sets, and consider when the trajectory \(\{f^i(x)\}_{i\in \mathbb {N}}\) of a point x in first visits one of the open sets \(\{B_i^n\}_{i,n\in \mathbb {N}}\) . We find that the system exhibits generic indecisiveness, that is for the typical sequence \(\{p_i\}_{i\in \mathbb {N}}\) , for each \(i\in \mathbb {N}\) , the trajectory \(\{f^i(x)\}_{i\in \mathbb {N}}\) will first visit \(B^n_i\) for infinitely many values of n.
