This article introduces a new deep learning approach to approximately solve the covering salesman problem (CSP). In this approach, given the city locations of a CSP as input, a deep neural network model is designed to directly output the solution. It is trained using the deep reinforcement learning without supervision. Specifically, in the model, we apply the multihead attention (MHA) to capture the structural patterns, and design a dynamic embedding to handle the dynamic patterns of the problem. Once the model is trained, it can generalize to various types of CSP tasks (different sizes and topologies) without the need of retraining. Through controlled experiments, the proposed approach shows desirable time complexity: it runs more than 20 times faster than the traditional heuristic solvers with a tiny gap of optimality. Moreover, it significantly outperforms the current state-of-the-art deep learning approaches for combinatorial optimization in the aspect of both training and inference. In comparison with traditional solvers, this approach is highly desirable for most of the challenging tasks in practice that are usually large scale and require quick decisions.