Group of covering tansformations of universal covering for Riemannian manifold is total discontinuous?

Question

Picture below is from Do Carmo's Riemannian Geometry. I use the definition of covering map. In this definition, if assuming $V_i\cap V_j =\emptyset ~(i\ne j)$ ($V_i$ is sheet), then I know how to get the red line, namely the $\Gamma$ is total discontinuous. But in Wiki or Munkres' Topology, the covering map don't need $V_i\cap V_j =\emptyset ~(i\ne j)$ (although, I feel it is right). So, I don't know how to get the red line.