Write down the definition of a covering space and a covering map. State and prove the path lifting property for covering spaces and state, without proof, the homotopy lifting property.

Suppose that a group $G$ is a group of homeomorphisms of a space $X$. Prove that, under conditions to be stated, the quotient map $X \rightarrow X / G$ is a covering map and that $\pi_{1}(X / G)$ is isomorphic to $G$. Give two examples in which this last result can be used to determine the fundamental group of a space.