Let \(G\) be a group acting on a topological space \(X\) by homeomorphisms, \(G\actson X\).
The action is free if \(g(x) = x\) for some \(x\in X\) implies \(g = e\).
The action is properly discontinuous if every \(x\in X\) has a neighborhood \(U\) such that for all \(g_1, g_2\in G\), \(g_1(U) \intersect g_2(U) \neq \emptyset\) implies \(g_1 = g_2\).
A properly discontinuous action in this sense is free. Other sources call an action properly discontinuous under the weaker condition that every \(x\in X\) has a neighborhood \(U\) with \(U \intersect g(U) \neq \emptyset\) for only finitely many \(g\in G\). An action satisfying the condition in the definition is called a covering space action in [@Hat02].