A Note on Properly Discontinuous Actions

This note is meant to clarify the relation between different commonly used definitions of proper discontinuity without the local compactness assumption for the underlying topological space. Much of the discussion applies to actions of nondiscrete topological groups, but, since my primary interest is geometric group theory, I will work only with discrete groups. Throughout this note, I will be working only with topological spaces which are 1st countable, since spaces most common in metric geometry, geometric topology, algebraic topology and geometric group theory satisfy this property. One advantage of this assumption is that if (xn) is a sequence converging to a point x ∈ X, then the subset {x} ∪ {xn : n ∈ N} is compact, which is not true if we work with nets instead of sequences. However, I will try to avoid the local compactness assumption whenever possible, since many spaces appearing in metric geometry and geometric group theory (e.g. asymptotic cones) and algebraic topology (e.g. CW complexes) are not locally compact. (Recall that topological space X is locally compact if every point has a basis of topology consisting of relatively compact subsets.) Lastly, all the material of this note is elementary and by no means original.