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Definition: If G is a group of homeomorphisms of X, the action of G on X is said to be properly discontinuous if for every x in X there is a neighborhood U of x such that g(U) is disjoint from U whenever g does not equal e. Here e is the identity element of G. It follows that g0(U) and g1(U) are disjoint whenever g0 does not equal g1, for otherwise U and g0^(-1)g1(U) would not be disjoint.
Source: Topology (second edition) by James R. Munkres