Properly discontinuous

In topology and related branches of mathematics, an action of a group G on a topological space X is called properly discontinuous if every element of X has a neighbourhood that moves outside itself under the action of any group element but the trivial element. The action of the deck transformation group of a cover is an example of such action.

The formal definition is as follows. Let a group G act on a topological space X by homeomophisms. This action is called properly discontinuous if, for every x in X, there is a neighborghood U of x such that