Properly discontinuous
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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 neighborhood 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 set of points at which G acts discontinuously is called the free regular set.
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Basic definition
The formal definition is as follows. Let a group G act on a topological space X by homeomorphisms. This action is called properly discontinuous if, for every x in X, there is a neighborhood U of x such that
- <math>\forall g \in G \quad (g \neq e) \Rightarrow (gU \cap U = \varnothing).</math>
The set U is called a nice neighborhood of x.
This narrow, basic definition fails when applied to a certain interesting case where one still wants to have a notion of discontinuity: the case where the stabilizer of the point x is non-trivial. Thus, the definition is frequently extended as below.
Definition with a non-trivial stabilizer
An extended definition is as follows. Consider a subgroup <math>H \subset G</math>. One then says that a set Y is precisely invariant under H in G if
- <math>\forall h \in H, \quad h(Y)=Y \;\mbox{ and }\;
\forall g \in G-H, \quad gY \cap Y = \varnothing.</math>
Then let <math>G_x</math> be the stabilizer of x in G. One says that G acts discontinuously at x in X if the stabilizer <math>G_x</math> is finite and there exists a neighborhood U of x that is precisely invariant under <math>G_x</math> in G. If G acts discontinuously at every point x in X, then one says that G acts properly discontinuously on X.
Definition as a locally finite set
Another common definition is in terms of a locally finite set. Given any x in X, let Gx be the orbit of x under the action of G. One then says that the orbit is locally finite if every compact subset K of X contains at most a finite number of points from the orbit Gx; that is, if
- <math>\mbox{card} (K\cap Gx) < \infty</math>
If the orbit Gx is locally finite for every x in X, then one says that the action of G on X is properly discontinuous.
Note that this alternate definition does not coincide with the basic definition if the stabilizer of x in G is non-trivial.