A series of related definitions, some weaker or stronger, are presented here.
There are several lemmas that follow from this definition, relating discontinuous action to ideas of discreteness and in particular to that of a discrete group.
Lemma: If G acts discontinuously then the orbits of the action have no accumulation points. That is, if is a sequence of distinct elements of G and then the sequence has no limit points. Conversely, if X is locally compact then an action that satisfies this condition is discontinuous.
Lemma: Assume that X is a locally compact Hausdorff space and let denote the group of self homeomorphisms of X endowed with the compact-open topology. If defines a discontinuous action then the image is a discrete subset of
The set U is called a nice neighborhood of x.
forall g in G-H, quad gY cap Y = varnothing.
Then let be the stabilizer of x in G. One says that G acts discontinuously at x in X if the stabilizer is finite and there exists a neighborhood U of x that is precisely invariant under in G. If G acts discontinuously at every point x in X, then one says that G acts properly discontinuously on X.
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.
Crash highlights importance of properly securing children ; One Maine expert says most children aren't properly secured in their car seats.
Jan 05, 2013; David Hench dhench@mainetodaycom Staff Writer<br />Portland Press Herald (Maine)<br />01-05-2013<br />Crash...
Patent Issued for Method and System for Motor Torque Control for Vehicles When a Current Sensor Is Not Operating Properly
Jan 09, 2013; According to news reporting originating from Alexandria, Virginia, by VerticalNews journalists, a patent by the inventors Jang,...