VersorIn mathematics, a versor is a quaternion of norm one (a unit quaternion). Each versor has the form where the r2 = −1 condition means that r is a unit-length vector quaternion (or that the first component of r is zero, and the last three components of r are a unit vector in 3 dimensions). The corresponding 3-dimensional rotation has the angle 2a about the axis r in axis–angle representation. In case a = π/2 (a right angle), then , and the resulting unit vector is termed a right versor.
Nowhere dense setIn mathematics, a subset of a topological space is called nowhere dense or rare if its closure has empty interior. In a very loose sense, it is a set whose elements are not tightly clustered (as defined by the topology on the space) anywhere. For example, the integers are nowhere dense among the reals, whereas the interval (0, 1) is not nowhere dense. A countable union of nowhere dense sets is called a meagre set. Meagre sets play an important role in the formulation of the , which is used in the proof of several fundamental results of functional analysis.
Dense-in-itselfIn general topology, a subset of a topological space is said to be dense-in-itself or crowded if has no isolated point. Equivalently, is dense-in-itself if every point of is a limit point of . Thus is dense-in-itself if and only if , where is the derived set of . A dense-in-itself closed set is called a perfect set. (In other words, a perfect set is a closed set without isolated point.) The notion of dense set is unrelated to dense-in-itself. This can sometimes be confusing, as "X is dense in X" (always true) is not the same as "X is dense-in-itself" (no isolated point).
