CONNECTEDNESS

in mathematics, fundamental topological property of sets that corresponds with the usual intuitive idea of having no breaks. It is of fundamental importance because it is one of the few properties of geometric figures that remains unchanged after homeomorphic transformationsthat is, transformations in which the figure is deformed without tearing or folding. A point is called a limit point of a set in the Euclidean plane if there is no minimum distance from that point to members of the set; for example, the set of all numbers less than one has one as a limit point. If a point is removed from a curve, any remaining points on either side of the break will lie at a finite distance from the break and thus cannot be limit points of the other side. Topologically, the concept of two sets' being separated means that neither set contains any limit points of the other set. If a single point is removed from a simple closed curve such as a circle or polygon, it remains connected; if any two points are removed it becomes separated. A figure-eight curve does not have this property because one point can be removed from each loop and the figure will remain connected. Whether or not a set remains connected after some of its points are removed is one of the principal ways of classifying figures in topology.