In mathematics, a Kline sphere characterization, named after John Robert Kline, is a

topological Topology (from the Greek words , and ) is the branch of mathematics concerned with the properties of a geometric object that are preserved under continuous deformations, such as stretching, twisting, crumpling, and bending; that is, wit ...

characterization of a two-dimensional

sphere A sphere (from Ancient Greek, Greek , ) is a surface (mathematics), surface analogous to the circle, a curve. In solid geometry, a sphere is the Locus (mathematics), set of points that are all at the same distance from a given point in three ...

in terms of what sort of subset separates it. Its proof was one of the first notable accomplishments of R. H. Bing; Bing gave an alternate proof using brick partitioning in his paper ''Complementary domains of continuous curves'' Bing, R.H., Complementary domains of continuous curves, ''Fund. Math.'' 36 (1949)
303-318
A

simple closed curve In mathematics, a curve (also called a curved line in older texts) is an object similar to a line, but that does not have to be straight. Intuitively, a curve may be thought of as the trace left by a moving point. This is the definition that ...

in a two-dimensional sphere (for instance, its equator) separates the sphere into two pieces upon removal. If one removes a pair of points from a sphere, however, the remainder is

connected Connected may refer to: Film and television * ''Connected'' (2008 film), a Hong Kong remake of the American movie ''Cellular'' * '' Connected: An Autoblogography About Love, Death & Technology'', a 2011 documentary film * ''Connected'' (2015 TV ...

. Kline's sphere characterization states that the converse is true: If a nondegenerate

locally connected In topology and other branches of mathematics, a topological space ''X'' is locally connected if every point admits a neighbourhood basis consisting of open connected sets. As a stronger notion, the space ''X'' is locally path connected if ev ...

metric Metric or metrical may refer to: Measuring * Metric system, an internationally adopted decimal system of measurement * An adjective indicating relation to measurement in general, or a noun describing a specific type of measurement Mathematics ...

continuum is separated by any simple closed curve but by no pair of points, then it is a two-dimensional sphere.

References

{{Reflist *Bing, R. H., The Kline sphere characterization problem, ''Bulletin of the American Mathematical Society'' 52 (1946)
644–653
Theorems in topology Topology