Kline sphere characterization
Encyclopedia
In mathematics, a Kline sphere characterization, named after John Robert Kline
, is a topological
characterization of a two-dimensional sphere
in terms of what sort of subset separates it. Its proof was one of the first notable accomplishments of R.H. Bing.
A simple closed curve
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
. Kline's sphere characterization states that the converse is true: If a nondegenerate locally connected
metric
continuum
is separated by any simple closed curve but by no pair of points, then it is a two-dimensional sphere.
John Robert Kline
John Robert Kline was a US-American Professor of Mathematics at the University of Pennsylvania from 1920 to 1955. A Ph.D. student of Robert Lee Moore, he was Guggenheim Fellow in 1925, later Chairman of the Department of Mathematics from 1933 to 1954 and Thomas A...
, is a topological
Topology
Topology is a major area of mathematics concerned with properties that are preserved under continuous deformations of objects, such as deformations that involve stretching, but no tearing or gluing...
characterization of a two-dimensional sphere
Sphere
A sphere is a perfectly round geometrical object in three-dimensional space, such as the shape of a round ball. Like a circle in two dimensions, a perfect sphere is completely symmetrical around its center, with all points on the surface lying the same distance r from the center point...
in terms of what sort of subset separates it. Its proof was one of the first notable accomplishments of R.H. Bing.
A simple closed curve
Curve
In mathematics, a curve is, generally speaking, an object similar to a line but which is not required to be straight...
in a two-dimensional sphere (for instance, its equator) separates
Connected space
In topology and related branches of mathematics, a connected space is a topological space that cannot be represented as the union of two or more disjoint nonempty open subsets. Connectedness is one of the principal topological properties that is used to distinguish topological spaces...
the sphere into two pieces upon removal. If one removes a pair of points from a sphere, however, the remainder is connected
Connected space
In topology and related branches of mathematics, a connected space is a topological space that cannot be represented as the union of two or more disjoint nonempty open subsets. Connectedness is one of the principal topological properties that is used to distinguish topological spaces...
. Kline's sphere characterization states that the converse is true: If a nondegenerate locally connected
Locally connected space
In topology and other branches of mathematics, a topological space X islocally connected if every point admits a neighbourhood basis consisting entirely of open, connected sets.-Background:...
metric
Metric space
In mathematics, a metric space is a set where a notion of distance between elements of the set is defined.The metric space which most closely corresponds to our intuitive understanding of space is the 3-dimensional Euclidean space...
continuum
Continuum (topology)
In the mathematical field of point-set topology, a continuum is a nonempty compact connected metric space, or less frequently, a compact connected Hausdorff topological space...
is separated by any simple closed curve but by no pair of points, then it is a two-dimensional sphere.