Kline sphere characterization
- Kline sphere characterization
The Kline Sphere Characterization is a topological characterization of the two-dimensional sphere in terms of what sorts of subsets separate it. Its proof was one of the first notable accomplishments of RH Bing.
It is well known that a simple closed curve in a two-dimensional sphere (for instance, its equator) separates the sphere into two pieces upon removal. However, if one removes a pair of points from a sphere, the remainder is connected. Kline's sphere characterization states that the converse is true: If a metric continuum is separated by any simple closed curve but by no pair of points, then it is a two-dimensional sphere.
References
Bing, R. H., "The Kline sphere characterization problem", Bull. Amer. Math. Soc. 52 (1946), 644 -- 653. Available at http://www.ams.org/bull/1946-52-08/S0002-9904-1946-08615-2/home.html
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