Sorgenfrey plane
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In topology, the Sorgenfrey plane is a frequently-cited counterexample to many otherwise plausible-sounding conjectures. It consists of the product of two copies of the real line R under the half-open interval topology.
A basis for the Sorgenfrey plane is therefore the set of rectangles that include the west edge, southwest corner, and south edge, and omit the southeast corner, east edge, northeast corner, north edge, and northwest corner. Open sets in the Sorgenfrey plane are unions of such rectangles.
The Sorgenfrey plane is an example of a space that is a product of Lindelöf spaces that is not itself a Lindelöf space. It is also an example of a space that is a product of normal spaces that is not itself normal. The so-called anti-diagonal 
is a discrete subset of this space, and this is a non-separable subset of the separable Sorgenfrey plane. It shows that separability does not inherit to subspaces.
Take the Sorgenfrey line to be the usual topology on a sub-basis of X. Then X is metrizable provided that X is connected and not mutually separated. Let 
be a continuous function where 
and 
converges to y. Take (a,b) to contain the southeast corner of the Sorgenfrey line. Let (a,b) = N(p,r) such that N(p,r) is open in the Sorgenfrey line. Then (a,b) is open in the topology of X. Now we can make the assumption that the southeast corner of Texas is an open set in the Sorgenfrey line and the southeast corner CANNOT be omitted from the topology.
It is named for its discoverer, American mathematician Robert Sorgenfrey.
[edit] References
- Lynn Arthur Steen and J. Arthur Seebach, Jr., Counterexamples in Topology. Springer-Verlag, New York, 1978. Reprinted by Dover Publications, New York, 1995. ISBN 0-486-68735-X (Dover edition).
