Sorgenfrey plane: Meaning (information, definition, explanation, facts)

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.

It is named for its discoverer, American mathematician Robert Sorgenfrey.

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 048668735X (Dover edition).