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 Sorgenfrey line, which is the real line R under the half-open interval topology. The Sorgenfrey line and plane are named for the American mathematician Robert Sorgenfrey.

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.

[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).