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[[Image:Sorgenfrey plane.png|thumb|An illustration of the anti-diagonal and an open rectangle in the Sorgenfrey plane that meets the anti-diagonal at a single point.]]


In [[topology]], the '''Sorgenfrey plane''' is a frequently-cited [[counterexample]] to many otherwise plausible-sounding conjectures.  It consists of the [[product space|product]] of two copies of the [[Sorgenfrey line]], which is the [[real line]] <math>\mathbb{R}</math> under the [[half-open interval topology]]. The Sorgenfrey line and plane are named for the [[USA|American]] mathematician [[Robert Sorgenfrey]].


A [[basis (topology)|basis]] for the Sorgenfrey plane, denoted <math>\mathbb{S}</math> from now on, is therefore the set of [[rectangle]]s 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 set]]s in <math>\mathbb{S}</math> are unions of such rectangles.
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<math>\mathbb{S}</math> is an example of a space that is a product of [[Lindelöf space]]s that is not itself a Lindelöf space. The so-called '''anti-diagonal''' <math>\Delta = \{(x, -x) \mid x \in \mathbb{R}\}</math> is an [[uncountable set|uncountable]] [[Discrete space|discrete]] subset of this space, and this is a non-[[Separable space|separable]] subset of the [[separable space]] <math>\mathbb{S}</math>. It shows that separability does not inherit to closed [[Topological subspace|subspaces]]. Note that <math>K = \{(x, -x) \mid x \in \mathbb{Q}\}</math> and <math>\Delta \setminus K</math> are closed sets that cannot be separated by open sets, showing that <math>\mathbb{S}</math> is not normal.  Thus it serves as a counterexample to the notion that the product of normal spaces is normal; in fact, it shows that even the finite product of perfectly normal spaces need not be normal.
 
==References==
* {{cite book
| first = John L.
| last = Kelley
| author-link = John L. Kelley
| year = 1955
| title = General Topology
| publisher = [[Van Nostrand Reinhold|van Nostrand]]
| isbn =
}}  Reprinted as {{cite book
| first = John L.
| last = Kelley
| author-link = John L. Kelley
| year = 1975
| title = General Topology
| publisher = [[Springer Science+Business Media|Springer-Verlag]]
| isbn = 0-387-90125-6
}}
* Robert Sorgenfrey, "On the topological product of paracompact spaces", ''[[Bull. Amer. Math. Soc.]]'' '''53''' (1947) 631–632.
* {{Cite book | last1=Steen | first1=Lynn Arthur | author1-link=Lynn Arthur Steen | last2=Seebach | first2=J. Arthur Jr. | author2-link=J. Arthur Seebach, Jr. | title=[[Counterexamples in Topology]] | origyear=1978 | publisher=[[Springer-Verlag]] | location=Berlin, New York | edition=[[Dover Publications|Dover]] reprint of 1978 | isbn=978-0-486-68735-3 | id={{MathSciNet|id=507446}} | year=1995 | postscript=<!-- Bot inserted parameter. Either remove it; or change its value to "." for the cite to end in a ".", as necessary. -->[[Category:Articles with inconsistent citation formats]]}}
 
[[Category:Topological spaces]]
 
{{topology-stub}}

Revision as of 04:35, 9 February 2014


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