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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. | |||
<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 11:55, 6 September 2013
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 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, denoted from now on, 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 are unions of such rectangles.
is an example of a space that is a product of Lindelöf spaces that is not itself a Lindelöf space. The so-called anti-diagonal is an uncountable discrete subset of this space, and this is a non-separable subset of the separable space . It shows that separability does not inherit to closed subspaces. Note that and are closed sets that cannot be separated by open sets, showing that 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
- 20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.
My blog: http://www.primaboinca.com/view_profile.php?userid=5889534 Reprinted as 20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.
My blog: http://www.primaboinca.com/view_profile.php?userid=5889534 - Robert Sorgenfrey, "On the topological product of paracompact spaces", Bull. Amer. Math. Soc. 53 (1947) 631–632.
- 20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.
My blog: http://www.primaboinca.com/view_profile.php?userid=5889534