Sorgenfrey plane

 ( Topological Spaces ) 

In topology, the Sorgenfrey plane is a frequently-cited counterexample to many otherwise plausible-sounding conjectures. It consists of the product space of two copies of the Sorgenfrey line, which is the real line $\backslash mathbb\{R\}$ 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, denoted $\backslash mathbb\{S\}$ from now on, is therefore the set of that include the west edge, southwest corner, and south edge, and omit the southeast corner, east edge, northeast corner, north edge, and northwest corner. in $\backslash mathbb\{S\}$ are unions of such rectangles.

$\backslash mathbb\{S\}$ 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 $\backslash Delta\; =\; \backslash \{(x,\; -x)\; \backslash mid\; x\; \backslash in\; \backslash mathbb\{R\}\backslash \}$ is an uncountable set Discrete space subset of this space, and this is a non-Separable space subset of the separable space $\backslash mathbb\{S\}$. It shows that separability does not inherit to closed subspaces. Note that $K\; =\; \backslash \{(x,\; -x)\; \backslash mid\; x\; \backslash in\; \backslash mathbb\{Q\}\backslash \}$ and $\backslash Delta\; \backslash setminus\; K$ are closed sets; it can be proved that they cannot be separated by open sets, showing that $\backslash mathbb\{S\}$ 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.

• List of topologies
• Moore plane

• Reprinted as
  John L. Kelley (1975). 9780387901251, Springer-Verlag. ISBN 9780387901251
• Robert Sorgenfrey, "On the topological product of paracompact spaces", Bull. Amer. Math. Soc. 53 (1947) 631–632.
• (1995). 9780486687353, Springer-Verlag. ISBN 9780486687353