Sorgenfrey plane

Definition
The Sorgenfrey plane is defined as the Cartesian product of two copies of the Sorgenfrey line, endowed with the product topology.

Properties it does not satisfy

 * Lindelof space: The Sorgenfrey plane is not Lindelof, even though the Sorgenfrey line is Lindelof.
 * Normal space: The Sogenfrey plane is not normal, even though the Sorgenfrey line is normal.
 * Hereditarily separable space: The anitdiagonal in the Sorgenfrey plane is a discrete uncountable set.

Properties it does satisfy

 * Separable space: This is because the Sorgenfrey line is separable, and a finite product of separable spaces is again separable.

Textbook references

 * , Page 198, Example 3, Chapter 4, Section 31