Sorgenfrey plane

This article describes a standard counterexample to some plausible but false implications. In other words, it lists a pathology that may be useful to keep in mind to avoid pitfalls in proofs
View other standard counterexamples in topology

Definition

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

Topological space properties

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. For full proof, refer: Sorgenfrey plane is not 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.