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
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.
- Topology (2nd edition) by James R. MunkresMore info, Page 198, Example 3, Chapter 4, Section 31