Sorgenfrey plane

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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

Properties it does satisfy

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

References

Textbook references

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