Sorgenfrey line

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 line is defined as follows: as a set, it is the real line, and its basis of open sets is taken as all the right-open, left-closed intervals, viz., sets of the form $[a,b)$. Equivalently, we can say that it is obtained by giving the lower limit topology corresponding to the usual ordering on the real line.

The product of two copies of the Sorgenfrey line is the Sorgenfrey plane, which is not normal. This gives an example of the fact that a product of two normal spaces need not be normal.

Topological space properties

Properties it does satisfy

• Totally disconnected space: The Sorgenfrey line is totally disconnected; given any two points, we can separate them by disjoint open sets.
• First-countable space
• Separable space
• Lindelof space
• Paracompact Hausdorff space
• Baire space
• Perfectly normal space
• Monotonically normal space

Properties it does not satisfy

• Sigma-compact space
• Locally compact space
• Metrizable space
• Elastic space

References

Textbook references

• Topology (2nd edition) by James R. Munkres^{More info}, Page 82, Chapter 2, Section 13 (the term Sorgenfrey line is not used, and the line is simply alluded to as the real numbers with the lower limit topology)