Sorgenfrey line

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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 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

Property Satisfied? Explanation Corollary properties satisfied/dissatisfied
Separation
normal space Yes satisfies: completely regular space, regular space, Hausdorff space, T1 space, Kolmogorov space
hereditarily normal space Yes satisfies: normal space
perfectly normal space Yes Sorgenfrey line is perfectly normal satisfies: perfect space, hereditarily normal space
monotonically normal space Yes Sorgenfrey line is monotonically normal satisfies: hereditarily normal space
Connectedness
totally disconnected space Yes dissatisfies: connected space
Countability
first-countable space Yes
separable space Yes
second-countable space No
Compactness
Lindelof space Yes
paracompact Hausdorff space Yes satisfies:normal space
sigma-compact space No
locally compact space No Sorgenfrey line is not locally compact
Miscellaneous
Baire space Yes Sorgenfrey line is Baire
elastic space No Sorgenfrey lineis not elastic dissatisfies: metrizable space

References

Textbook references

  • Topology (2nd edition) by James R. MunkresMore 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)