Sorgenfrey line
From Topospaces
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 . 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. 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)