Lower limit topology
In mathematics, the lower limit topology or right half-open interval topology is a topology defined on, the set of real numbers; it is different from the standard topology on , where a and b are real numbers.
The resulting topological space is called the Sorgenfrey line after Robert Sorgenfrey or the arrow and is sometimes written, or. Like the Cantor set and the long line, the Sorgenfrey line often serves as a useful counterexample to many otherwise plausible-sounding conjectures in general topology.
The product of with itself is also a useful counterexample, known as the Sorgenfrey plane.
In complete analogy, one can also define the upper limit topology, or left half-open interval topology.
Properties
- The lower limit topology is finer than the standard topology on the real numbers. The reason is that every open interval can be written as a union of half-open intervals.
- For any real and, the interval is clopen in . Furthermore, for all real, the sets and are also clopen. This shows that the Sorgenfrey line is totally disconnected.
- Any compact subset of must be an at most countable set. To see this, consider a non-empty compact subset. Fix an, consider the following open cover of :
- The name "lower limit topology" comes from the following fact: a sequence in converges to the limit if and only if it "approaches from the right", meaning for every there exists an index such that. The Sorgenfrey line can thus be used to study right-sided limits: if is a function, then the ordinary right-sided limit of at is the same as the usual limit of at when the domain is equipped with the lower limit topology and the codomain carries the standard topology.
- In terms of separation axioms, is a perfectly normal Hausdorff space.
- In terms of countability axioms, is first-countable and separable, but not second-countable.
- In terms of compactness properties, is Lindelöf and paracompact, but not σ-compact nor locally compact.
- is not metrizable, since separable metric spaces are second-countable. However, the topology of a Sorgenfrey line is generated by a quasimetric.
- is a Baire space.
- does not have any connected compactifications.