Lower limit topology

In mathematics, the lower limit topology or right half-open interval topology is a topology defined on the set R of real numbers; it is different from the standard topology on R and has a number of interesting properties. It is the topology generated by the basis of all half-open intervals [a,b), where a and b are real numbers.

The resulting topological space, sometimes written R_l and called the Sorgenfrey line after Robert Sorgenfrey , often serves as a useful counterexample in general topology, like the Cantor set and the long line. The product of R_l 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 (has more open sets) than the standard topology on the real numbers (which is generated by the open intervals). The reason is that every open interval can be written as a union of (infinitely many) half-open intervals.

For any real a and b, the interval [a, b) is clopen in R_l (i.e. both open and closed). Furthermore, for all real a, the sets {x ∈ R : x < a} and {x ∈ R : x ≥ a} are also clopen. This shows that the Sorgenfrey line is totally disconnected.

The name "lower limit topology" comes from the following fact: a sequence (or net) (x_α) in R_l converges to the limit L iff it "approaches L from the right", meaning for every ε>0 there exists an index α₀ such that for all α > α₀: L ≤ x_α < L + ε. The Sorgenfrey line can thus be used to study right-sided limits: if f : R → R is a function, then the ordinary right-sided limit of f at x (when both domain and codomain carry the standard topology) is the same as the limit of f at x when the domain is equipped with the lower limit topology and the codomain carries the standard topology.

In terms of separation axioms, R_l is a perfectly normal Hausdorff space.

It is first-countable and separable, but not second-countable (and hence not metrizable, by Urysohn's metrization theorem).

In terms of compactness, R_l is Lindelöf and paracompact, but not σ-compact or locally compact.

The Sorgenfrey line is a Baire space [1].

References

• Lynn Arthur Steen and J. Arthur Seebach, Jr., Counterexamples in Topology. Springer-Verlag, New York, 1978. Reprinted by Dover Publications, New York, 1995. ISBN 0-486-68735-X (Dover edition).