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 intervalsa,b), where a and b are real numbers.
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 Rl (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) (xα) in Rl converges to the limit Liff it "approaches L from the right", meaning for every ε>0 there exists an index α0 such that for all α > α0: L ≤ xα < 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.
