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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 Rl and called the Sorgenfrey line, often serves as a useful counterexample in general topology, like the Cantor set and the long line.

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 (infintely 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 {xR : x < a} and {xR : xa} are also clopen. This shows that the Sorgenfrey line is totally disconnected.

Rl is a regular Hausdorff space. It is separable and first countable but not second countable (and hence not metrizable).

Rl is Lindelöf and paracompact but not sigma-compact or locally compact.

A sequence (or net) (xα) in Rl 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: Lxα < L + ε. The Sorgenfrey line can thus be used to study right-sided limits: if f : RR 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.

References

• Lynn Arthur Steen and J. Arthur Seebach, Jr., Counterexamples in Topology (ISBN 0-486-68735-X)