The Sorgenfrey line, named after Robert Sorgenfrey
(1915-1996), is the lower limit topology
on the real numbers
This topology has as its basis the collection of all half-open intervals [a, b) where a < b. The collection of open sets in R that comprises this topological space is therefore the collection of unions of half-open intervals in R, sometimes written R L
(Actually it's written with a fancy lowercase l that is not reproducible in my html, and which I guess stands for 'lower limit'.)
To show this is a topological space, we need to show that the unions and finite intersections of all open sets are themselves open sets, which we can do as follows:
R and the empty set are open by stipulation.
The union of any number of unions of half-open intervals is trivially a union of half-open intervals.
Take two sets, U and V which are unions of half-open intervals, and their intersection, W. Denoting the half-open intervals whose union gives U and V by U1, etc. and V1, etc. and using 'U' to denote union and '^' to denote intersection, we can write W as
a union of intersections of half-open intervals, as follows:
(U1 ^ V1) U (U1 ^ V2) U ... U (U2 ^ V1) U ...
Consider two half-open intervals, [a,b) and [c,d), and call their intersection A. We have the following cases:
- b <= c,
then A is the empty set.
c < b <= d,
if a < c, A is [c,b)
otherwise A is [a,b).
b > d
A is empty if a >= d,
A = [a,d) when c <= a < d,
A = [c,b) should it be the case that a < c.
Hence in all the available cases, the intersection of two half-open intervals is either empty or itself a half-open interval, and so W is a union of half-open intervals. Since we can decompose any finite intersection into a set of such pairwise intersections, we have satisfied the requirements for the open sets of a topology and shown that R L is a topological space.
None of which is what is interesting about the Sorgenfrey line itself, which apparently has something to do with the fact that while it is itself a normal space (yeah, right), though non-metrizable, its self-product, R L x R L, sometimes known as the Sorgenfrey plane, is not normal.