The Sorgenfrey line, named after
Robert Sorgenfrey (19151996), is the
lower limit topology on the
real numbers,
R.
This topology has as its basis the collection of all halfopen 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 halfopen 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 halfopen intervals is trivially a union of halfopen intervals.
Take two sets, U and V which are unions of halfopen intervals, and their intersection, W. Denoting the halfopen intervals whose union gives U and V by U_{1}, etc. and V_{1}, etc. and using 'U' to denote union and '^' to denote intersection, we can write W as
a union of intersections of halfopen intervals, as follows:
(U_{1} ^ V_{1}) U (U_{1} ^ V_{2}) U ... U (U_{2} ^ V_{1}) U ...
Consider two halfopen 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 halfopen intervals is either empty or itself a halfopen interval, and so W is a union of halfopen 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 nonmetrizable, its selfproduct, R _{L} x R _{L}, sometimes known as the Sorgenfrey plane, is not normal.