Sorgenfrey line (idea)
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|The Sorgenfrey line, named after Robert Sorgenfrey (1915-1996), is the lower limit topology on the real numbers, R.
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.
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.