In mathematics, the lower limit topology or right half-open interval topology is a topology defined on R {\displaystyle \mathbb {R} } , the set of real numbers; it is different from the standard topology on R {\displaystyle \mathbb {R} } (generated by the open intervals) 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 is called the Sorgenfrey line after Robert Sorgenfrey or the arrow and is sometimes written R l {\displaystyle \mathbb {R} _{l}} , R LL {\displaystyle \mathbb {R} _{\text{LL}}} or R bad {\displaystyle \mathbb {R} _{\text{bad}}} . Like the Cantor set and the long line, the Sorgenfrey line often serves as a useful counterexample to many otherwise plausible-sounding conjectures in general topology. The product of R l {\displaystyle \mathbb {R} _{l}} with itself is also a useful counterexample, known as the Sorgenfrey plane. 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 (countably infinite) union of half-open intervals. For any real a {\displaystyle a} and b {\displaystyle b} , the interval [ a , b ) {\displaystyle [a,b)} is clopen in R l {\displaystyle \mathbb {R} _{l}} (i.e., both open and closed). Furthermore, for all real a {\displaystyle a} , the sets { x ∈ R : x < a } {\displaystyle \{x\in \mathbb {R} :x<a\}} and { x ∈ R : x ≥ a } {\displaystyle \{x\in \mathbb {R} :x\geq a\}} are also clopen. This shows that the Sorgenfrey line is totally disconnected. Any compact subset of R l {\displaystyle \mathbb {R} _{l}} must be an at most countable set. To see this, consider a non-empty compact subset C ⊆ R l {\displaystyle C\subseteq \mathbb {R} _{l}} . Fix an x ∈ C {\displaystyle x\in C} , consider the following open cover of C {\displaystyle C} :
{ [ x , + ∞ ) } ∪ { ( − ∞ , x − 1 n ) | n ∈ N } . {\displaystyle {\bigl \{}[x,+\infty ){\bigr \}}\cup {\Bigl \{}{\bigl (}-\infty ,x-{\tfrac {1}{n}}{\bigr )}\,{\Big |}\,n\in \mathbb {N} {\Bigr \}}.}
… excerpt ends here. Continue reading the full article.
