In
mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, the lower limit topology or right half-open interval topology is a
topology
In mathematics, topology (from the Greek words , and ) is concerned with the properties of a geometric object that are preserved under continuous deformations, such as stretching, twisting, crumpling, and bending; that is, without closing ...
defined on the set
of
real numbers
In mathematics, a real number is a number that can be used to measure a ''continuous'' one-dimensional quantity such as a distance, duration or temperature. Here, ''continuous'' means that values can have arbitrarily small variations. Every r ...
; it is different from the standard topology on
(generated by the
open interval
In mathematics, a (real) interval is a set of real numbers that contains all real numbers lying between any two numbers of the set. For example, the set of numbers satisfying is an interval which contains , , and all numbers in between. Othe ...
s) and has a number of interesting properties. It is the topology generated by the
basis of all
half-open interval
In mathematics, a (real) interval is a set of real numbers that contains all real numbers lying between any two numbers of the set. For example, the set of numbers satisfying is an interval which contains , , and all numbers in between. Other ...
s
''a'',''b''), where ''a'' and ''b'' are real numbers.
The resulting topological space">/nowiki>''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 . Like the Cantor set and the long line (topology), long line, the Sorgenfrey line often serves as a useful counterexample to many otherwise plausible-sounding conjectures in general topology
In mathematics, general topology is the branch of topology that deals with the basic set-theoretic definitions and constructions used in topology. It is the foundation of most other branches of topology, including differential topology, geome ...
.
The product of 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 and , the interval _and_closed_set.html" "title="open_set.html" "title="clopen_set.html" "title=",b) is clopen set">clopen