In
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; it is different from the standard topology on
(generated by the
open intervals) and has a number of interesting properties. It is the topology generated by the
basis
Basis may refer to:
Finance and accounting
* Adjusted basis, the net cost of an asset after adjusting for various tax-related items
*Basis point, 0.01%, often used in the context of interest rates
* Basis trading, a trading strategy consisting ...
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. Othe ...
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, geometri ...
.
The product
Product may refer to:
Business
* Product (business), an item that serves as a solution to a specific consumer problem.
* Product (project management), a deliverable or set of deliverables that contribute to a business solution
Mathematics
* Produ ...
of with itself is also a useful counterexample, known as the Sorgenfrey plane
In topology, the Sorgenfrey plane is a frequently-cited counterexample to many otherwise plausible-sounding conjectures. It consists of the product of two copies of the Sorgenfrey line, which is the real line \mathbb under the half-open interval ...
.
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