Lower limit topology
Encyclopedia
In mathematics
, the lower limit topology or right half-open interval topology is a topology
defined on the set R of real numbers; it is different from the standard topology on R 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
, sometimes written Rl and called the Sorgenfrey line after Robert Sorgenfrey
, often serves as a useful counterexample in general topology
, like the Cantor set
and the long line
.
The product of Rl 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.
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
, the lower limit topology or right half-open interval topology is a topology
Topological space
Topological spaces are mathematical structures that allow the formal definition of concepts such as convergence, connectedness, and continuity. They appear in virtually every branch of modern mathematics and are a central unifying notion...
defined on the set R of real numbers; it is different from the standard topology on R and has a number of interesting properties. It is the topology generated by the basis of all half-open intervals
The resulting topological space
Topological space
Topological spaces are mathematical structures that allow the formal definition of concepts such as convergence, connectedness, and continuity. They appear in virtually every branch of modern mathematics and are a central unifying notion...
, sometimes written Rl and called the Sorgenfrey line after Robert Sorgenfrey
Robert Sorgenfrey
Robert Henry Sorgenfrey was an American mathematician and Professor Emeritus of Mathematics at the University of California, Los Angeles....
, often serves as a useful counterexample in general topology
General topology
In mathematics, general topology or point-set topology is the branch of topology which studies properties of topological spaces and structures defined on them...
, like the Cantor set
Cantor set
In mathematics, the Cantor set is a set of points lying on a single line segment that has a number of remarkable and deep properties. It was discovered in 1875 by Henry John Stephen Smith and introduced by German mathematician Georg Cantor in 1883....
and the long line
Long line (topology)
In topology, the long line is a topological space somewhat similar to the real line, but in a certain way "longer". It behaves locally just like the real line, but has different large-scale properties. Therefore it serves as one of the basic counterexamples of topology...
.
The product of Rl 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 and b, the interval
[ a, b) is clopenClopen setIn topology, a clopen set in a topological space is a set which is both open and closed. That this is possible for a set is not as counter-intuitive as it might seem if the terms open and closed were thought of as antonyms; in fact they are not...
in Rl (i.e., both openOpen setThe concept of an open set is fundamental to many areas of mathematics, especially point-set topology and metric topology. Intuitively speaking, a set U is open if any point x in U can be "moved" a small amount in any direction and still be in the set U...
and closedClosed setIn geometry, topology, and related branches of mathematics, a closed set is a set whose complement is an open set. In a topological space, a closed set can be defined as a set which contains all its limit points...
). Furthermore, for all real a, the sets {x ∈ R : x < a} and {x ∈ R : x ≥ a} are also clopen. This shows that the Sorgenfrey line is totally disconnected.
- The name "lower limit topology" comes from the following fact: a sequence (or net) (xα) in Rl converges to the limit L iffIFFIFF, Iff or iff may refer to:Technology/Science:* Identification friend or foe, an electronic radio-based identification system using transponders...
it "approaches L from the right", meaning for every ε>0 there exists an index α0 such that for all α > α0: L ≤ xα < L + ε. The Sorgenfrey line can thus be used to study right-sided limits: if f : R → R is a functionFunction (mathematics)In mathematics, a function associates one quantity, the argument of the function, also known as the input, with another quantity, the value of the function, also known as the output. A function assigns exactly one output to each input. The argument and the value may be real numbers, but they can...
, then the ordinary right-sided limit of f at x (when both domain and codomain carry the standard topology) is the same as the limit of f at x when the domain is equipped with the lower limit topology and the codomain carries the standard topology.
- In terms of separation axioms, Rl is a perfectly normal Hausdorff space.
- In terms of countability axiomsAxiom of countabilityIn mathematics, an axiom of countability is a property of certain mathematical objects that requires the existence of a countable set with certain properties, while without it such sets might not exist....
, it is first-countableFirst-countable spaceIn topology, a branch of mathematics, a first-countable space is a topological space satisfying the "first axiom of countability". Specifically, a space X is said to be first-countable if each point has a countable neighbourhood basis...
and separable, but not second-countableSecond-countable spaceIn topology, a second-countable space, also called a completely separable space, is a topological space satisfying the second axiom of countability. A space is said to be second-countable if its topology has a countable base...
.
- In terms of compactness properties, Rl is LindelöfLindelöf spaceIn mathematics, a Lindelöf space is a topological space in which every open cover has a countable subcover. The Lindelöf property is a weakening of the more commonly used notion of compactness, which requires the existence of a finite subcover....
and paracompact, but not σ-compact nor locally compact.
- Rl is not metrizable, since separable metric spaces are second-countable. However, the topology of a Sorgenfrey line is generated by a premetric.
- Rl is a Baire spaceBaire spaceIn mathematics, a Baire space is a topological space which, intuitively speaking, is very large and has "enough" points for certain limit processes. It is named in honor of René-Louis Baire who introduced the concept.- Motivation :...
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