half-open interval topology on:
[Wikipedia]
[Google]
[Amazon]
In
mathematics
Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...
, the lower limit topology or right half-open interval topology is a
topology
Topology (from the Greek language, Greek words , and ) is the branch of mathematics concerned with the properties of a Mathematical object, geometric object that are preserved under Continuous function, continuous Deformation theory, deformat ...
defined on
, the set of
real numbers
In mathematics, a real number is a number that can be used to measurement, measure a continuous variable, continuous one-dimensional quantity such as a time, duration or temperature. Here, ''continuous'' means that pairs of values can have arbi ...
; it is different from the standard topology on
(generated by the
open interval
In mathematics, a real interval is the set (mathematics), set of all real numbers lying between two fixed endpoints with no "gaps". Each endpoint is either a real number or positive or negative infinity, indicating the interval extends without ...
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 the set of all real numbers lying between two fixed endpoints with no "gaps". Each endpoint is either a real number or positive or negative infinity, indicating the interval extends without a bound. A real in ...
s
/nowiki>''a'',''b''), where ''a'' and ''b'' are real numbers.
The resulting topological space is called the Sorgenfrey line after Robert Sorgenfrey">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.
The product space, product 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 inter ...
.
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). Furthermore, for all real
, the sets
and
are also clopen. This shows that the Sorgenfrey line is
closed set">closed). Furthermore, for all real
, the sets
and
are also clopen. This shows that the Sorgenfrey line is
compact subset of
must be an at most countable set">totally disconnected.
* Any compact space">compact subset of
must be an at most countable set. To see this, consider a non-empty compact subset
. Fix an
, consider the following open cover of
:
::
:Since
is compact, this cover has a finite subcover, and hence there exists a real number
such that the interval
contains no point of
apart from
. This is true for all
. Now choose a rational number
. Since the intervals
, parametrized by
, are pairwise disjoint, the function
is injective, and so
is at most countable.
* The name "lower limit topology" comes from the following fact: a sequence (or
net (topology), net)
in
converges to the limit
if and only if
In logic and related fields such as mathematics and philosophy, "if and only if" (often shortened as "iff") is paraphrased by the biconditional, a logical connective between statements. The biconditional is true in two cases, where either bo ...
it "approaches
from the right", meaning for every
there exists an index
such that
. The Sorgenfrey line can thus be used to study
right-sided limits: if
is a
function
Function or functionality may refer to:
Computing
* Function key, a type of key on computer keyboards
* Function model, a structured representation of processes in a system
* Function object or functor or functionoid, a concept of object-orie ...
, then the ordinary right-sided limit of
at
(when the codomain carries the standard topology) is the same as the usual limit of
at
when the domain is equipped with the lower limit topology and the codomain carries the standard topology.
* In terms of
separation axioms
In topology and related fields of mathematics, there are several restrictions that one often makes on the kinds of topological spaces that one wishes to consider. Some of these restrictions are given by the separation axioms. These are sometimes ...
,
is a
perfectly normal Hausdorff space.
* In terms of
countability axioms,
is
first-countable
In 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 (local base ...
and
separable, but not
second-countable
In topology, a second-countable space, also called a completely separable space, is a topological space whose topology has a countable base. More explicitly, a topological space T is second-countable if there exists some countable collection \mat ...
.
* In terms of compactness properties,
is
Lindelöf and
paracompact
In mathematics, a paracompact space is a topological space in which every open cover has an open Cover (topology)#Refinement, refinement that is locally finite collection, locally finite. These spaces were introduced by . Every compact space is par ...
, but not
σ-compact nor
locally compact
In topology and related branches of mathematics, a topological space is called locally compact if, roughly speaking, each small portion of the space looks like a small portion of a compact space. More precisely, it is a topological space in which e ...
.
*
is not
metrizable
In topology and related areas of mathematics, a metrizable space is a topological space that is homeomorphic to a metric space. That is, a topological space (X, \tau) is said to be metrizable if there is a metric d : X \times X \to , \infty) suc ...
, since separable metric spaces are second-countable. However, the topology of a Sorgenfrey line is generated by a
quasimetric.
*
is a
Baire space
In mathematics, a topological space X is said to be a Baire space if countable unions of closed sets with empty interior also have empty interior.
According to the Baire category theorem, compact Hausdorff spaces and complete metric spaces are ...
.
*
does not have any connected compactifications.
[Adam Emeryk, Władysław Kulpa. The Sorgenfrey line has no connected compactification. ''Comm. Math. Univ. Carolinae'' 18 (1977), 483–487.]
See also
*
List of topologies
The following is a list of named topologies or topological spaces, many of which are counterexamples in topology and related branches of mathematics. This is not a list of properties that a topology or topological space might possess; for that, ...
References
* {{Citation , last1=Steen , first1=Lynn Arthur , author1-link=Lynn Arthur Steen , last2=Seebach , first2=J. Arthur Jr. , author2-link=J. Arthur Seebach, Jr. , title=
Counterexamples in Topology
''Counterexamples in Topology'' (1970, 2nd ed. 1978) is a book on mathematics by topologists Lynn Steen and J. Arthur Seebach, Jr.
In the process of working on problems like the metrization problem, topologists (including Steen and Seebach) ...
, orig-year=1978 , publisher=
Springer-Verlag
Springer Science+Business Media, commonly known as Springer, is a German multinational publishing company of books, e-books and peer-reviewed journals in science, humanities, technical and medical (STM) publishing.
Originally founded in 1842 in ...
, location=Berlin, New York , edition=
Dover
Dover ( ) is a town and major ferry port in Kent, southeast England. It faces France across the Strait of Dover, the narrowest part of the English Channel at from Cap Gris Nez in France. It lies southeast of Canterbury and east of Maidstone. ...
reprint of 1978 , isbn=978-0-486-68735-3 , mr=507446 , year=1995
Topological spaces