Half-open Interval Topology
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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 \mathbb of real numbers; it is different from the standard topology on \mathbb (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 /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 \mathbb_l. 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 \mathbb_l 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 a and b, the interval
clopen_ In_topology,_a_clopen_set_(a_portmanteau_of_closed-open_set)_in_a_topological_space_is_a_set_which_is_both_open_and_closed.__That_this_is_possible_may_seem_counter-intuitive,_as_the_common_meanings_of__and__are_antonyms,_but_their_mathematical_de_...
_in_\mathbb_l_(i.e.,_both_ clopen_ In_topology,_a_clopen_set_(a_portmanteau_of_closed-open_set)_in_a_topological_space_is_a_set_which_is_both_open_and_closed.__That_this_is_possible_may_seem_counter-intuitive,_as_the_common_meanings_of__and__are_antonyms,_but_their_mathematical_de_...
_in_\mathbb_l_(i.e.,_both_open_set">open_ Open_or_OPEN_may_refer_to: _Music *__Open_(band),_Australian_pop/rock_band *__The_Open_(band),_English_indie_rock_band *__''Open''_(Blues_Image_album),_1969 *__''Open''_(Gotthard_album),_1999 *__''Open''_(Cowboy_Junkies_album),_2001 *__''Open''_(_...
_and_closed_set.html" ;"title="open_set.html" "title="clopen_set.html" "title=",b) is clopen set">clopen In topology, a clopen set (a portmanteau of closed-open set) in a topological space is a set which is both open and closed. That this is possible may seem counter-intuitive, as the common meanings of and are antonyms, but their mathematical de ...
 in \mathbb_l (i.e., both open set">open Open or OPEN may refer to: Music * Open (band), Australian pop/rock band * The Open (band), English indie rock band * ''Open'' (Blues Image album), 1969 * ''Open'' (Gotthard album), 1999 * ''Open'' (Cowboy Junkies album), 2001 * ''Open'' ( ...
 and closed set">closed). Furthermore, for all real a, the sets \ and \ are also clopen. This shows that the Sorgenfrey line is totally disconnected. * Any compact space, compact subset of \mathbb_l must be an at most countable set. To see this, consider a non-empty compact subset C\subseteq\mathbb_l. Fix an x \in C, consider the following open cover of C: :: \bigl\ \cup \Bigl\. :Since C is compact, this cover has a finite subcover, and hence there exists a real number a(x) such that the interval (a(x), x] contains no point of C apart from x. This is true for all x\in C. Now choose a rational number q(x) \in (a(x), x]\cap\mathbb. Since the intervals (a(x), x], parametrized by x \in C, are pairwise disjoint, the function q: C \to \mathbb is injective, and so C is at most countable. * The name "lower limit topology" comes from the following fact: a sequence (or net (topology), net) (x_\alpha) in \mathbb_l converges to the limit L
if and only if In logic and related fields such as mathematics and philosophy, "if and only if" (shortened as "iff") is a biconditional logical connective between statements, where either both statements are true or both are false. The connective is b ...
it "approaches L from the right", meaning for every \epsilon>0 there exists an index \alpha_0 such that \forall\alpha \geq \alpha_0 : L \leq x_\alpha < L+\epsilon. The Sorgenfrey line can thus be used to study
right-sided limit In calculus, a one-sided limit refers to either one of the two limits of a function f(x) of a real variable x as x approaches a specified point either from the left or from the right. The limit as x decreases in value approaching a (x approaches ...
s: if f: \mathbb \to \mathbb 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-oriente ...
, then the ordinary right-sided limit of f at x (when the codomain carries the standard topology) is the same as the usual 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, \mathbb_l is a
perfectly normal Hausdorff space In topology and related branches of mathematics, a normal space is a topological space ''X'' that satisfies Axiom T4: every two disjoint closed sets of ''X'' have disjoint open neighborhoods. A normal Hausdorff space is also called a T4 space. T ...
. * In terms of countability axioms, \mathbb_l 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 terms of compactness properties, \mathbb_l is Lindelöf and
paracompact In mathematics, a paracompact space is a topological space in which every open cover has an open refinement that is locally finite. These spaces were introduced by . Every compact space is paracompact. Every paracompact Hausdorff space is normal ...
, but not σ-compact nor locally compact. * \mathbb_l 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, \mathcal) is said to be metrizable if there is a metric d : X \times X \to , \infty) s ...
, since separable metric spaces are second-countable. However, the topology of a Sorgenfrey line is generated by a
quasimetric In mathematics, a metric space is a set together with a notion of ''distance'' between its elements, usually called points. The distance is measured by a function called a metric or distance function. Metric spaces are the most general settin ...
. * \mathbb_l 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 e ...
. * \mathbb_l 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) h ...
, 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 ...
, location=Berlin, New York , edition= Dover reprint of 1978 , isbn=978-0-486-68735-3 , mr=507446 , year=1995 Topological spaces