Editing Interval (mathematics) (section)

==Definitions and terminology==
An ''interval'' is a [[subset]] of the [[real number]]s that contains all real numbers lying between any two numbers of the subset. In particular, the [[empty set]] <math>\varnothing</math> and the entire set of real numbers <math>\R</math> are both intervals.

The ''endpoints'' of an interval are its [[supremum]], and its [[infimum]], if they exist as real numbers.<ref name="bertsekas">{{cite book
 | last = Bertsekas | first = Dimitri P.
 | title = Network Optimization: Continuous and Discrete Methods
 | year = 1998
 | url = https://books.google.com/books?id=qUUxEAAAQBAJ&pg=PA409
 | page = 409
 | publisher = Athena Scientific
 | isbn = 1-886529-02-7
}}</ref> If the infimum does not exist, one says often that the corresponding endpoint is <math>-\infty.</math> Similarly, if the supremum does not exist, one says that the corresponding endpoint is <math>+\infty.</math>

Intervals are completely determined by their endpoints and whether each endpoint belong to the interval. This is a consequence of the [[least-upper-bound property]] of the real numbers. This characterization is used to specify intervals by mean of ''{{vanchor|interval notation}}'', which is described below.

An '''''{{visible anchor|open interval}}''''' does not include any endpoint, and is indicated with parentheses.<ref name="strichartz">{{cite book
 | last = Strichartz | first = Robert S.
 | title = The Way of Analysis
 | year = 2000
 | url = https://books.google.com/books?id=Yix09oVvI1IC&pg=PA86
 | page = 86
 | publisher = Jones & Bartlett Publishers
 | isbn = 0-7637-1497-6
}}</ref> For example, <math>(0, 1) = \{x \mid 0 < x < 1\}</math> is the interval of all real numbers greater than {{math|0}} and less than {{math|1}}. (This interval can also be denoted by {{math|]0, 1[}}, see below). The open interval {{math|{{open-open|0, +∞}}}} consists of real numbers greater than {{math|0}}, i.e., positive real numbers. The open intervals have thus one of the forms
:<math>\begin{align}
 (a,b)        &= \{x\in\mathbb R \mid a<x<b\}, \\
 (-\infty, b) &= \{x\in\mathbb R \mid x<b\}, \\
 (a, +\infty) &= \{x\in\mathbb R \mid a<x\}, \\
 (-\infty, +\infty) &= \R, \\
 (a,a)&=\emptyset,
\end{align}</math>
where <math>a</math> and <math>b</math> are real numbers such that <math>a< b.</math> In the last case, the resulting interval is the [[empty set]] and does not depend on {{tmath|a}}. The open intervals are those intervals that are [[open set]]s for the usual [[topological space|topology]] on the real numbers.

A '''''{{visible anchor|closed interval}}''''' is an interval that includes all its endpoints and is denoted with square brackets.<ref name="strichartz" /> For example, {{closed-closed|0, 1}} means greater than or equal to {{math|0}} and less than or equal to {{math|1}}. Closed intervals have one of the following forms in which {{mvar|a}} and {{mvar|b}} are real numbers such that <math>a<  b\colon</math>
:<math>\begin{align}
  \;[a,b]        &= \{x\in\mathbb R \mid a\le x\le b\}, \\
 \left(-\infty, b\right] &= \{x\in\mathbb R \mid x\le b\}, \\
 \left[a, +\infty\right) &= \{x\in\mathbb R \mid a\le x\}, \\
 (-\infty, +\infty) &= \R,\\
 \left[a,a\right]&=\{a\}.
\end{align}</math>
The closed intervals are those intervals that are [[closed set]]s for the usual [[topological space|topology]] on the real numbers.

A ''{{visible anchor|half-open interval}}'' has two endpoints and includes only one of them. It is said ''left-open'' or ''right-open'' depending on whether the excluded endpoint is on the left or on the right. These intervals are denoted by mixing notations for open and closed intervals.<ref name=":2">{{Cite web|last=Weisstein|first=Eric W.|title=Interval|url=https://mathworld.wolfram.com/Interval.html|access-date=2020-08-23|website=mathworld.wolfram.com|language=en}}</ref> For example, {{open-closed|0, 1}} means greater than {{math|0}} and less than or equal to {{math|1}}, while {{closed-open|0, 1}} means greater than or equal to {{math|0}} and less than {{math|1}}. The half-open intervals have the form
:<math>\begin{align}
 \left(a,b\right]       &= \{x\in\R \mid a<x\le b\}, \\
 \left[a,b\right)       &= \{x\in\R \mid a\le x<b\}. \\
\end{align}</math>

In summary, a set of the real numbers is an interval, if and only if it is an open interval, a closed interval, or a half-open interval. The only intervals that appear twice in the above classification are {{tmath|\emptyset}} and {{tmath|\R}} that are both open and closed.<ref name="eom">{{eom|title=Interval and segment}}</ref><ref name="tao">{{cite book
 | last = Tao | first = Terence | author-link = Terence Tao
 | title = Analysis I
 | year = 2016
 | url = https://books.google.com/books?id=ecTsDAAAQBAJ&pg=PA212
 | page = 212
 | edition = 3
 | series = Texts and Readings in Mathematics
 | volume = 37
 | publisher = Springer
 | location = Singapore
 | isbn = 978-981-10-1789-6
 | issn = 2366-8725
 | doi = 10.1007/978-981-10-1789-6
 | lccn = 2016940817
}} See Definition 9.1.1.</ref>

A ''{{visible anchor|degenerate interval}}'' is any [[singleton set|set consisting of a single real number]] (i.e., an interval of the form {{closed-closed|''a'', ''a''}}).<ref name="cramer">{{cite book
 | last = Cramér | first = Harald
 | title = Mathematical Methods of Statistics
 | year = 1999
 | url = https://books.google.com/books?id=CRTKKaJO0DYC&pg=PA11
 | page = 11
 | publisher = Princeton University Press
| isbn = 0691005478
 }}</ref> Some authors include the empty set in this definition. A real interval that is neither empty nor degenerate is said to be ''proper'', and has infinitely many elements.

{{anchor|bounded interval|unbounded interval|half-bounded interval|finite interval}}An interval is said to be ''left-bounded'' or ''right-bounded'', if there is some real number that is, respectively, smaller than or larger than all its elements. An interval is said to be ''bounded'', if it is both left- and right-bounded; and is said to be ''unbounded'' otherwise. Intervals that are bounded at only one end are said to be ''half-bounded''. The empty set is bounded, and the set of all reals is the only interval that is unbounded at both ends. Bounded intervals are also commonly known as ''finite intervals''.

Bounded intervals are [[bounded set]]s, in the sense that their [[diameter]] (which is equal to the [[absolute difference]] between the endpoints) is finite.  The diameter may be called the ''length'', ''width'', ''measure'', ''range'', or ''size'' of the interval. The size of unbounded intervals is usually defined as {{math|+∞}}, and the size of the empty interval may be defined as {{math|0}} (or left undefined).

The ''centre'' ([[midpoint]]) of a bounded interval with endpoints {{mvar|a}} and {{mvar|b}} is {{math|(''a'' + ''b'')/2}}, and its ''radius'' is the half-length {{math|{{mabs|''a'' − ''b''}}/2}}. These concepts are undefined for empty or unbounded intervals.

An interval is said to be ''left-open'' if and only if it contains no [[minimum]] (an element that is smaller than all other elements); ''right-open'' if it contains no [[maximum]]; and ''open'' if it contains neither. The interval {{math|{{closed-open|0, 1}} {{=}} {{mset|''x'' | 0 ≤ ''x'' &lt; 1}}}}, for example, is left-closed and right-open. The empty set and the set of all reals are both open and closed intervals, while the set of non-negative reals, is a closed interval that is right-open but not left-open. The open intervals are [[open set]]s of the real line in its standard [[point-set topology|topology]], and form a [[base (topology)|base]] of the open sets.

An interval is said to be ''left-closed'' if it has a minimum element or is left-unbounded, ''right-closed'' if it has a maximum or is right unbounded; it is simply ''closed'' if it is both left-closed and right closed. So, the closed intervals coincide with the [[closed set]]s in that topology.

The ''interior'' of an interval {{mvar|I}} is the largest open interval that is contained in {{mvar|I}}; it is also the set of points in {{mvar|I}} which are not endpoints of {{mvar|I}}. The ''closure'' of {{mvar|I}} is the smallest closed interval that contains {{mvar|I}}; which is also the set {{mvar|I}} augmented with its finite endpoints.

For any set {{mvar|X}} of real numbers, the ''interval enclosure'' or ''interval span'' of {{mvar|X}} is the unique interval that contains {{mvar|X}}, and does not properly contain any other interval that also contains {{mvar|X}}.

An interval {{mvar|I}} is a ''subinterval'' of interval {{mvar|J}} if {{mvar|I}} is a [[subset]] of {{mvar|J}}. An interval {{mvar|I}} is a ''proper subinterval'' of {{mvar|J}} if {{mvar|I}} is a [[proper subset]] of {{mvar|J}}.

However, there is conflicting terminology for the terms ''segment'' and ''interval'', which have been employed in the literature in two essentially opposite ways, resulting in ambiguity when these terms are used. The ''Encyclopedia of Mathematics''<ref>{{Cite web|url=https://www.encyclopediaofmath.org/index.php/Interval_and_segment|title=Interval and segment - Encyclopedia of Mathematics|website=encyclopediaofmath.org|access-date=2016-11-12|url-status=live|archive-url=https://web.archive.org/web/20141226211146/http://www.encyclopediaofmath.org/index.php/Interval_and_segment|archive-date=2014-12-26}}</ref> defines ''interval'' (without a qualifier) to exclude both endpoints (i.e., open interval) and ''segment'' to include both endpoints (i.e., closed interval), while Rudin's ''Principles of Mathematical Analysis''<ref>{{Cite book|title=Principles of Mathematical Analysis|url=https://archive.org/details/principlesmathem00rudi_663|url-access=limited|last=Rudin|first=Walter|publisher=McGraw-Hill|year=1976|isbn=0-07-054235-X|location=New York|pages=[https://archive.org/details/principlesmathem00rudi_663/page/n39 31]}}</ref> calls sets of the form [''a'', ''b''] ''intervals'' and sets of the form (''a'', ''b'') ''segments'' throughout. These terms tend to appear in older works; modern texts increasingly favor the term ''interval'' (qualified by ''open'', ''closed'', or ''half-open''), regardless of whether endpoints are included.