6602:
634:
2277:
392:
1901:
1276:
42:
7836:, In: Research Association of Applied Geometry (RAAG) Memoirs, Ggujutsu Bunken Fukuy-kai. Tokyo, Japan, 1958, Vol. 2, pp. 29โ46 (547-564); reprinted in Japan Journal on Industrial and Applied Mathematics, 2009, Vol. 26, No. 2-3, pp. 126โ143.
3339:
The intersection of any collection of intervals is always an interval. The union of two intervals is an interval if and only if they have a non-empty intersection or an open end-point of one interval is a closed end-point of the other, for example
629:{\displaystyle {\begin{aligned}(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 )&=\mathbb {R} ,\end{aligned}}}
1621:, 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
3265:
3172:
2272:{\displaystyle {\begin{aligned}(a,b)={\mathopen {]}}a,b{\mathclose {(a,b]={\mathopen {]}}a,b{\mathclose {]}}&=\{x\in \mathbb {R} \mid a<x\leq b\},\\={\mathopen {}}&=\{x\in \mathbb {R} \mid a\leq x\leq b\}.\end{aligned}}}
3079:
187:
Unless explicitly otherwise specified, all intervals considered in this article are real intervals, that is, intervals of real numbers. Notable generalizations are summarized in a section below possibly with links to separate
994:
1906:
3334:
5818:
5452:
7024:
4282:
4056:
ball (without a well-defined center or radius), a half-space can be taken as analogous to a half-bounded interval, with its boundary plane as the (degenerate) sphere corresponding to the finite endpoint.
6851:
5229:
918:
5397:
5091:
4502:
of half-bounded intervals (with an empty intersection taken to mean the whole real line), and the intersection of any number of half-bounded intervals is a (possibly empty) interval. Generalized to
4959:
4887:
4815:
5160:
999:
397:
5025:
4743:
1884:
To indicate that one of the endpoints is to be excluded from the set, the corresponding square bracket can be either replaced with a parenthesis, or reversed. Both notations are described in
6745:
5338:
362:
2904:
6900:. In a reconfiguration published in 1956 by M Warmus, the axis of "balanced intervals" is used along with the axis of intervals that reduce to a point. Instead of the direct sum
3709:
157:
consists of computing with intervals instead of real numbers for providing a guaranteed enclosure of the result of a numerical computation, even in the presence of uncertainties of
5684:
5537:
5274:
3947:
5750:
3896:
3662:
776:
2939:
of an interval is the union of the interval and the set of its finite endpoints, and hence is also an interval. (The latter also follows from the fact that the closure of every
4489:
4102:
3409:
2561:
5506:
846:
5938:
5905:
5871:
5846:
5616:
4634:
2876:
2792:
6800:
6390:
6361:
5581:
3433:
2840:
2818:
2587:
1322:
943:
6927:
6547:
6425:
6154:
4669:
2701:
An integer interval that has a finite lower or upper endpoint always includes that endpoint. Therefore, the exclusion of endpoints can be explicitly denoted by writing
273:
247:
7060:
5652:
964:
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. For example,
706:
6567:
6508:
5713:
4011:
3584:
5303:
4389:
4359:
3844:
1416:
1384:
3178:
3085:
4186:
4156:
732:
6301:
6080:
6013:
5478:
2931:
6481:
6325:
6275:
6244:
6224:
6201:
6178:
6116:
6057:
6037:
5978:
5958:
4520:
4457:
4433:
4413:
4332:
4303:
4217:
4130:
4035:
3749:
3729:
677:
657:
3979:
1357:
2953:
1271:{\displaystyle {\begin{aligned}\left(a,b\right]&=\{x\in \mathbb {R} \mid a<x\leq b\},\\\left&=\{x\in \mathbb {R} \mid x\leq b\}.\end{aligned}}}
1487:. 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
3271:
1422:, the result will not be an interval, since it is not even a subset of the real numbers. Instead, the result can be seen as an interval in the
7332:
1433:
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.
5762:
277:
Intervals are completely determined by their endpoints and whether each endpoint belong to the interval. This is a consequence of the
5403:
1780:, which have been employed in the literature in two essentially opposite ways, resulting in ambiguity when these terms are used. The
4041:, a ball is the set of points whose distance from the center is less than the radius. In the 2-dimensional case, a ball is called a
6943:
6666:
4222:
5340:
Actually, the intervals with single or no endpoints are the same as the intervals with two endpoints in the larger preordered set
6808:
6638:
6619:
7871:
132:
5166:
2679:, for example, it is used to formally define a subrange type, most frequently used to specify lower and upper bounds of valid
1475:, if there is some real number that is, respectively, smaller than or larger than all its elements. An interval is said to be
853:
93:. A real interval can contain neither endpoint, either endpoint, or both endpoints, excluding any endpoint which is infinite.
7829:
7717:
7543:
7514:
7264:
5346:
5031:
1460:). Some authors include the empty set in this definition. A real interval that is neither empty nor degenerate is said to be
7859:
6645:
4893:
4821:
4749:
923:
The closed intervals are those intervals that are closed sets for the usual topology on the real numbers. The empty set and
7069:
5097:
4965:
4677:
2691:
6652:
7758:
7370:
7177:
7147:
6720:
6685:
6777:
7068:, provides the plane with a multiplicative structure having some analogies to ordinary complex arithmetic, such as
5715:
Unlike in the case of the real line, a convex set of a preordered set need not be an interval. For example, in the
2355:
Both notations may overlap with other uses of parentheses and brackets in mathematics. For instance, the notation
1418:
are neither an open set nor a closed set. If one allows an endpoint in the closed side to be an infinity (such as
223:, if they exist as real numbers. If the infimum does not exist, one says often that the corresponding endpoint is
6634:
5308:
305:
7707:
6086:, such a component is always unique. That is, the convex components of a subset of a totally ordered set form a
85:
lying between two fixed endpoints with no "gaps". Each endpoint is either a real number or positive or negative
7884:
7308:
6699:
6623:
4547:
7919:
7753:. Sigma Series in Pure Mathematics. Vol. 6 (Revised and completed ed.). Berlin: Heldermann Verlag.
7228:
2881:
30:
This article is about intervals of real numbers and some generalizations. For intervals in order theory, see
17:
6698:
Intervals can be associated with points of the plane, and hence regions of intervals can be associated with
3667:
5657:
5511:
5235:
3905:
7218:
5721:
3857:
7223:
3631:
2676:
740:
4462:
4075:
3343:
2843:
2534:
278:
136:
6886:
5552:
5457:
defined by adding new smallest and greatest elements (even if there were ones), which are subsets of
4499:
2462:
35:
7324:
5483:
3751:
are integers. Depending on the context, either endpoint may or may not be included in the interval.
822:
7111:
7086:
5914:
5878:
5851:
5826:
5589:
4607:
4053:
3779:
3613:
2856:
2772:
7362:
7251:. Texts and Readings in Mathematics. Vol. 37 (3 ed.). Singapore: Springer. p. 212.
6869:
of this algebra is the condensed interval . If interval is not in one of the ideals, then it has
6783:
6659:
6369:
6340:
5560:
3416:
2823:
2801:
2570:
1327:
1292:
926:
7096:
6612:
4583:
799:
is an interval that includes all its endpoints and is denoted with square brackets. For example,
31:
6903:
6513:
6395:
6124:
6098:
A generalization of the characterizations of the real intervals follows. For a non-empty subset
1812:
throughout. These terms tend to appear in older works; modern texts increasingly favor the term
7850:
7449:
6870:
6758:
6278:
4639:
2520:
1885:
252:
226:
7246:
7137:
7032:
5625:
2563:
The context affects some of the above definitions and terminology. For instance, the interval
682:
7924:
7298:
7167:
7101:
6552:
6493:
5689:
4594:
4070:
4049:
3984:
3542:
3260:{\displaystyle \operatorname {cl} (-\infty ,a)=\operatorname {cl} (-\infty ,a]=(-\infty ,a],}
3167:{\displaystyle \operatorname {cl} (a,+\infty )=\operatorname {cl} [a,+\infty )=[a,+\infty ),}
2528:
1289:, but an interval that is a closed set need not be a closed interval. For example, intervals
128:
109:
7354:
5282:
4364:
778:
which is a degenerate interval (see below). The open intervals are those intervals that are
7794:
7768:
7727:
7681:
7613:
7467:
6585:
6364:
6335:
4337:
3817:
2672:
2380:
1892:
1389:
7776:
7735:
7689:
7621:
7553:
8:
7355:
6854:
6573:
6083:
5716:
4199:
Allowing for a mix of open, closed, and infinite endpoints, the
Cartesian product of any
4165:
4135:
4105:
2795:
1503:
711:
154:
140:
7471:
6283:
6062:
5995:
5460:
2913:
7669:
7601:
7483:
7457:
6890:
6451:
6310:
6260:
6229:
6209:
6186:
6163:
6101:
6087:
6042:
6022:
5963:
5943:
5540:
4571:
4505:
4442:
4418:
4398:
4317:
4288:
4202:
4189:
4115:
4042:
4020:
3847:
3734:
3714:
3495:
topologies in the real line coincide, which is the standard topology of the real line.
3074:{\displaystyle \operatorname {cl} (a,b)=\operatorname {cl} (a,b]=\operatorname {cl} =,}
2936:
1648:
if it is both left-closed and right closed. So, the closed intervals coincide with the
1626:
1423:
662:
642:
249:
Similarly, if the supremum does not exist, one says that the corresponding endpoint is
7880:
3952:
2589:
is closed in the realm of ordinary reals, but not in the realm of the extended reals.
7929:
7892:
7868:
7754:
7713:
7661:
7640:"A direct proof that a linearly ordered space is hereditarily collection-wise normal"
7593:
7539:
7510:
7366:
7304:
7278:
7270:
7260:
7173:
7143:
6930:
6581:
4543:
4109:
3851:
3783:
3771:
The dyadic intervals consequently have a structure that reflects that of an infinite
3761:
Each dyadic interval is contained in exactly one dyadic interval of twice the length.
2944:
2384:
783:
78:
7487:
7479:
7772:
7731:
7685:
7651:
7617:
7583:
7549:
7531:
7475:
7252:
7065:
6878:
6866:
6773:
6702:
of the plane. Generally, an interval in mathematics corresponds to an ordered pair
6119:
4526:, an intersection of half-spaces (of arbitrary orientation) is (the interior of) a
2415:
2400:
2376:
281:
of the real numbers. This characterization is used to specify intervals by mean of
177:
7826:
7875:
7863:
7856:
7833:
7764:
7723:
7677:
7609:
7506:
6572:
The concepts of convex sets and convex components are used in a proof that every
6484:
6441:
5985:
5753:
4527:
4038:
3791:
3488:
2940:
2766:
2684:
1630:
7790:
3778:
Dyadic intervals are relevant to several areas of numerical analysis, including
3767:
If two open dyadic intervals overlap, then one of them is a subset of the other.
7809:
7091:
7081:
6882:
6772:
are allowed. Then, the collection of all intervals can be identified with the
6715:
6577:
6247:
5908:
5584:
4602:
4598:
4559:
4531:
4193:
4066:
3492:
2680:
2392:
2388:
1427:
302:
does not include any endpoint, and is indicated with parentheses. For example,
162:
7895:
7535:
7256:
7194:
7913:
7791:
Review of "Interval analysis in the extended interval space" by Edgar
Kaucher
7703:
7665:
7597:
7274:
7116:
6445:
4563:
1869:
1762:
1445:
148:
105:
7440:
2523:
endpoint to indicate that there is no bound in that direction. For example,
2445:; namely, the set of all real numbers that are either less than or equal to
386:, i.e., positive real numbers. The open intervals are thus one of the forms
208:
that contains all real numbers lying between any two numbers of the subset.
7805:
7106:
6934:
6802:
with itself, where addition and multiplication are defined component-wise.
4523:
3764:
Each dyadic interval is spanned by two dyadic intervals of half the length.
3436:
3329:{\displaystyle \operatorname {cl} (-\infty ,+\infty )=(-\infty ,\infty ).}
1788:(without a qualifier) to exclude both endpoints (i.e., open interval) and
7242:
6488:
4542:
An open interval is a connected open set of real numbers. Generalized to
3772:
3464:
2907:
2372:
1495:
205:
169:
90:
82:
70:
7462:
7673:
7605:
6626: in this section. Unsourced material may be challenged and removed.
6328:
6016:
2850:
1649:
1526:
of the interval. The size of unbounded intervals is usually defined as
1282:
158:
7827:"Theory of interval algebra and its application to numerical analysis"
1714:, and does not properly contain any other interval that also contains
7900:
7386:
4567:
4159:
3440:
2319:
1888:
1873:
1483:
otherwise. Intervals that are bounded at only one end are said to be
1286:
735:
104:, and all numbers in between is an interval, denoted and called the
7656:
7639:
7588:
7571:
6601:
5546:
7387:"Why is American and French notation different for open intervals (
6897:
6874:
4014:
3981:
is the corresponding closed ball, and the interval's two endpoints
2695:
2352:, all four notations are usually taken to represent the empty set.
1622:
1542:
1499:
779:
216:
173:
144:
86:
7420:
7282:
3758:
The length of a dyadic interval is always an integer power of two.
3604:
are both non-empty (and have non-empty interiors), if and only if
1506:
between the endpoints) is finite. The diameter may be called the
6896:
Every interval can be considered a symmetric interval around its
5813:{\displaystyle \mathbb {Q} =\{x\in \mathbb {Q} \mid x^{2}<2\}}
3787:
2606:
2396:
1792:
to include both endpoints (i.e., closed interval), while Rudin's
1599:
1591:
1583:. These concepts are undefined for empty or unbounded intervals.
220:
6761:, this restriction is discarded, and "reversed intervals" where
2461:
In some contexts, an interval may be defined as a subset of the
6255:
5447:{\displaystyle -\infty <x<\infty \qquad (\forall x\in X)}
3899:
1868:
of the interval. In countries where numbers are written with a
1738:
201:
7530:. Studies in Economic Theory. Vol. 14. Berlin: Springer.
4334:
is the result of replacing any non-degenerate interval factor
7019:{\displaystyle z={\tfrac {1}{2}}(x+y)+{\tfrac {1}{2}}(x-y)j,}
5981:
4277:{\displaystyle I=I_{1}\times I_{2}\times \cdots \times I_{n}}
2368:
6747:
of real numbers with itself, where it is often assumed that
4361:
by a degenerate interval consisting of a finite endpoint of
956:
has two endpoints and includes only one of them. It is said
811:. Closed intervals have one of the following forms in which
41:
6846:{\displaystyle (\mathbb {R} \oplus \mathbb {R} ,+,\times )}
1479:, if it is both left- and right-bounded; and is said to be
4577:
2519:
Even in the context of the ordinary reals, one may use an
2414:
too is occasionally used for ordered pairs, especially in
7509:, Miodrag Petkoviฤ, Ljiljana Petkoviฤ, Wiley-VCH, 1998,
5224:{\displaystyle (-\infty ,b]=\{x\in X\mid x\lesssim b\},}
1772:
However, there is conflicting terminology for the terms
1644:
if it has a maximum or is right unbounded; it is simply
913:{\displaystyle =\{x\in \mathbb {R} \mid a\leq x\leq b\}}
5392:{\displaystyle {\bar {X}}=X\sqcup \{-\infty ,\infty \}}
5086:{\displaystyle [a,\infty )=\{x\in X\mid a\lesssim x\},}
4546:
in general, a non-empty connected open set is called a
4065:
A finite interval is (the interior of) a 1-dimensional
1530:, and the size of the empty interval may be defined as
6984:
6954:
4954:{\displaystyle [a,b)=\{x\in X\mid a\lesssim x<b\},}
4882:{\displaystyle (a,b]=\{x\in X\mid a<x\lesssim b\},}
4810:{\displaystyle =\{x\in X\mid a\lesssim x\lesssim b\},}
3910:
3862:
3672:
3636:
1594:(an element that is smaller than all other elements);
945:
are the only intervals that are both open and closed.
116:; the set of all real numbers is an interval, denoted
7570:
Heath, R. W.; Lutzer, David J.; Zenor, P. L. (1973).
7139:
7064:
This linear mapping of the plane, which amounts of a
7035:
6946:
6906:
6811:
6786:
6723:
6555:
6516:
6496:
6454:
6398:
6372:
6343:
6313:
6286:
6263:
6232:
6212:
6189:
6166:
6127:
6104:
6065:
6045:
6025:
5998:
5966:
5946:
5917:
5881:
5854:
5829:
5765:
5724:
5692:
5660:
5628:
5592:
5563:
5514:
5486:
5463:
5406:
5349:
5311:
5285:
5238:
5169:
5100:
5034:
4968:
4896:
4824:
4752:
4680:
4642:
4610:
4593:
The concept of intervals can be defined in arbitrary
4508:
4465:
4445:
4421:
4401:
4367:
4340:
4320:
4291:
4225:
4205:
4168:
4138:
4118:
4078:
4023:
3987:
3955:
3908:
3860:
3820:
3737:
3717:
3670:
3634:
3545:
3419:
3346:
3274:
3181:
3088:
2956:
2916:
2884:
2859:
2826:
2804:
2775:
2573:
2537:
1904:
1392:
1360:
1330:
1295:
997:
929:
856:
825:
743:
714:
685:
665:
645:
395:
308:
255:
229:
7325:"Interval and segment - Encyclopedia of Mathematics"
6889:
determined by the axes, or ideals in this case. The
2842:
is also an interval. This is one formulation of the
2647:, is sometimes used to indicate the interval of all
2512:
denotes the set of all ordinary real numbers, while
6929:the ring of intervals has been identified with the
5155:{\displaystyle (-\infty ,b)=\{x\in X\mid x<b\},}
1879:
96:For example, the set of real numbers consisting of
7054:
7018:
6921:
6845:
6794:
6739:
6561:
6541:
6502:
6475:
6419:
6384:
6355:
6319:
6295:
6269:
6238:
6218:
6195:
6172:
6148:
6110:
6074:
6051:
6031:
6007:
5972:
5952:
5932:
5899:
5865:
5840:
5812:
5744:
5707:
5678:
5646:
5610:
5575:
5531:
5500:
5472:
5446:
5391:
5332:
5297:
5268:
5223:
5154:
5085:
5020:{\displaystyle (a,\infty )=\{x\in X\mid a<x\},}
5019:
4953:
4881:
4809:
4737:
4663:
4628:
4514:
4483:
4451:
4427:
4407:
4383:
4353:
4326:
4297:
4276:
4211:
4180:
4150:
4124:
4096:
4029:
4005:
3973:
3941:
3890:
3838:
3743:
3723:
3703:
3656:
3578:
3427:
3403:
3328:
3259:
3166:
3073:
2925:
2898:
2870:
2834:
2812:
2786:
2690:Another way to interpret integer intervals are as
2581:
2555:
2271:
1663:is the largest open interval that is contained in
1640:if it has a minimum element or is left-unbounded,
1410:
1378:
1351:
1316:
1270:
937:
912:
840:
770:
726:
700:
671:
651:
628:
356:
267:
241:
180:. The notation of integer intervals is considered
7890:
7576:Transactions of the American Mathematical Society
5547:Convex sets and convex components in order theory
4738:{\displaystyle (a,b)=\{x\in X\mid a<x<b\},}
1828:), regardless of whether endpoints are included.
734:in the first case, the resulting interval is the
364:is the interval of all real numbers greater than
7911:
7644:Proceedings of the American Mathematical Society
7569:
7507:Complex interval arithmetic and its applications
4459:are the faces that consist of a single point of
3754:Dyadic intervals have the following properties:
2947:is a connected subset.) In other words, we have
2794:It follows that the image of an interval by any
2508:are all meaningful and distinct. In particular,
7172:. Jones & Bartlett Publishers. p. 86.
6740:{\displaystyle \mathbb {R} \times \mathbb {R} }
3790:. Another way to represent such a structure is
3628:is a bounded real interval whose endpoints are
1876:may be used as a separator to avoid ambiguity.
168:Intervals are likewise defined on an arbitrary
4498:Any finite interval can be constructed as the
4060:
2433:to denote the complement of the interval
1683:is the smallest closed interval that contains
191:
6082:but such components need not be unique. In a
2465:, the set of all real numbers augmented with
53:on the number line. All numbers greater than
7161:
7159:
5807:
5774:
5386:
5371:
5215:
5191:
5146:
5122:
5077:
5053:
5011:
4987:
4945:
4915:
4873:
4843:
4801:
4771:
4729:
4699:
4000:
3988:
3586:and the elements that are greater than
3570:
3564:
2259:
2227:
2169:
2137:
2079:
2047:
1989:
1957:
1258:
1232:
1193:
1167:
1128:
1096:
1060:
1028:
907:
875:
576:
550:
515:
489:
454:
422:
351:
327:
131:. For example, they occur implicitly in the
89:, indicating the interval extends without a
6483:Actually, every Tychonoff space that has a
5333:{\displaystyle x\lesssim y\not \lesssim x.}
357:{\displaystyle (0,1)=\{x\mid 0<x<1\}}
139:asserts that the image of an interval by a
7303:. Princeton University Press. p. 11.
7165:
1831:
7748:
7742:
7655:
7587:
7461:
7156:
7135:
6824:
6816:
6788:
6733:
6725:
6686:Learn how and when to remove this message
6303:the following conditions are equivalent.
6156:the following conditions are equivalent.
6059:is contained in some convex component of
5856:
5831:
5784:
5767:
5729:
5519:
5494:
4468:
4081:
3421:
2892:
2861:
2828:
2806:
2777:
2575:
2540:
2237:
2147:
2057:
1967:
1242:
1177:
1106:
1038:
931:
885:
615:
560:
499:
432:
7633:
7631:
7565:
7563:
7129:
5848:since there is no square root of two in
4435:itself and all faces of its facets. The
988:. The half-open intervals have the form
372:. (This interval can also be denoted by
40:
7702:
7696:
7438:
4578:Intervals in posets and preordered sets
2899:{\displaystyle X\subseteq \mathbb {R} }
2757:are rarely used for integer intervals.
1545:) of a bounded interval with endpoints
14:
7912:
7869:Interval computations research centers
7810:Review of "Calculus of Approximations"
7296:
6591:
6435:
3704:{\displaystyle {\tfrac {j+1}{2^{n}}},}
3612:. This is an interval version of the
3467:are the closed bounded intervals
2476:In this interpretation, the notations
1446:set consisting of a single real number
382:consists of real numbers greater than
181:
133:epsilon-delta definition of continuity
7891:
7637:
7628:
7560:
7528:Independence, additivity, uncertainty
7352:
7290:
7192:
5679:{\displaystyle x\lesssim z\lesssim y}
5532:{\displaystyle {\bar {\mathbb {R} }}}
5269:{\displaystyle (-\infty ,\infty )=X,}
3942:{\displaystyle {\tfrac {1}{2}}(b-a).}
3531:: respectively, the elements of
2456:
1710:is the unique interval that contains
1691:augmented with its finite endpoints.
1606:if it contains neither. The interval
27:All numbers between two given numbers
7525:
7519:
6624:adding citations to reliable sources
6595:
5745:{\displaystyle (\mathbb {Q} ,\leq )}
4671:one similarly defines the intervals
4553:
3891:{\displaystyle {\tfrac {1}{2}}(a+b)}
3443:are the open bounded intervals
2733:. Alternate-bracket notations like
2592:
2421:Some authors such as Yves Tillรฉ use
1464:, and has infinitely many elements.
7426:
7357:Principles of Mathematical Analysis
7241:
6877:, the algebra of intervals forms a
4493:
3657:{\displaystyle {\tfrac {j}{2^{n}}}}
3619:
2878:The interval enclosure of a subset
1794:Principles of Mathematical Analysis
771:{\displaystyle (a,a)=\varnothing ,}
151:are defined over an interval; etc.
24:
7361:. New York: McGraw-Hill. pp.
7300:Mathematical Methods of Statistics
7235:
7211:
7142:. Athena Scientific. p. 409.
5823:is convex, but not an interval of
5429:
5422:
5410:
5383:
5377:
5251:
5245:
5176:
5107:
5044:
4978:
3804:
3317:
3311:
3296:
3287:
3242:
3221:
3194:
3155:
3134:
3107:
1667:; it is also the set of points in
1343:
1302:
1211:
1152:
601:
592:
537:
470:
259:
233:
25:
7941:
7840:
7186:
4562:can be defined as regions of the
4530:, or in the 2-dimensional case a
4484:{\displaystyle \mathbb {R} ^{n}.}
4097:{\displaystyle \mathbb {R} ^{n},}
3404:{\displaystyle (a,b)\cup =(a,c].}
2556:{\displaystyle \mathbb {R} _{+}.}
1864:. The two numbers are called the
1625:of the real line in its standard
762:
7429:, p. 214, See Lemma 9.1.12.
6600:
4597:or more generally, in arbitrary
2765:The intervals are precisely the
1880:Including or excluding endpoints
1836:The interval of numbers between
376:, see below). The open interval
7819:
7799:
7783:
7500:
7480:10.1070/IM2002v066n02ABEH000381
7432:
7335:from the original on 2014-12-26
6611:needs additional citations for
6510:is embeddable into the product
6430:
6093:
5425:
4108:hyperrectangle (or box) is the
2334:denotes the singleton set
1448:(i.e., an interval of the form
980:means greater than or equal to
803:means greater than or equal to
65:fall within that open interval.
7885:Wolfram Demonstrations Project
7379:
7346:
7317:
7166:Strichartz, Robert S. (2000).
7136:Bertsekas, Dimitri P. (1998).
7007:
6995:
6977:
6965:
6885:of this ring consists of four
6840:
6812:
6530:
6517:
6467:
6455:
6140:
6128:
5894:
5882:
5739:
5725:
5605:
5593:
5523:
5501:{\displaystyle X=\mathbb {R} }
5441:
5426:
5356:
5254:
5239:
5185:
5170:
5116:
5101:
5047:
5035:
4981:
4969:
4909:
4897:
4837:
4825:
4765:
4753:
4693:
4681:
4623:
4611:
4588:
3968:
3956:
3933:
3921:
3885:
3873:
3833:
3821:
3558:
3546:
3510:into three disjoint intervals
3395:
3383:
3377:
3365:
3359:
3347:
3320:
3305:
3299:
3281:
3251:
3236:
3230:
3215:
3203:
3188:
3158:
3143:
3137:
3122:
3110:
3095:
3065:
3053:
3047:
3035:
3023:
3011:
2999:
2987:
2975:
2963:
2449:, or greater than or equal to
2215:
2199:
2191:
2179:
2125:
2109:
2101:
2089:
2035:
2019:
2011:
1999:
1945:
1929:
1921:
1909:
1590:if and only if it contains no
1405:
1393:
1373:
1361:
1346:
1331:
1311:
1296:
869:
857:
841:{\displaystyle a\leq b\colon }
756:
744:
604:
586:
540:
525:
479:
464:
412:
400:
321:
309:
13:
1:
7857:Interval computations website
7712:(2 ed.). Prentice Hall.
7572:"Monotonically normal spaces"
7122:
6893:of this group is quadrant I.
6448:of the closed unit intervals
5933:{\displaystyle Y\subseteq X.}
5900:{\displaystyle (X,\lesssim )}
5866:{\displaystyle \mathbb {Q} .}
5841:{\displaystyle \mathbb {Q} ,}
5611:{\displaystyle (X,\lesssim )}
4629:{\displaystyle (X,\lesssim )}
2871:{\displaystyle \mathbb {R} .}
2787:{\displaystyle \mathbb {R} .}
2760:
2516:denotes the extended reals.
120:; and any single real number
6795:{\displaystyle \mathbb {R} }
6385:{\displaystyle F\subseteq L}
6356:{\displaystyle I\subseteq L}
5576:{\displaystyle A\subseteq X}
3506:defines a partition of
3428:{\displaystyle \mathbb {R} }
2835:{\displaystyle \mathbb {R} }
2813:{\displaystyle \mathbb {R} }
2582:{\displaystyle \mathbb {R} }
1352:{\displaystyle [a,+\infty )}
1317:{\displaystyle (-\infty ,b]}
938:{\displaystyle \mathbb {R} }
289:, which is described below.
182:in the special section below
127:Intervals are ubiquitous in
7:
7749:Engelking, Ryszard (1989).
7224:Encyclopedia of Mathematics
7075:
6937:through the identification
6635:"Interval" mathematics
6226:is a connected subset when
5988:of this poset is called an
4061:Multi-dimensional intervals
3949:The closed finite interval
3608:is in the interior of
2849:The intervals are also the
2692:sets defined by enumeration
2367:is often used to denote an
1782:Encyclopedia of Mathematics
1671:which are not endpoints of
1281:Every closed interval is a
819:are real numbers such that
679:are real numbers such that
192:Definitions and terminology
10:
7946:
7851:American Scientist article
6922:{\displaystyle R\oplus R,}
6542:{\displaystyle ^{\kappa }}
6420:{\displaystyle S=I\cap F.}
6331:and an (order-)convex set.
6149:{\displaystyle (L,\leq ),}
5550:
4581:
4537:
2844:intermediate value theorem
1636:An interval is said to be
1586:An interval is said to be
1498:, in the sense that their
1467:An interval is said to be
972:and less than or equal to
807:and less than or equal to
279:least-upper-bound property
186:
137:intermediate value theorem
124:is an interval, denoted .
29:
7853:provides an introduction.
7812:from Mathematical Reviews
7536:10.1007/978-3-540-24757-9
7257:10.1007/978-981-10-1789-6
6873:. Endowed with the usual
6569:copies of the intervals.
5553:convex set (order theory)
4664:{\displaystyle a,b\in X,}
268:{\displaystyle +\infty .}
242:{\displaystyle -\infty .}
36:Interval (disambiguation)
7450:Izvestiya RAN. Ser. Mat.
7445:-adic spectral analysis"
7439:Kozyrev, Sergey (2002).
7112:Partition of an interval
7055:{\displaystyle j^{2}=1.}
5647:{\displaystyle x,y\in A}
3846:is a 1-dimensional open
3814:An open finite interval
3809:
3780:adaptive mesh refinement
3535:that are less than
2403:introduced the notation
1796:calls sets of the form
1687:; which is also the set
701:{\displaystyle a\leq b.}
112:is an interval, denoted
7638:Steen, Lynn A. (1970).
7297:Cramรฉr, Harald (1999).
7097:Interval finite element
6805:The direct sum algebra
6562:{\displaystyle \kappa }
6503:{\displaystyle \kappa }
5708:{\displaystyle z\in A.}
4584:interval (order theory)
4284:is sometimes called an
4006:{\displaystyle \{a,b\}}
3579:{\displaystyle =\{x\},}
2659:included. The notation
1832:Notations for intervals
1502:(which is equal to the
215:of an interval are its
32:Interval (order theory)
7353:Rudin, Walter (1976).
7329:encyclopediaofmath.org
7219:"Interval and segment"
7056:
7020:
6923:
6871:multiplicative inverse
6847:
6796:
6759:mathematical structure
6741:
6563:
6543:
6504:
6477:
6421:
6386:
6357:
6321:
6297:
6271:
6240:
6220:
6197:
6174:
6150:
6112:
6076:
6053:
6033:
6009:
5974:
5954:
5934:
5901:
5867:
5842:
5814:
5746:
5709:
5680:
5648:
5612:
5577:
5533:
5502:
5474:
5448:
5393:
5334:
5299:
5298:{\displaystyle x<y}
5270:
5225:
5156:
5087:
5021:
4955:
4883:
4811:
4739:
4665:
4630:
4595:partially ordered sets
4516:
4485:
4453:
4429:
4409:
4385:
4384:{\displaystyle I_{k}.}
4355:
4328:
4299:
4278:
4213:
4182:
4152:
4132:finite intervals. For
4126:
4098:
4052:is taken as a kind of
4031:
4007:
3975:
3943:
3892:
3840:
3745:
3725:
3705:
3658:
3580:
3429:
3405:
3330:
3261:
3168:
3075:
2927:
2900:
2872:
2836:
2814:
2788:
2583:
2557:
2273:
1886:International standard
1800:and sets of the form (
1494:Bounded intervals are
1412:
1380:
1353:
1318:
1272:
939:
914:
842:
772:
728:
702:
673:
653:
630:
358:
269:
243:
66:
34:. For other uses, see
7407:hsm.stackexchange.com
7287:See Definition 9.1.1.
7199:mathworld.wolfram.com
7102:Interval (statistics)
7057:
7021:
6924:
6848:
6797:
6742:
6564:
6544:
6505:
6478:
6444:is embeddable into a
6422:
6387:
6358:
6322:
6298:
6272:
6241:
6221:
6198:
6175:
6151:
6113:
6077:
6054:
6034:
6010:
5975:
5955:
5935:
5902:
5868:
5843:
5815:
5747:
5710:
5681:
5649:
5613:
5578:
5534:
5503:
5475:
5449:
5394:
5335:
5300:
5271:
5226:
5157:
5088:
5022:
4956:
4884:
4812:
4740:
4666:
4631:
4517:
4486:
4454:
4430:
4410:
4386:
4356:
4354:{\displaystyle I_{k}}
4329:
4305:-dimensional interval
4300:
4279:
4214:
4183:
4153:
4127:
4099:
4071:real coordinate space
4032:
4013:form a 0-dimensional
4008:
3976:
3944:
3893:
3841:
3839:{\displaystyle (a,b)}
3746:
3726:
3706:
3659:
3581:
3539:, the singleton
3487:. In particular, the
3430:
3406:
3331:
3262:
3169:
3076:
2928:
2901:
2873:
2837:
2815:
2789:
2673:programming languages
2584:
2558:
2529:positive real numbers
2463:extended real numbers
2274:
1698:of real numbers, the
1534:(or left undefined).
1413:
1411:{\displaystyle [a,b)}
1381:
1379:{\displaystyle (a,b]}
1354:
1319:
1273:
940:
915:
843:
786:on the real numbers.
773:
729:
703:
674:
654:
631:
359:
270:
244:
129:mathematical analysis
110:positive real numbers
44:
7920:Sets of real numbers
7795:Mathematical Reviews
7033:
6944:
6904:
6861:โ R } and { :
6809:
6784:
6721:
6620:improve this article
6586:monotonically normal
6553:
6514:
6494:
6452:
6396:
6370:
6341:
6311:
6284:
6261:
6246:is endowed with the
6230:
6210:
6187:
6164:
6125:
6102:
6063:
6043:
6023:
6019:, any convex set of
5996:
5964:
5944:
5915:
5879:
5852:
5827:
5763:
5722:
5690:
5658:
5626:
5590:
5561:
5512:
5484:
5461:
5404:
5347:
5309:
5283:
5236:
5167:
5098:
5032:
4966:
4894:
4822:
4750:
4678:
4640:
4608:
4506:
4463:
4443:
4419:
4399:
4365:
4338:
4318:
4314:of such an interval
4289:
4223:
4203:
4166:
4136:
4116:
4076:
4021:
3985:
3953:
3906:
3858:
3818:
3735:
3715:
3668:
3632:
3614:trichotomy principle
3543:
3502:of an interval
3417:
3344:
3272:
3179:
3086:
2954:
2914:
2882:
2857:
2824:
2802:
2773:
2571:
2535:
1902:
1893:set builder notation
1390:
1358:
1328:
1293:
995:
927:
854:
823:
741:
712:
683:
663:
643:
393:
306:
253:
227:
7849:by Brian Hayes: An
7526:Vind, Karl (2003).
7472:2002IzMat..66..367K
7441:"Wavelet theory as
7193:Weisstein, Eric W.
7169:The Way of Analysis
7070:polar decomposition
6592:Topological algebra
6574:totally ordered set
6436:In general topology
6084:totally ordered set
5984:under inclusion. A
5940:The convex sets of
5717:totally ordered set
4181:{\displaystyle n=3}
4151:{\displaystyle n=2}
2796:continuous function
2391:, or (sometimes) a
2371:in set theory, the
1852:, is often denoted
1569:is the half-length
1504:absolute difference
1440:degenerate interval
968:means greater than
727:{\displaystyle a=b}
155:Interval arithmetic
141:continuous function
7893:Weisstein, Eric W.
7874:2007-02-03 at the
7862:2006-03-02 at the
7832:2012-03-09 at the
7789:Kaj Madsen (1979)
7052:
7016:
6993:
6963:
6931:hyperbolic numbers
6919:
6891:identity component
6843:
6792:
6757:. For purposes of
6737:
6559:
6539:
6500:
6473:
6417:
6382:
6353:
6317:
6296:{\displaystyle L,}
6293:
6267:
6236:
6216:
6193:
6170:
6146:
6108:
6075:{\displaystyle Y,}
6072:
6049:
6029:
6008:{\displaystyle Y.}
6005:
5970:
5950:
5930:
5897:
5863:
5838:
5810:
5742:
5705:
5676:
5644:
5608:
5573:
5541:extended real line
5529:
5498:
5473:{\displaystyle X.}
5470:
5444:
5389:
5330:
5295:
5266:
5221:
5152:
5083:
5017:
4951:
4879:
4807:
4735:
4661:
4626:
4544:topological spaces
4512:
4481:
4449:
4425:
4405:
4381:
4351:
4324:
4295:
4274:
4209:
4190:rectangular cuboid
4178:
4148:
4122:
4094:
4027:
4003:
3971:
3939:
3919:
3888:
3871:
3836:
3741:
3721:
3701:
3696:
3654:
3652:
3576:
3425:
3401:
3326:
3257:
3164:
3071:
2926:{\displaystyle X.}
2923:
2896:
2868:
2832:
2810:
2784:
2579:
2553:
2531:, also written as
2457:Infinite endpoints
2269:
2267:
1751:proper subinterval
1700:interval enclosure
1652:in that topology.
1633:of the open sets.
1598:if it contains no
1426:, which occurs in
1424:extended real line
1408:
1376:
1349:
1314:
1268:
1266:
952:half-open interval
935:
910:
838:
768:
724:
698:
669:
649:
626:
624:
354:
265:
239:
67:
7881:Interval Notation
7719:978-0-13-181629-9
7704:Munkres, James R.
7545:978-3-540-41683-8
7515:978-3-527-40134-5
7266:978-981-10-1789-6
6992:
6962:
6933:by M. Warmus and
6696:
6695:
6688:
6670:
6582:completely normal
6576:endowed with the
6476:{\displaystyle .}
6320:{\displaystyle S}
6270:{\displaystyle S}
6239:{\displaystyle L}
6219:{\displaystyle I}
6196:{\displaystyle I}
6173:{\displaystyle I}
6111:{\displaystyle I}
6052:{\displaystyle Y}
6032:{\displaystyle X}
5973:{\displaystyle Y}
5953:{\displaystyle X}
5526:
5359:
4636:and two elements
4554:Complex intervals
4515:{\displaystyle n}
4452:{\displaystyle I}
4428:{\displaystyle I}
4408:{\displaystyle I}
4327:{\displaystyle I}
4298:{\displaystyle n}
4212:{\displaystyle n}
4125:{\displaystyle n}
4110:Cartesian product
4069:. Generalized to
4030:{\displaystyle n}
4017:. Generalized to
3918:
3870:
3784:multigrid methods
3744:{\displaystyle n}
3724:{\displaystyle j}
3695:
3651:
3498:Any element
2945:topological space
2593:Integer intervals
2385:analytic geometry
672:{\displaystyle b}
652:{\displaystyle a}
285:interval notation
108:; the set of all
16:(Redirected from
7937:
7906:
7905:
7883:by George Beck,
7847:A Lucid Interval
7813:
7803:
7797:
7787:
7781:
7780:
7751:General topology
7746:
7740:
7739:
7700:
7694:
7693:
7659:
7635:
7626:
7625:
7591:
7567:
7558:
7557:
7523:
7517:
7504:
7498:
7497:
7495:
7494:
7465:
7444:
7436:
7430:
7424:
7418:
7417:
7415:
7413:
7383:
7377:
7376:
7360:
7350:
7344:
7343:
7341:
7340:
7321:
7315:
7314:
7294:
7288:
7286:
7239:
7233:
7232:
7215:
7209:
7208:
7206:
7205:
7190:
7184:
7183:
7163:
7154:
7153:
7133:
7066:ring isomorphism
7061:
7059:
7058:
7053:
7045:
7044:
7025:
7023:
7022:
7017:
6994:
6985:
6964:
6955:
6928:
6926:
6925:
6920:
6879:topological ring
6867:identity element
6852:
6850:
6849:
6844:
6827:
6819:
6801:
6799:
6798:
6793:
6791:
6774:topological ring
6771:
6756:
6746:
6744:
6743:
6738:
6736:
6728:
6713:
6691:
6684:
6680:
6677:
6671:
6669:
6628:
6604:
6596:
6568:
6566:
6565:
6560:
6548:
6546:
6545:
6540:
6538:
6537:
6509:
6507:
6506:
6501:
6482:
6480:
6479:
6474:
6426:
6424:
6423:
6418:
6391:
6389:
6388:
6383:
6362:
6360:
6359:
6354:
6326:
6324:
6323:
6318:
6302:
6300:
6299:
6294:
6276:
6274:
6273:
6268:
6245:
6243:
6242:
6237:
6225:
6223:
6222:
6217:
6203:is order-convex.
6202:
6200:
6199:
6194:
6179:
6177:
6176:
6171:
6155:
6153:
6152:
6147:
6120:linear continuum
6117:
6115:
6114:
6109:
6081:
6079:
6078:
6073:
6058:
6056:
6055:
6050:
6038:
6036:
6035:
6030:
6014:
6012:
6011:
6006:
5990:convex component
5979:
5977:
5976:
5971:
5959:
5957:
5956:
5951:
5939:
5937:
5936:
5931:
5906:
5904:
5903:
5898:
5872:
5870:
5869:
5864:
5859:
5847:
5845:
5844:
5839:
5834:
5819:
5817:
5816:
5811:
5800:
5799:
5787:
5770:
5754:rational numbers
5751:
5749:
5748:
5743:
5732:
5714:
5712:
5711:
5706:
5685:
5683:
5682:
5677:
5653:
5651:
5650:
5645:
5617:
5615:
5614:
5609:
5582:
5580:
5579:
5574:
5538:
5536:
5535:
5530:
5528:
5527:
5522:
5517:
5507:
5505:
5504:
5499:
5497:
5479:
5477:
5476:
5471:
5453:
5451:
5450:
5445:
5398:
5396:
5395:
5390:
5361:
5360:
5352:
5339:
5337:
5336:
5331:
5304:
5302:
5301:
5296:
5275:
5273:
5272:
5267:
5230:
5228:
5227:
5222:
5161:
5159:
5158:
5153:
5092:
5090:
5089:
5084:
5026:
5024:
5023:
5018:
4960:
4958:
4957:
4952:
4888:
4886:
4885:
4880:
4816:
4814:
4813:
4808:
4744:
4742:
4741:
4736:
4670:
4668:
4667:
4662:
4635:
4633:
4632:
4627:
4521:
4519:
4518:
4513:
4494:Convex polytopes
4490:
4488:
4487:
4482:
4477:
4476:
4471:
4458:
4456:
4455:
4450:
4434:
4432:
4431:
4426:
4414:
4412:
4411:
4406:
4390:
4388:
4387:
4382:
4377:
4376:
4360:
4358:
4357:
4352:
4350:
4349:
4333:
4331:
4330:
4325:
4304:
4302:
4301:
4296:
4283:
4281:
4280:
4275:
4273:
4272:
4254:
4253:
4241:
4240:
4218:
4216:
4215:
4210:
4192:(also called a "
4187:
4185:
4184:
4179:
4157:
4155:
4154:
4149:
4131:
4129:
4128:
4123:
4103:
4101:
4100:
4095:
4090:
4089:
4084:
4036:
4034:
4033:
4028:
4012:
4010:
4009:
4004:
3980:
3978:
3977:
3974:{\displaystyle }
3972:
3948:
3946:
3945:
3940:
3920:
3911:
3897:
3895:
3894:
3889:
3872:
3863:
3845:
3843:
3842:
3837:
3800:
3788:wavelet analysis
3750:
3748:
3747:
3742:
3730:
3728:
3727:
3722:
3710:
3708:
3707:
3702:
3697:
3694:
3693:
3684:
3673:
3663:
3661:
3660:
3655:
3653:
3650:
3649:
3637:
3620:Dyadic intervals
3611:
3607:
3600:
3593:
3589:
3585:
3583:
3582:
3577:
3538:
3534:
3527:
3520:
3513:
3509:
3505:
3501:
3486:
3462:
3434:
3432:
3431:
3426:
3424:
3410:
3408:
3407:
3402:
3335:
3333:
3332:
3327:
3266:
3264:
3263:
3258:
3173:
3171:
3170:
3165:
3080:
3078:
3077:
3072:
2941:connected subset
2932:
2930:
2929:
2924:
2905:
2903:
2902:
2897:
2895:
2877:
2875:
2874:
2869:
2864:
2841:
2839:
2838:
2833:
2831:
2819:
2817:
2816:
2811:
2809:
2793:
2791:
2790:
2785:
2780:
2756:
2744:
2732:
2721:
2711:
2671:is used in some
2670:
2658:
2654:
2646:
2636:
2624:
2609:, the notation โฆ
2604:
2600:
2588:
2586:
2585:
2580:
2578:
2566:
2562:
2560:
2559:
2554:
2549:
2548:
2543:
2526:
2515:
2514:[โโ,โ+โ]
2511:
2507:
2499:
2491:
2483:
2472:
2468:
2452:
2448:
2444:
2432:
2416:computer science
2413:
2366:
2351:
2341:
2333:
2317:
2305:
2293:
2278:
2276:
2275:
2270:
2268:
2240:
2219:
2218:
2203:
2202:
2150:
2129:
2128:
2113:
2112:
2060:
2039:
2038:
2023:
2022:
1970:
1949:
1948:
1933:
1932:
1863:
1851:
1847:
1843:
1839:
1768:
1760:
1756:
1748:
1744:
1736:
1732:
1724:
1717:
1713:
1709:
1697:
1690:
1686:
1682:
1674:
1670:
1666:
1662:
1620:
1610:
1582:
1580:
1564:
1552:
1548:
1533:
1529:
1489:finite intervals
1459:
1442:
1441:
1421:
1417:
1415:
1414:
1409:
1385:
1383:
1382:
1377:
1356:
1355:
1350:
1323:
1321:
1320:
1315:
1277:
1275:
1274:
1269:
1267:
1245:
1224:
1220:
1180:
1159:
1155:
1109:
1088:
1084:
1041:
1020:
1016:
987:
983:
979:
975:
971:
967:
954:
953:
944:
942:
941:
936:
934:
919:
917:
916:
911:
888:
847:
845:
844:
839:
818:
814:
810:
806:
802:
796:
795:
777:
775:
774:
769:
733:
731:
730:
725:
707:
705:
704:
699:
678:
676:
675:
670:
658:
656:
655:
650:
635:
633:
632:
627:
625:
618:
563:
502:
435:
385:
381:
380:
375:
371:
367:
363:
361:
360:
355:
299:
298:
287:
286:
274:
272:
271:
266:
248:
246:
245:
240:
178:rational numbers
143:is an interval;
123:
119:
115:
103:
99:
21:
7945:
7944:
7940:
7939:
7938:
7936:
7935:
7934:
7910:
7909:
7876:Wayback Machine
7864:Wayback Machine
7843:
7834:Wayback Machine
7822:
7817:
7816:
7804:
7800:
7788:
7784:
7761:
7747:
7743:
7720:
7701:
7697:
7657:10.2307/2037311
7636:
7629:
7589:10.2307/1996713
7568:
7561:
7546:
7524:
7520:
7505:
7501:
7492:
7490:
7463:math-ph/0012019
7442:
7437:
7433:
7425:
7421:
7411:
7409:
7385:
7384:
7380:
7373:
7351:
7347:
7338:
7336:
7323:
7322:
7318:
7311:
7295:
7291:
7267:
7240:
7236:
7217:
7216:
7212:
7203:
7201:
7191:
7187:
7180:
7164:
7157:
7150:
7134:
7130:
7125:
7078:
7040:
7036:
7034:
7031:
7030:
6983:
6953:
6945:
6942:
6941:
6905:
6902:
6901:
6823:
6815:
6810:
6807:
6806:
6787:
6785:
6782:
6781:
6762:
6748:
6732:
6724:
6722:
6719:
6718:
6714:taken from the
6703:
6692:
6681:
6675:
6672:
6629:
6627:
6617:
6605:
6594:
6554:
6551:
6550:
6533:
6529:
6515:
6512:
6511:
6495:
6492:
6491:
6453:
6450:
6449:
6442:Tychonoff space
6438:
6433:
6397:
6394:
6393:
6371:
6368:
6367:
6342:
6339:
6338:
6312:
6309:
6308:
6285:
6282:
6281:
6262:
6259:
6258:
6231:
6228:
6227:
6211:
6208:
6207:
6188:
6185:
6184:
6180:is an interval.
6165:
6162:
6161:
6126:
6123:
6122:
6103:
6100:
6099:
6096:
6064:
6061:
6060:
6044:
6041:
6040:
6024:
6021:
6020:
5997:
5994:
5993:
5986:maximal element
5965:
5962:
5961:
5945:
5942:
5941:
5916:
5913:
5912:
5880:
5877:
5876:
5855:
5853:
5850:
5849:
5830:
5828:
5825:
5824:
5795:
5791:
5783:
5766:
5764:
5761:
5760:
5728:
5723:
5720:
5719:
5691:
5688:
5687:
5659:
5656:
5655:
5627:
5624:
5623:
5591:
5588:
5587:
5562:
5559:
5558:
5555:
5549:
5518:
5516:
5515:
5513:
5510:
5509:
5493:
5485:
5482:
5481:
5480:In the case of
5462:
5459:
5458:
5405:
5402:
5401:
5351:
5350:
5348:
5345:
5344:
5310:
5307:
5306:
5284:
5281:
5280:
5237:
5234:
5233:
5168:
5165:
5164:
5099:
5096:
5095:
5033:
5030:
5029:
4967:
4964:
4963:
4895:
4892:
4891:
4823:
4820:
4819:
4751:
4748:
4747:
4679:
4676:
4675:
4641:
4638:
4637:
4609:
4606:
4605:
4599:preordered sets
4591:
4586:
4580:
4560:complex numbers
4556:
4540:
4528:convex polytope
4507:
4504:
4503:
4496:
4472:
4467:
4466:
4464:
4461:
4460:
4444:
4441:
4440:
4420:
4417:
4416:
4400:
4397:
4396:
4372:
4368:
4366:
4363:
4362:
4345:
4341:
4339:
4336:
4335:
4319:
4316:
4315:
4290:
4287:
4286:
4268:
4264:
4249:
4245:
4236:
4232:
4224:
4221:
4220:
4204:
4201:
4200:
4167:
4164:
4163:
4137:
4134:
4133:
4117:
4114:
4113:
4085:
4080:
4079:
4077:
4074:
4073:
4063:
4039:Euclidean space
4022:
4019:
4018:
3986:
3983:
3982:
3954:
3951:
3950:
3909:
3907:
3904:
3903:
3861:
3859:
3856:
3855:
3819:
3816:
3815:
3812:
3807:
3805:Generalizations
3795:
3792:p-adic analysis
3736:
3733:
3732:
3716:
3713:
3712:
3689:
3685:
3674:
3671:
3669:
3666:
3665:
3645:
3641:
3635:
3633:
3630:
3629:
3626:dyadic interval
3622:
3609:
3605:
3603:
3598:
3596:
3591:
3587:
3544:
3541:
3540:
3536:
3532:
3530:
3525:
3523:
3518:
3516:
3511:
3507:
3503:
3499:
3468:
3444:
3435:is viewed as a
3420:
3418:
3415:
3414:
3345:
3342:
3341:
3273:
3270:
3269:
3180:
3177:
3176:
3087:
3084:
3083:
2955:
2952:
2951:
2915:
2912:
2911:
2891:
2883:
2880:
2879:
2860:
2858:
2855:
2854:
2827:
2825:
2822:
2821:
2805:
2803:
2800:
2799:
2776:
2774:
2771:
2770:
2763:
2746:
2734:
2723:
2713:
2702:
2660:
2656:
2652:
2638:
2626:
2614:
2602:
2598:
2595:
2574:
2572:
2569:
2568:
2564:
2544:
2539:
2538:
2536:
2533:
2532:
2524:
2513:
2509:
2501:
2493:
2485:
2477:
2470:
2466:
2459:
2450:
2446:
2434:
2422:
2404:
2356:
2343:
2335:
2323:
2318:represents the
2307:
2295:
2283:
2266:
2265:
2236:
2220:
2214:
2213:
2198:
2197:
2176:
2175:
2146:
2130:
2124:
2123:
2108:
2107:
2086:
2085:
2056:
2040:
2034:
2033:
2018:
2017:
1996:
1995:
1966:
1950:
1944:
1943:
1928:
1927:
1905:
1903:
1900:
1899:
1882:
1853:
1849:
1845:
1841:
1837:
1834:
1766:
1758:
1754:
1746:
1742:
1734:
1730:
1722:
1715:
1711:
1707:
1695:
1688:
1684:
1680:
1672:
1668:
1664:
1660:
1659:of an interval
1608:
1607:
1572:
1570:
1554:
1550:
1546:
1531:
1527:
1449:
1439:
1438:
1430:, for example.
1419:
1391:
1388:
1387:
1359:
1329:
1326:
1325:
1294:
1291:
1290:
1265:
1264:
1241:
1225:
1207:
1203:
1200:
1199:
1176:
1160:
1142:
1138:
1135:
1134:
1105:
1089:
1074:
1070:
1067:
1066:
1037:
1021:
1006:
1002:
998:
996:
993:
992:
985:
981:
977:
973:
969:
965:
951:
950:
930:
928:
925:
924:
884:
855:
852:
851:
824:
821:
820:
816:
812:
808:
804:
800:
794:closed interval
793:
792:
742:
739:
738:
713:
710:
709:
684:
681:
680:
664:
661:
660:
644:
641:
640:
623:
622:
614:
607:
583:
582:
559:
543:
522:
521:
498:
482:
461:
460:
431:
415:
396:
394:
391:
390:
383:
378:
377:
373:
369:
365:
307:
304:
303:
296:
295:
284:
283:
254:
251:
250:
228:
225:
224:
194:
189:
170:totally ordered
163:rounding errors
121:
117:
113:
101:
97:
39:
28:
23:
22:
15:
12:
11:
5:
7943:
7933:
7932:
7927:
7922:
7908:
7907:
7888:
7878:
7866:
7854:
7842:
7841:External links
7839:
7838:
7837:
7821:
7818:
7815:
7814:
7798:
7782:
7759:
7741:
7718:
7695:
7650:(4): 727โ728.
7627:
7559:
7544:
7518:
7499:
7456:(2): 149โ158.
7431:
7419:
7378:
7371:
7345:
7316:
7309:
7289:
7265:
7234:
7210:
7185:
7178:
7155:
7148:
7127:
7126:
7124:
7121:
7120:
7119:
7114:
7109:
7104:
7099:
7094:
7092:Interval graph
7089:
7084:
7082:Arc (geometry)
7077:
7074:
7051:
7048:
7043:
7039:
7027:
7026:
7015:
7012:
7009:
7006:
7003:
7000:
6997:
6991:
6988:
6982:
6979:
6976:
6973:
6970:
6967:
6961:
6958:
6952:
6949:
6918:
6915:
6912:
6909:
6883:group of units
6842:
6839:
6836:
6833:
6830:
6826:
6822:
6818:
6814:
6790:
6776:formed by the
6735:
6731:
6727:
6716:direct product
6694:
6693:
6676:September 2023
6608:
6606:
6599:
6593:
6590:
6578:order topology
6558:
6536:
6532:
6528:
6525:
6522:
6519:
6499:
6472:
6469:
6466:
6463:
6460:
6457:
6437:
6434:
6432:
6429:
6428:
6427:
6416:
6413:
6410:
6407:
6404:
6401:
6381:
6378:
6375:
6352:
6349:
6346:
6332:
6316:
6292:
6289:
6266:
6252:
6251:
6248:order topology
6235:
6215:
6204:
6192:
6181:
6169:
6145:
6142:
6139:
6136:
6133:
6130:
6107:
6095:
6092:
6071:
6068:
6048:
6028:
6004:
6001:
5969:
5949:
5929:
5926:
5923:
5920:
5909:preordered set
5896:
5893:
5890:
5887:
5884:
5862:
5858:
5837:
5833:
5821:
5820:
5809:
5806:
5803:
5798:
5794:
5790:
5786:
5782:
5779:
5776:
5773:
5769:
5741:
5738:
5735:
5731:
5727:
5704:
5701:
5698:
5695:
5675:
5672:
5669:
5666:
5663:
5643:
5640:
5637:
5634:
5631:
5620:(order-)convex
5607:
5604:
5601:
5598:
5595:
5585:preordered set
5572:
5569:
5566:
5551:Main article:
5548:
5545:
5525:
5521:
5496:
5492:
5489:
5469:
5466:
5455:
5454:
5443:
5440:
5437:
5434:
5431:
5428:
5424:
5421:
5418:
5415:
5412:
5409:
5399:
5388:
5385:
5382:
5379:
5376:
5373:
5370:
5367:
5364:
5358:
5355:
5329:
5326:
5323:
5320:
5317:
5314:
5294:
5291:
5288:
5277:
5276:
5265:
5262:
5259:
5256:
5253:
5250:
5247:
5244:
5241:
5231:
5220:
5217:
5214:
5211:
5208:
5205:
5202:
5199:
5196:
5193:
5190:
5187:
5184:
5181:
5178:
5175:
5172:
5162:
5151:
5148:
5145:
5142:
5139:
5136:
5133:
5130:
5127:
5124:
5121:
5118:
5115:
5112:
5109:
5106:
5103:
5093:
5082:
5079:
5076:
5073:
5070:
5067:
5064:
5061:
5058:
5055:
5052:
5049:
5046:
5043:
5040:
5037:
5027:
5016:
5013:
5010:
5007:
5004:
5001:
4998:
4995:
4992:
4989:
4986:
4983:
4980:
4977:
4974:
4971:
4961:
4950:
4947:
4944:
4941:
4938:
4935:
4932:
4929:
4926:
4923:
4920:
4917:
4914:
4911:
4908:
4905:
4902:
4899:
4889:
4878:
4875:
4872:
4869:
4866:
4863:
4860:
4857:
4854:
4851:
4848:
4845:
4842:
4839:
4836:
4833:
4830:
4827:
4817:
4806:
4803:
4800:
4797:
4794:
4791:
4788:
4785:
4782:
4779:
4776:
4773:
4770:
4767:
4764:
4761:
4758:
4755:
4745:
4734:
4731:
4728:
4725:
4722:
4719:
4716:
4713:
4710:
4707:
4704:
4701:
4698:
4695:
4692:
4689:
4686:
4683:
4660:
4657:
4654:
4651:
4648:
4645:
4625:
4622:
4619:
4616:
4613:
4603:preordered set
4590:
4587:
4582:Main article:
4579:
4576:
4555:
4552:
4539:
4536:
4532:convex polygon
4511:
4495:
4492:
4480:
4475:
4470:
4448:
4424:
4404:
4380:
4375:
4371:
4348:
4344:
4323:
4294:
4271:
4267:
4263:
4260:
4257:
4252:
4248:
4244:
4239:
4235:
4231:
4228:
4208:
4177:
4174:
4171:
4147:
4144:
4141:
4121:
4093:
4088:
4083:
4067:hyperrectangle
4062:
4059:
4026:
4002:
3999:
3996:
3993:
3990:
3970:
3967:
3964:
3961:
3958:
3938:
3935:
3932:
3929:
3926:
3923:
3917:
3914:
3887:
3884:
3881:
3878:
3875:
3869:
3866:
3835:
3832:
3829:
3826:
3823:
3811:
3808:
3806:
3803:
3769:
3768:
3765:
3762:
3759:
3740:
3720:
3700:
3692:
3688:
3683:
3680:
3677:
3648:
3644:
3640:
3621:
3618:
3601:
3594:
3575:
3572:
3569:
3566:
3563:
3560:
3557:
3554:
3551:
3548:
3528:
3521:
3514:
3423:
3400:
3397:
3394:
3391:
3388:
3385:
3382:
3379:
3376:
3373:
3370:
3367:
3364:
3361:
3358:
3355:
3352:
3349:
3337:
3336:
3325:
3322:
3319:
3316:
3313:
3310:
3307:
3304:
3301:
3298:
3295:
3292:
3289:
3286:
3283:
3280:
3277:
3267:
3256:
3253:
3250:
3247:
3244:
3241:
3238:
3235:
3232:
3229:
3226:
3223:
3220:
3217:
3214:
3211:
3208:
3205:
3202:
3199:
3196:
3193:
3190:
3187:
3184:
3174:
3163:
3160:
3157:
3154:
3151:
3148:
3145:
3142:
3139:
3136:
3133:
3130:
3127:
3124:
3121:
3118:
3115:
3112:
3109:
3106:
3103:
3100:
3097:
3094:
3091:
3081:
3070:
3067:
3064:
3061:
3058:
3055:
3052:
3049:
3046:
3043:
3040:
3037:
3034:
3031:
3028:
3025:
3022:
3019:
3016:
3013:
3010:
3007:
3004:
3001:
2998:
2995:
2992:
2989:
2986:
2983:
2980:
2977:
2974:
2971:
2968:
2965:
2962:
2959:
2922:
2919:
2894:
2890:
2887:
2867:
2863:
2851:convex subsets
2830:
2808:
2783:
2779:
2762:
2759:
2594:
2591:
2577:
2552:
2547:
2542:
2527:is the set of
2458:
2455:
2399:. That is why
2393:complex number
2389:linear algebra
2282:Each interval
2280:
2279:
2264:
2261:
2258:
2255:
2252:
2249:
2246:
2243:
2239:
2235:
2232:
2229:
2226:
2223:
2221:
2217:
2212:
2209:
2206:
2201:
2196:
2193:
2190:
2187:
2184:
2181:
2178:
2177:
2174:
2171:
2168:
2165:
2162:
2159:
2156:
2153:
2149:
2145:
2142:
2139:
2136:
2133:
2131:
2127:
2122:
2119:
2116:
2111:
2106:
2103:
2100:
2097:
2094:
2091:
2088:
2087:
2084:
2081:
2078:
2075:
2072:
2069:
2066:
2063:
2059:
2055:
2052:
2049:
2046:
2043:
2041:
2037:
2032:
2029:
2026:
2021:
2016:
2013:
2010:
2007:
2004:
2001:
1998:
1997:
1994:
1991:
1988:
1985:
1982:
1979:
1976:
1973:
1969:
1965:
1962:
1959:
1956:
1953:
1951:
1947:
1942:
1939:
1936:
1931:
1926:
1923:
1920:
1917:
1914:
1911:
1908:
1907:
1881:
1878:
1833:
1830:
1816:(qualified by
1745:. An interval
1428:measure theory
1407:
1404:
1401:
1398:
1395:
1375:
1372:
1369:
1366:
1363:
1348:
1345:
1342:
1339:
1336:
1333:
1313:
1310:
1307:
1304:
1301:
1298:
1279:
1278:
1263:
1260:
1257:
1254:
1251:
1248:
1244:
1240:
1237:
1234:
1231:
1228:
1226:
1223:
1219:
1216:
1213:
1210:
1206:
1202:
1201:
1198:
1195:
1192:
1189:
1186:
1183:
1179:
1175:
1172:
1169:
1166:
1163:
1161:
1158:
1154:
1151:
1148:
1145:
1141:
1137:
1136:
1133:
1130:
1127:
1124:
1121:
1118:
1115:
1112:
1108:
1104:
1101:
1098:
1095:
1092:
1090:
1087:
1083:
1080:
1077:
1073:
1069:
1068:
1065:
1062:
1059:
1056:
1053:
1050:
1047:
1044:
1040:
1036:
1033:
1030:
1027:
1024:
1022:
1019:
1015:
1012:
1009:
1005:
1001:
1000:
984:and less than
933:
921:
920:
909:
906:
903:
900:
897:
894:
891:
887:
883:
880:
877:
874:
871:
868:
865:
862:
859:
837:
834:
831:
828:
801:[0, 1]
782:for the usual
767:
764:
761:
758:
755:
752:
749:
746:
723:
720:
717:
697:
694:
691:
688:
668:
648:
637:
636:
621:
617:
613:
610:
608:
606:
603:
600:
597:
594:
591:
588:
585:
584:
581:
578:
575:
572:
569:
566:
562:
558:
555:
552:
549:
546:
544:
542:
539:
536:
533:
530:
527:
524:
523:
520:
517:
514:
511:
508:
505:
501:
497:
494:
491:
488:
485:
483:
481:
478:
475:
472:
469:
466:
463:
462:
459:
456:
453:
450:
447:
444:
441:
438:
434:
430:
427:
424:
421:
418:
416:
414:
411:
408:
405:
402:
399:
398:
368:and less than
353:
350:
347:
344:
341:
338:
335:
332:
329:
326:
323:
320:
317:
314:
311:
264:
261:
258:
238:
235:
232:
193:
190:
149:real functions
57:and less than
26:
9:
6:
4:
3:
2:
7942:
7931:
7928:
7926:
7923:
7921:
7918:
7917:
7915:
7903:
7902:
7897:
7894:
7889:
7886:
7882:
7879:
7877:
7873:
7870:
7867:
7865:
7861:
7858:
7855:
7852:
7848:
7845:
7844:
7835:
7831:
7828:
7824:
7823:
7811:
7807:
7802:
7796:
7792:
7786:
7778:
7774:
7770:
7766:
7762:
7760:3-88538-006-4
7756:
7752:
7745:
7737:
7733:
7729:
7725:
7721:
7715:
7711:
7710:
7705:
7699:
7691:
7687:
7683:
7679:
7675:
7671:
7667:
7663:
7658:
7653:
7649:
7645:
7641:
7634:
7632:
7623:
7619:
7615:
7611:
7607:
7603:
7599:
7595:
7590:
7585:
7581:
7577:
7573:
7566:
7564:
7555:
7551:
7547:
7541:
7537:
7533:
7529:
7522:
7516:
7512:
7508:
7503:
7489:
7485:
7481:
7477:
7473:
7469:
7464:
7459:
7455:
7452:
7451:
7446:
7435:
7428:
7423:
7408:
7404:
7402:
7398:
7394:
7390:
7382:
7374:
7372:0-07-054235-X
7368:
7364:
7359:
7358:
7349:
7334:
7330:
7326:
7320:
7312:
7306:
7302:
7301:
7293:
7284:
7280:
7276:
7272:
7268:
7262:
7258:
7254:
7250:
7249:
7244:
7238:
7230:
7226:
7225:
7220:
7214:
7200:
7196:
7189:
7181:
7179:0-7637-1497-6
7175:
7171:
7170:
7162:
7160:
7151:
7149:1-886529-02-7
7145:
7141:
7140:
7132:
7128:
7118:
7117:Unit interval
7115:
7113:
7110:
7108:
7105:
7103:
7100:
7098:
7095:
7093:
7090:
7088:
7085:
7083:
7080:
7079:
7073:
7071:
7067:
7062:
7049:
7046:
7041:
7037:
7013:
7010:
7004:
7001:
6998:
6989:
6986:
6980:
6974:
6971:
6968:
6959:
6956:
6950:
6947:
6940:
6939:
6938:
6936:
6932:
6916:
6913:
6910:
6907:
6899:
6894:
6892:
6888:
6884:
6880:
6876:
6872:
6868:
6864:
6860:
6856:
6837:
6834:
6831:
6828:
6820:
6803:
6779:
6775:
6769:
6765:
6760:
6755:
6751:
6729:
6717:
6711:
6707:
6701:
6690:
6687:
6679:
6668:
6665:
6661:
6658:
6654:
6651:
6647:
6644:
6640:
6637: โ
6636:
6632:
6631:Find sources:
6625:
6621:
6615:
6614:
6609:This section
6607:
6603:
6598:
6597:
6589:
6587:
6584:or moreover,
6583:
6579:
6575:
6570:
6556:
6534:
6526:
6523:
6520:
6497:
6490:
6486:
6470:
6464:
6461:
6458:
6447:
6446:product space
6443:
6414:
6411:
6408:
6405:
6402:
6399:
6379:
6376:
6373:
6366:
6350:
6347:
6344:
6337:
6333:
6330:
6314:
6306:
6305:
6304:
6290:
6287:
6280:
6264:
6257:
6249:
6233:
6213:
6205:
6190:
6182:
6167:
6159:
6158:
6157:
6143:
6137:
6134:
6131:
6121:
6105:
6091:
6089:
6085:
6069:
6066:
6046:
6039:contained in
6026:
6018:
6002:
5999:
5991:
5987:
5983:
5967:
5960:contained in
5947:
5927:
5924:
5921:
5918:
5910:
5891:
5888:
5885:
5873:
5860:
5835:
5804:
5801:
5796:
5792:
5788:
5780:
5777:
5771:
5759:
5758:
5757:
5755:
5736:
5733:
5718:
5702:
5699:
5696:
5693:
5673:
5670:
5667:
5664:
5661:
5641:
5638:
5635:
5632:
5629:
5622:if for every
5621:
5602:
5599:
5596:
5586:
5570:
5567:
5564:
5554:
5544:
5542:
5508:one may take
5490:
5487:
5467:
5464:
5438:
5435:
5432:
5419:
5416:
5413:
5407:
5400:
5380:
5374:
5368:
5365:
5362:
5353:
5343:
5342:
5341:
5327:
5324:
5321:
5318:
5315:
5312:
5292:
5289:
5286:
5263:
5260:
5257:
5248:
5242:
5232:
5218:
5212:
5209:
5206:
5203:
5200:
5197:
5194:
5188:
5182:
5179:
5173:
5163:
5149:
5143:
5140:
5137:
5134:
5131:
5128:
5125:
5119:
5113:
5110:
5104:
5094:
5080:
5074:
5071:
5068:
5065:
5062:
5059:
5056:
5050:
5041:
5038:
5028:
5014:
5008:
5005:
5002:
4999:
4996:
4993:
4990:
4984:
4975:
4972:
4962:
4948:
4942:
4939:
4936:
4933:
4930:
4927:
4924:
4921:
4918:
4912:
4906:
4903:
4900:
4890:
4876:
4870:
4867:
4864:
4861:
4858:
4855:
4852:
4849:
4846:
4840:
4834:
4831:
4828:
4818:
4804:
4798:
4795:
4792:
4789:
4786:
4783:
4780:
4777:
4774:
4768:
4762:
4759:
4756:
4746:
4732:
4726:
4723:
4720:
4717:
4714:
4711:
4708:
4705:
4702:
4696:
4690:
4687:
4684:
4674:
4673:
4672:
4658:
4655:
4652:
4649:
4646:
4643:
4620:
4617:
4614:
4604:
4600:
4596:
4585:
4575:
4573:
4569:
4565:
4564:complex plane
4561:
4558:Intervals of
4551:
4549:
4545:
4535:
4533:
4529:
4525:
4522:-dimensional
4509:
4501:
4491:
4478:
4473:
4446:
4438:
4422:
4402:
4394:
4378:
4373:
4369:
4346:
4342:
4321:
4313:
4308:
4306:
4292:
4269:
4265:
4261:
4258:
4255:
4250:
4246:
4242:
4237:
4233:
4229:
4226:
4206:
4197:
4195:
4191:
4175:
4172:
4169:
4161:
4145:
4142:
4139:
4119:
4111:
4107:
4091:
4086:
4072:
4068:
4058:
4055:
4051:
4046:
4044:
4040:
4037:-dimensional
4024:
4016:
3997:
3994:
3991:
3965:
3962:
3959:
3936:
3930:
3927:
3924:
3915:
3912:
3901:
3882:
3879:
3876:
3867:
3864:
3853:
3849:
3830:
3827:
3824:
3802:
3798:
3793:
3789:
3785:
3781:
3776:
3774:
3766:
3763:
3760:
3757:
3756:
3755:
3752:
3738:
3718:
3698:
3690:
3686:
3681:
3678:
3675:
3646:
3642:
3638:
3627:
3617:
3615:
3573:
3567:
3561:
3555:
3552:
3549:
3496:
3494:
3490:
3484:
3480:
3476:
3472:
3466:
3460:
3456:
3452:
3448:
3442:
3438:
3411:
3398:
3392:
3389:
3386:
3380:
3374:
3371:
3368:
3362:
3356:
3353:
3350:
3323:
3314:
3308:
3302:
3293:
3290:
3284:
3278:
3275:
3268:
3254:
3248:
3245:
3239:
3233:
3227:
3224:
3218:
3212:
3209:
3206:
3200:
3197:
3191:
3185:
3182:
3175:
3161:
3152:
3149:
3146:
3140:
3131:
3128:
3125:
3119:
3116:
3113:
3104:
3101:
3098:
3092:
3089:
3082:
3068:
3062:
3059:
3056:
3050:
3044:
3041:
3038:
3032:
3029:
3026:
3020:
3017:
3014:
3008:
3005:
3002:
2996:
2993:
2990:
2984:
2981:
2978:
2972:
2969:
2966:
2960:
2957:
2950:
2949:
2948:
2946:
2942:
2938:
2933:
2920:
2917:
2909:
2888:
2885:
2865:
2852:
2847:
2845:
2797:
2781:
2768:
2758:
2754:
2750:
2742:
2738:
2730:
2726:
2720:
2716:
2709:
2705:
2699:
2697:
2693:
2688:
2686:
2682:
2678:
2674:
2668:
2664:
2650:
2645:
2641:
2634:
2630:
2622:
2618:
2612:
2608:
2590:
2567: =
2550:
2545:
2530:
2522:
2517:
2505:
2497:
2489:
2481:
2474:
2464:
2454:
2442:
2438:
2430:
2426:
2419:
2417:
2412:
2408:
2402:
2398:
2394:
2390:
2386:
2382:
2378:
2374:
2370:
2364:
2360:
2353:
2350:
2346:
2339:
2331:
2327:
2321:
2315:
2311:
2303:
2299:
2291:
2287:
2262:
2256:
2253:
2250:
2247:
2244:
2241:
2233:
2230:
2224:
2222:
2210:
2207:
2204:
2194:
2188:
2185:
2182:
2172:
2166:
2163:
2160:
2157:
2154:
2151:
2143:
2140:
2134:
2132:
2120:
2117:
2114:
2104:
2098:
2095:
2092:
2082:
2076:
2073:
2070:
2067:
2064:
2061:
2053:
2050:
2044:
2042:
2030:
2027:
2024:
2014:
2008:
2005:
2002:
1992:
1986:
1983:
1980:
1977:
1974:
1971:
1963:
1960:
1954:
1952:
1940:
1937:
1934:
1924:
1918:
1915:
1912:
1898:
1897:
1896:
1894:
1890:
1887:
1877:
1875:
1871:
1870:decimal comma
1867:
1861:
1857:
1829:
1827:
1823:
1819:
1815:
1811:
1807:
1803:
1799:
1795:
1791:
1787:
1783:
1779:
1775:
1770:
1764:
1763:proper subset
1752:
1740:
1728:
1719:
1705:
1704:interval span
1701:
1692:
1678:
1658:
1653:
1651:
1647:
1643:
1639:
1634:
1632:
1629:, and form a
1628:
1624:
1618:
1614:
1605:
1601:
1597:
1593:
1589:
1584:
1579:
1575:
1568:
1562:
1558:
1544:
1540:
1535:
1525:
1521:
1517:
1513:
1509:
1505:
1501:
1497:
1492:
1490:
1486:
1482:
1478:
1474:
1473:right-bounded
1470:
1465:
1463:
1457:
1453:
1447:
1443:
1434:
1431:
1429:
1425:
1402:
1399:
1396:
1370:
1367:
1364:
1340:
1337:
1334:
1308:
1305:
1299:
1288:
1284:
1261:
1255:
1252:
1249:
1246:
1238:
1235:
1229:
1227:
1221:
1217:
1214:
1208:
1204:
1196:
1190:
1187:
1184:
1181:
1173:
1170:
1164:
1162:
1156:
1149:
1146:
1143:
1139:
1131:
1125:
1122:
1119:
1116:
1113:
1110:
1102:
1099:
1093:
1091:
1085:
1081:
1078:
1075:
1071:
1063:
1057:
1054:
1051:
1048:
1045:
1042:
1034:
1031:
1025:
1023:
1017:
1013:
1010:
1007:
1003:
991:
990:
989:
963:
959:
955:
946:
904:
901:
898:
895:
892:
889:
881:
878:
872:
866:
863:
860:
850:
849:
848:
835:
832:
829:
826:
798:
797:
787:
785:
781:
765:
759:
753:
750:
747:
737:
721:
718:
715:
695:
692:
689:
686:
666:
646:
619:
611:
609:
598:
595:
589:
579:
573:
570:
567:
564:
556:
553:
547:
545:
534:
531:
528:
518:
512:
509:
506:
503:
495:
492:
486:
484:
476:
473:
467:
457:
451:
448:
445:
442:
439:
436:
428:
425:
419:
417:
409:
406:
403:
389:
388:
387:
348:
345:
342:
339:
336:
333:
330:
324:
318:
315:
312:
301:
300:
297:open interval
290:
288:
280:
275:
262:
256:
236:
230:
222:
218:
214:
209:
207:
203:
199:
185:
183:
179:
175:
172:set, such as
171:
166:
164:
160:
156:
152:
150:
146:
142:
138:
134:
130:
125:
111:
107:
106:unit interval
94:
92:
88:
84:
80:
76:
75:real interval
72:
64:
60:
56:
52:
48:
45:The addition
43:
37:
33:
19:
18:Open interval
7925:Order theory
7899:
7846:
7820:Bibliography
7806:D. H. Lehmer
7801:
7785:
7750:
7744:
7708:
7698:
7647:
7643:
7579:
7575:
7527:
7521:
7502:
7491:. Retrieved
7453:
7448:
7434:
7422:
7410:. Retrieved
7406:
7400:
7396:
7392:
7388:
7381:
7356:
7348:
7337:. Retrieved
7328:
7319:
7299:
7292:
7247:
7243:Tao, Terence
7237:
7222:
7213:
7202:. Retrieved
7198:
7188:
7168:
7138:
7131:
7107:Line segment
7063:
7028:
6935:D. H. Lehmer
6895:
6862:
6858:
6857:, { :
6804:
6767:
6763:
6753:
6749:
6709:
6705:
6697:
6682:
6673:
6663:
6656:
6649:
6642:
6630:
6618:Please help
6613:verification
6610:
6571:
6439:
6431:Applications
6334:There is an
6253:
6097:
5989:
5874:
5822:
5619:
5556:
5456:
5278:
4592:
4557:
4541:
4524:affine space
4500:intersection
4497:
4436:
4392:
4311:
4309:
4285:
4198:
4106:axis-aligned
4064:
4047:
3813:
3796:
3777:
3770:
3753:
3625:
3623:
3590:. The parts
3497:
3482:
3478:
3474:
3470:
3465:closed balls
3458:
3454:
3450:
3446:
3437:metric space
3412:
3338:
2934:
2906:is also the
2848:
2764:
2752:
2748:
2740:
2736:
2728:
2724:
2718:
2714:
2707:
2703:
2700:
2689:
2666:
2662:
2648:
2643:
2639:
2632:
2628:
2620:
2616:
2610:
2596:
2518:
2503:
2495:
2487:
2479:
2475:
2460:
2440:
2436:
2428:
2424:
2420:
2410:
2406:
2369:ordered pair
2362:
2358:
2354:
2348:
2344:
2337:
2329:
2325:
2313:
2309:
2301:
2297:
2289:
2285:
2281:
1883:
1865:
1859:
1855:
1844:, including
1835:
1825:
1821:
1817:
1813:
1809:
1805:
1801:
1797:
1793:
1789:
1785:
1781:
1777:
1773:
1771:
1750:
1729:of interval
1726:
1721:An interval
1720:
1703:
1699:
1694:For any set
1693:
1676:
1656:
1654:
1645:
1642:right-closed
1641:
1637:
1635:
1616:
1612:
1603:
1595:
1587:
1585:
1577:
1573:
1566:
1560:
1556:
1538:
1536:
1523:
1519:
1515:
1511:
1507:
1496:bounded sets
1493:
1488:
1485:half-bounded
1484:
1480:
1476:
1472:
1469:left-bounded
1468:
1466:
1461:
1455:
1451:
1437:
1435:
1432:
1280:
961:
957:
949:
947:
922:
791:
790:
788:
638:
294:
293:
291:
282:
276:
212:
210:
206:real numbers
197:
195:
167:
153:
126:
95:
83:real numbers
74:
68:
62:
58:
54:
50:
46:
7825:T. Sunaga,
7582:: 481โ493.
7395:) vs. ]
6865:โ R }. The
6489:cardinality
4589:Definitions
4568:rectangular
4219:intervals,
3773:binary tree
2908:convex hull
2769:subsets of
2373:coordinates
1891:. Thus, in
1727:subinterval
1650:closed sets
1638:left-closed
71:mathematics
7914:Categories
7896:"Interval"
7777:0684.54001
7736:0951.54001
7690:0189.53103
7622:0269.54009
7554:1080.91001
7493:2012-04-05
7427:Tao (2016)
7339:2016-11-12
7310:0691005478
7283:2016940817
7248:Analysis I
7204:2020-08-23
7195:"Interval"
7123:References
7087:Inequality
6778:direct sum
6646:newspapers
6392:such that
6329:sublattice
6094:Properties
6017:Zorn lemma
5756:, the set
5654:and every
5539:to be the
4188:this is a
4158:this is a
4054:degenerate
4050:half-space
3463:, and its
3441:open balls
2761:Properties
2698:notation.
2322:, whereas
1609:[0, 1)
1596:right-open
1565:, and its
1420:(0,+โ]
1283:closed set
978:[0, 1)
966:(0, 1]
962:right-open
219:, and its
159:input data
7901:MathWorld
7666:0002-9939
7598:0002-9947
7275:2366-8725
7229:EMS Press
7002:−
6911:⊕
6887:quadrants
6838:×
6821:⊕
6730:×
6557:κ
6535:κ
6498:κ
6409:∩
6377:⊆
6348:⊆
6138:≤
6088:partition
5922:⊆
5892:≲
5789:∣
5781:∈
5737:≤
5697:∈
5671:≲
5665:≲
5639:∈
5603:≲
5568:⊆
5557:A subset
5524:¯
5436:∈
5430:∀
5423:∞
5411:∞
5408:−
5384:∞
5378:∞
5375:−
5369:⊔
5357:¯
5316:≲
5252:∞
5246:∞
5243:−
5210:≲
5204:∣
5198:∈
5177:∞
5174:−
5135:∣
5129:∈
5108:∞
5105:−
5072:≲
5066:∣
5060:∈
5045:∞
5000:∣
4994:∈
4979:∞
4934:≲
4928:∣
4922:∈
4868:≲
4856:∣
4850:∈
4796:≲
4790:≲
4784:∣
4778:∈
4712:∣
4706:∈
4653:∈
4621:≲
4566:, either
4415:comprise
4262:×
4259:⋯
4256:×
4243:×
4160:rectangle
3928:−
3363:∪
3318:∞
3312:∞
3309:−
3297:∞
3288:∞
3285:−
3279:
3243:∞
3240:−
3222:∞
3219:−
3213:
3195:∞
3192:−
3186:
3156:∞
3135:∞
3120:
3108:∞
3093:
3033:
3009:
2985:
2961:
2889:⊆
2767:connected
2498:,โ+โ]
2478:[โโ,
2320:empty set
2254:≤
2248:≤
2242:∣
2234:∈
2164:≤
2152:∣
2144:∈
2068:≤
2062:∣
2054:∈
1972:∣
1964:∈
1889:ISO 31-11
1874:semicolon
1866:endpoints
1826:half-open
1798:intervals
1623:open sets
1588:left-open
1481:unbounded
1344:∞
1303:∞
1300:−
1287:real line
1253:≤
1247:∣
1239:∈
1212:∞
1209:−
1188:≤
1182:∣
1174:∈
1153:∞
1117:≤
1111:∣
1103:∈
1055:≤
1043:∣
1035:∈
958:left-open
902:≤
896:≤
890:∣
882:∈
836::
830:≤
780:open sets
763:∅
736:empty set
690:≤
602:∞
593:∞
590:−
565:∣
557:∈
538:∞
504:∣
496:∈
471:∞
468:−
437:∣
429:∈
334:∣
260:∞
234:∞
231:−
213:endpoints
188:articles.
145:integrals
7930:Topology
7872:Archived
7860:Archived
7830:Archived
7709:Topology
7706:(2000).
7488:16796699
7412:28 April
7333:Archived
7245:(2016).
7076:See also
6898:midpoint
6875:topology
6853:has two
6766:−
6307:The set
6206:The set
6183:The set
6160:The set
5911:and let
5686:we have
5322:≴
4601:. For a
4572:circular
2696:ellipsis
2694:, using
2651:between
2649:integers
2637:or just
2607:integers
2565:(โโ,โ+โ)
2521:infinite
2510:(โโ,โ+โ)
2401:Bourbaki
2342:. When
1814:interval
1810:segments
1786:interval
1784:defines
1778:interval
1657:interior
1627:topology
1581:|/2
1543:midpoint
1500:diameter
976:, while
784:topology
217:supremum
198:interval
174:integers
87:infinity
7808:(1956)
7769:1039321
7728:0464128
7682:0257985
7674:2037311
7614:0372826
7606:1996713
7468:Bibcode
7403:[?"
7231:, 2001
6700:regions
6660:scholar
6279:lattice
6015:By the
5980:form a
5583:of the
4538:Domains
4437:corners
3850:with a
2937:closure
2727:+โ1 ..
2717:+โ1 ..
2681:indices
2525:(0,โ+โ)
2397:algebra
1790:segment
1774:segment
1677:closure
1619:< 1}
1600:maximum
1592:minimum
1516:measure
1477:bounded
1444:is any
1285:of the
379:(0, +โ)
221:infimum
204:of the
118:(โโ, โ)
81:of all
77:is the
7775:
7767:
7757:
7734:
7726:
7716:
7688:
7680:
7672:
7664:
7620:
7612:
7604:
7596:
7552:
7542:
7513:
7486:
7369:
7307:
7281:
7273:
7263:
7176:
7146:
7029:where
6881:. The
6855:ideals
6770:< 0
6662:
6655:
6648:
6641:
6633:
6440:Every
6365:filter
6363:and a
6256:subset
6254:For a
5305:means
5279:where
4548:domain
4162:; for
4015:sphere
3900:radius
3898:and a
3852:center
3711:where
3489:metric
3439:, its
2722:, or
2683:of an
2677:Pascal
2613:โง, or
2500:, and
2381:vector
2306:, and
1822:closed
1739:subset
1675:. The
1646:closed
1615:| 0 โค
1602:; and
1571:|
1567:radius
1539:centre
1508:length
1462:proper
639:where
374:]0, 1[
202:subset
135:; the
114:(0, โ)
7793:from
7670:JSTOR
7602:JSTOR
7484:S2CID
7458:arXiv
6752:>
6667:JSTOR
6653:books
6336:ideal
6327:is a
6277:of a
6118:of a
5982:poset
5907:be a
4393:faces
4312:facet
4048:If a
3810:Balls
3794:(for
3493:order
3485:]
3469:[
2943:of a
2798:from
2735:[
2685:array
2675:; in
2669:]
2661:[
2623:]
2615:[
2597:When
2506:,โ+โ)
2502:[
2494:[
2490:]
2486:(โโ,
2482:]
2377:point
2375:of a
2347:>
2332:]
2324:[
2316:]
2296:[
1862:]
1854:[
1824:, or
1761:is a
1749:is a
1737:is a
1725:is a
1522:, or
1520:range
1512:width
1458:]
1450:[
708:When
200:is a
91:bound
73:, a
7755:ISBN
7714:ISBN
7662:ISSN
7594:ISSN
7540:ISBN
7511:ISBN
7414:2018
7367:ISBN
7305:ISBN
7279:LCCN
7271:ISSN
7261:ISBN
7174:ISBN
7144:ISBN
6639:news
6485:base
5875:Let
5802:<
5420:<
5414:<
5290:<
5141:<
5006:<
4940:<
4862:<
4724:<
4718:<
4391:The
4196:").
4043:disk
3848:ball
3786:and
3731:and
3664:and
3597:and
3491:and
2935:The
2655:and
2611:a, b
2605:are
2601:and
2469:and
2387:and
2158:<
2074:<
1984:<
1978:<
1872:, a
1848:and
1840:and
1818:open
1776:and
1655:The
1631:base
1604:open
1549:and
1537:The
1524:size
1386:and
1324:and
1123:<
1049:<
815:and
659:and
571:<
510:<
449:<
443:<
346:<
340:<
211:The
161:and
7773:Zbl
7732:Zbl
7686:Zbl
7652:doi
7618:Zbl
7584:doi
7580:178
7550:Zbl
7532:doi
7476:doi
7253:doi
6780:of
6622:by
6580:is
6549:of
6487:of
5992:of
5752:of
5618:is
4570:or
4439:of
4395:of
4194:box
4112:of
4104:an
3902:of
3854:at
3801:).
3799:= 2
3413:If
2910:of
2853:of
2820:to
2751:..
2745:or
2739:..
2731:โโ1
2712:,
2710:โโ1
2706:..
2665:..
2642:..
2631:..
2625:or
2619:..
2395:in
2383:in
2379:or
1765:of
1757:if
1753:of
1741:of
1733:if
1706:of
1702:or
1679:of
1611:= {
1563:)/2
1553:is
1471:or
960:or
292:An
196:An
176:or
147:of
79:set
69:In
7916::
7898:.
7771:.
7765:MR
7763:.
7730:.
7724:MR
7722:.
7684:.
7678:MR
7676:.
7668:.
7660:.
7648:24
7646:.
7642:.
7630:^
7616:.
7610:MR
7608:.
7600:.
7592:.
7578:.
7574:.
7562:^
7548:.
7538:.
7482:.
7474:.
7466:.
7454:66
7447:.
7405:.
7399:,
7391:,
7365:.
7363:31
7331:.
7327:.
7277:.
7269:.
7259:.
7227:,
7221:,
7197:.
7158:^
7072:.
7050:1.
6708:,
6588:.
6090:.
5543:.
4574:.
4550:.
4534:.
4310:A
4307:.
4045:.
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