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Topic: Open intervals

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	Interval (mathematics) Summary
		Interval notation is the notation in which permitted values for a variable are expressed as ranging over a certain interval; "5 < x < 9" is an example of the application of interval notation.
		Intervals using the round brackets (or) as in the general interval (a,b) or specific examples (-1,3) and (2,4) are called open intervals and the endpoints are not included in the set.
		Intervals using both square and round brackets [ and) or (and ] as in the general intervals (a,b] and [a,b) or specific examples [-1,3) and (2,4] are called half-closed intervals or half-open intervals.
	www.bookrags.com /Interval_(mathematics) (1439 words)

	Interval (mathematics) - Wikipedia, the free encyclopedia
		Interval notation is the notation in which permitted values for a variable are expressed as ranging over a a certain interval; "5 < x < 9" is an example of the application of interval notation.
		Intervals of type (1), (5), (7), (9) and (11) are called open intervals (because they are open sets) and intervals (2), (6), (8), (9), (10) and (11) closed intervals (because they are closed sets).
		Interval arithmetic, also called interval mathematics, interval analysis, and interval computation, was introduced by mathematicians in the 1950s and 1960s as an approach to putting bounds on rounding errors in mathematical computation.
	en.wikipedia.org /wiki/Interval_(mathematics) (1087 words)

	PlanetMath: interval
		The logic is here that a empty circle represent a point not belonging to the interval, while a filled circle represents a point belonging to the interval.
		In [1,2], an open interval is always called a segment, and a closed interval is called simply an interval.
		This is version 11 of interval, born on 2003-07-14, modified 2006-03-24.
	planetmath.org /encyclopedia/Interval.html (437 words)

	PlanetMath: basis (topology)
		A basis for the usual topology of the real line is given by the set of open intervals since every open set can be expressed as a union of open intervals.
		For instance, the set of all open intervals with rational endpoints and the set of all intervals whose length is a power of
		More generally, the set of open balls forms a basis for the topology on a metric space.
	planetmath.org /encyclopedia/BasisTopologicalSpace.html (328 words)

	Base (topology) - Wikipedia, the free encyclopedia
		In mathematics, a base (or basis) B for a topological space X with topology T is a collection of open sets in T such that every open set in T can be written as a union of elements of B.
		For example, the open intervals with rational endpoints are also a base for the real numbers, as are the open intervals with irrational endpoints, but these two sets are completely disjoint and both properly contained in the base of all open intervals.
		An example of a collection of open sets which is not a base is the set S of all semi-infinite intervals of the forms (â’â, a) and ( a, â), where a is a real number.
	en.wikipedia.org /wiki/Base_(topology) (783 words)

	Interval (mathematics) [Definition] (Site not responding. Last check: 2007-10-17)
		Interval notation is where the permitted values for a variable In computer science and mathematics, a variable is a symbol denoting a quantity or symbolic representation.
		Intervals are precisely the connected In topology and related branches of mathematics, a topological space X is said to be disconnected if it is the union of two disjoint nonempty open sets.
		Interval arithmetic, also called interval mathematics, interval analysis, and interval computation, was introduced by mathematicians in the 1950s Events and trends Technology United States tests the first fusion bomb.
	www.wikimirror.com /Interval_(mathematics) (2543 words)

	Interval (mathematics) Information - TextSheet.com (Site not responding. Last check: 2007-10-17)
		For example, the interval "(10,20)" stands for all real numbers between 10 and 20, not including 10 or 20.
		In higher mathematics, a formal definition is the following: An interval is a subset S of a totally ordered set T with the property that whenever x and y are in S and x
		Intervals of R are of the following eleven different types (where a and b are real numbers, with a
	forum.top5miami.com /encyclopedia/i/in/interval__mathematics_.html (688 words)

	Body
		These are subsets of the closed interval [0,1] and of measure greater or equal to 0 but less than 1.
		We shall enumerate the open intervals to be deleted.
		The open intervals are ordered from the left to the right by the second indices.
	www.math.nus.edu.sg /~matngtb/Calculus/Cantor/cantor.htm (2458 words)

	PlanetMath: basis (topology)
		A basis for the usual topology of the real line is given by the set of open intervals since every open set can be expressed as a union of open intervals.
		For instance, the set of all open intervals with rational endpoints and the set of all intervals whose length is a power of
		More generally, the set of open balls forms a basis for the topology on a metric space.
	www.planetmath.org /encyclopedia/Base2.html (327 words)

	Remarks on Proving The Fundamental Theorem of Algebra
		An open interval is the set of points which lie strictly between two distinct points, called the "endpoints" of the interval.
		For each point, c, in an open interval, there are points of the open interval which are greater than c and (other) points of the interval less than c.
		For each point, c, in an open interval, there is a closed interval about c, contained (as a subset) in the open interval.
	www.cut-the-knot.com /fta/brodie.shtml (1433 words)

	[No title] (Site not responding. Last check: 2007-10-17)
		In the study, closed birth intervals are defined as the length of the interval between the birth of the index child and the birth of the subsequent child; open birth intervals, the length of a birth interval of about five years.
		Analysis of closed birth intervals and life table estimates both show that age of the mother at the birth of the index child is one of the important determinants of the birth interval.
		A natural choice then is to assume that the open birth intervals would have ended with a subsequent birth, because the time of each of the reference periods, as the proportion of those who have completed such intervals after remaining without having any birth for the period would be equal to the open birth intervals.
	www.un.org /Depts/escap/pop/journal/v11n4dn.htm (3366 words)

	open - Definitions from Dictionary.com
		Open marriage, one in which the partners sleep with whomever they please, is from 1972.
		See also: be an open secret, bring (something) out into the open, open-air, opener, opening, openly, open-minded, open-plan, in the open, in the open air, keep/have an open mind, open on to, open to, open up, the open sea, with open arms
		NOTE: When a defect, hazard, or condition is open such that a reasonable person under the circumstances should have recognized the danger posed by it, a defendant is usually relieved of liability for failure to warn.
	dictionary.reference.com /browse/open (6323 words)

	Analysis WebNotes: Chapter 02, Class 07
		The idea of open intervals is generalized in this section to an important class of subsets of the real line, called open sets.
		Open sets will be used a lot in Chapters 3, 5 and 6.
		The union of a family of open sets is open.
	www.math.unl.edu /~webnotes/classes/class07/class07.htm (198 words)

	Math Forum - Ask Dr. Math
		Date: 08/01/98 at 04:51:29 From: Michael Sabroski Subject: Formal Definition of a Limit The formal definition of the limit states: Let f be a function that is defined on an open interval containing c, except possibly at c itself, and let L be a real number.
		All this means is that you can take a tiny interval around 3 and find a tiny interval around 5 such that every x in the little interval around 5 lands in the little interval around 3 when you take f(x).
		In this case, open intervals can be thought of as "everything that is a distance less than _____ away from something." Say you want everything that is a distance less than.0001 away from 5.
	mathforum.org /library/drmath/view/51933.html (712 words)

	Remarks on Proving The Fundamental Theorem of Algebra
		An open interval is the set of points which lie strictly between two distinct points, called the "endpoints" of the interval.
		For each point, c, in an open interval, there are points of the open interval which are greater than c and (other) points of the interval less than c.
		In summary, it is impossible for the union of two disjoint open intervals to form an interval.
	www.cut-the-knot.org /fta/brodie.shtml (1369 words)

	Ca2+-Dependent Gating Mechanisms for dSlo, a Large-Conductance Ca2+-Activated K+ (BK) Channel -- Moss et al. 76 (6): ...
		(17,528 fitted intervals) were described by the sum of six significant exponential components: 0.03 ms (0.538), 0.08 ms (0.288), 0.37 ms (0.085), 3.80 ms (0.016), 24.20 ms (0.062), and 71.27 ms (0.011).
		FIGURE 5 The numbers of significant open (A) and closed (B) exponential components are plotted against the number of fitted intervals for 23 open and 23 closed dwell-time distributions from dSlo.
		The difference plots are the square root of the numbers of predicted intervals minus the square root of the numbers of observed intervals.
	www.biophysj.org /cgi/content/full/76/6/3099 (8718 words)

	[No title] (Site not responding. Last check: 2007-10-17)
		For example, two intervals, i and j, might be indistinguishable if their beginning points are at most a moment apart, and likewise for their end points.
		ÃÃOpen and Closed Nature of IntervalsÄÄ: Although intervals are taken in the theory as primitive, that is there are no definitions about the ending©points for intervals, the axiomatisation allows the expression of the "open" and "closed" nature of intervals.
		The above interpretation of the "open" and "closed" nature of primitive intervals is in fact in line with the conventional meaning of the open and closed nature for point©based intervals.
	www.gre.ac.uk /~j.ma/papers/paper8.doc (4162 words)

	Base (topology) (Site not responding. Last check: 2007-10-17)
		In topology, a base (or basis) B for a topological space X with topology T is a collection of open sets in T such that every open set in T can be written as a union of elements of B.
		In contrast to the basis of linear algebra, the elements of a base need not be independent; that is, it may be possible to write one as a union of some of the others.
		An example of a collection of open sets which is not a basis is the set S of all semi-infinite intervals of the forms (−∞, a) and ( a,∞), where a is a real number.
	www.portaljuice.com /base__topology_.html (675 words)

	Lower limit topology (Site not responding. Last check: 2007-10-17)
		In mathematics, the lower limit topology or righthalf-open interval topology is a topology defined on theset R of real numbers ; it is different from the standardtopology on R and has a number of interesting properties.
		It is the topology generated by the basis of all half-open intervals [ a, b), where a and b are real numbers.
		The lower limit topology is finer (has more open sets) than thestandard topology on the real numbers (which is generated by the open intervals).
	www.therfcc.org /lower-limit-topology-218999.html (253 words)

	Desensitization of diliganded mouse muscle nicotinic acetylcholine receptor channels -- Elenes and Auerbach 541 (2): ...
		On the time scale of ~0.1 ms to ~1 h, apparent open intervals are described by a single exponential component, and shut intervals associated with desensitization are described by the sum of four or five exponential components.
		An important objective was to quantify and model the durations of the open and shut intervals arising from the main population of AChRs.
		ACh, the apparent openings occurred in clusters and the pattern of closed intervals in the record was complex ( Fig.
	jp.physoc.org /cgi/content/full/541/2/367 (7580 words)

	Gating Kinetics of Single Large-Conductance Ca2+-activated K+ Channels in High Ca2+ Suggest a Two-Tiered Allosteric ...
		The durations of open (A–D) and closed (E–H) intervals were log binned at a resolution of 25 bins/decade and plotted as the square-root of the number of intervals in each bin.
		Adjacent open and closed intervals were binned as pairs, with the logs of the open and closed interval durations locating the bins on the x and y axes, respectively.
		The deficit of intervals at position 1 and the excess of intervals at position 2 are consistently seen.
	www.jgp.org /cgi/content/full/114/1/93 (7762 words)

	A. Intervals
		If the boundary is included in the interval, the interval is said to be closed on that side; if it is excluded it is said to be open on that side.
		If an interval is closed on both sides, it is said to be a closed interval.
		If it is open on both sides it is said to be an open interval.
	keiapl.info /jhelp/phrases/intervals.htm (441 words)

	ON INTERVALS
		The operations you can perform on intervals should in no way depend on whether you choose to store (under the covers) the open bounds, the closed bounds or one of each.
		With a TO and a FROM attribute to represent the interval over which something is believed to hold true, you have to know which bounds are represented by the TO and FROM values.
		The interval [a,a] contains just the point a; the intervals (a,a], [a,a) and (a,a) are all illegal, and contain no points at all.
	www.dbdebunk.com /page/page/1132623.htm (414 words)

	Overlapping intervals (Site not responding. Last check: 2007-10-17)
		This is for "open intervals," where the endpoints are not part of the interval, so "touching" is not considered an overlap.
		Although there are four ways for two intervals to overlap, they can only fail to overlap in two ways: either i1 comes before i2, or i2 comes before i1.
		To test for rectangle overlap, computer graphics programmers use this interval test twice, once for the X coordinates and again for the Y coordinates.
	rgrjr.dyndns.org /emacs/overlap.html (313 words)

	Exercises 1
		Draw the graphs of continuous maps which show that any two closed intervals are homeomorphic (topologically equivalent).
		Prove that any two open intervals are homeomorphic.
		Draw the graph of a continuous function from the open interval (0, 1) onto the closed interval [0, 1].
	www-groups.dcs.st-and.ac.uk /~john/MT4522/Tutorials/T1.html (275 words)

	[No title] (Site not responding. Last check: 2007-10-17)
		This is what we Ô ‘XÉ* Ôwould expect intuitively: tÃÃ1ÄÄ and tÃÃ2ÄÄ are closed at p and tÃÃ3ÄÄ is open at p; m(tÃÃ1ÄÄ, tÃÃ2ÄÄ) and m(tÃÃ2ÄÄ, tÃÃ3ÄÄ)Ô@É* p-¿+¿+°°!°" °¸"q-¡ @ÔÔ ‘X Ôassert tÃÃ1ÄÄ = [p, X}, tÃÃ2ÄÄ = {X, p], t3 = (p, Y} are consecutive intervals, where "{" represents either open or closed.
		A significant feature of this approach is that, in common with Allen's system, intervals do not need to be defined as point pairs.
		The formulation of axioms by means of a single relation allows a graphical representation of the temporal database entities, and this in turn allows an efficient consistency checker in terms of a search for graphical cycles.
	www.gre.ac.uk /~j.ma/papers/paper5.doc (1161 words)

	Manifolds with Boundaries (Site not responding. Last check: 2007-10-17)
		As a way of understanding what a boundary is, consider regions which are intervals on the real line.
		If intervals always contained thier endpoints (or never contained endpoints), this notion could be supported by Regions.
		has a boundary consisting of two pieces, which are half open intervals on the curve separating points inside the region from those outside the region.
	www.research.ibm.com /nao/Primer/ManifoldBoundaryPrimer.html (297 words)

	pelago / intelligent design : : : intervals
		Intervals is a collection of current links to important news and resources, provided by pelago, to help unite the web and business communities.
		But some companies and government organizations are taking their commitment to open source a step further by actively participating in the open-source community.
		intervals is a powerful application that can enhance your web presence.
	www.pelagodesign.com /intervals (844 words)

	Exercises 1
		Draw the graphs of continuous maps which show that any two closed intervals are homeomorphic (topologically equivalent).
		Prove that any two open intervals are homeomorphic.
		Draw the graph of a continuous function from the open interval (0, 1) onto the closed interval [0, 1].
	www-history.mcs.st-and.ac.uk /~john/MT4522/Tutorials/T1.html (295 words)

	Fractal - Wikipedia, the free encyclopedia
		This recursive nature is clear in these examples â” take a branch from a tree or a frond from a fern and you will see it is a miniature replica of the whole: not identical, but similar in nature.
		A relatively simple class of examples is the Cantor sets, in which short and then shorter (open) intervals are struck out of the unit interval [0, 1], leaving a set that might (or might not) actually be self-similar under enlargement, and might (or might not) have dimension d that has 0 < d < 1.
		A simple recipe, such as excluding the digit 7 from decimal expansions, is self-similar under 10-fold enlargement, and also has dimension log 9/log 10 (this value is the same, no matter what logarithmic base is chosen), showing the connection of the two concepts.
	en.wikipedia.org /wiki/Fractal (1860 words)

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