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Topic: Closed set

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In the News (Wed 19 Jun 19)

  Closed set - Wikipedia, the free encyclopedia
In topology and related branches of mathematics, a closed set is a set whose complement is open.
The notion of closed set is defined above in terms of open sets, a concept that makes sense for topological spaces, as well as for other spaces that carry topological structures, such as metric spaces, differentiable manifolds, uniform spaces, and gauge spaces.
We have seen twice that whether a set is closed is relative; it depends on the space that it's embedded in.
en.wikipedia.org /wiki/Closed_set   (572 words)

 Encyclopedia: Closed set
In mathematics, the closure of a set S consists of all points which are intuitively close to S. A point which is in the closure of S is a point of closure of S. The notion of closure is in many ways dual to the notion of interior.
In mathematics the term countable set is used to describe the size of a set, e.
In mathematics, an Fσ set (said F-sigma set) is a countable union of closed sets.
www.nationmaster.com /encyclopedia/Closed-set   (1544 words)

 Encyclopedia topic: Closed set   (Site not responding. Last check: 2007-10-06)
An alternative characterization of closed sets is available via sequence (Serial arrangement in which things follow in logical order or a recurrent pattern) s and net (An open fabric of string or rope or wire woven together at regular intervals) s.
Any intersection (A junction where one street or road crosses another) of arbitrarily many closed sets is closed, and any union (The state of being joined or united or linked) of finitely (additional info and facts about finitely) many closed sets is closed.
The intersection property also allows one to define the closure (Termination of operations) of a set A in a space X, which is defined as the smallest closed subset of X that is a superset (additional info and facts about superset) of A.
www.absoluteastronomy.com /encyclopedia/c/cl/closed_set.htm   (682 words)

 PlanetMath: closed set in a compact space is compact
closed set in a compact space is compact
"closed set in a compact space is compact" is owned by mathcam.
This is version 6 of closed set in a compact space is compact, born on 2003-04-11, modified 2003-06-19.
planetmath.org /encyclopedia/AClosedSetInACompactSpaceIsCompact.html   (105 words)

 Open and closed maps - Wikipedia, the free encyclopedia
Likewise, a closed map is a function which maps closed sets to closed sets.
The floor function from R to Z is open and closed (because Z carries the discrete topology).
In the first two cases, being open or closed is merely a sufficient condition for the result to follow.
en.wikipedia.org /wiki/Open_map   (702 words)

 PlanetMath: club
If a set is both closed and unbounded then it is a club set.
club, closed, unbounded, closed unbounded, closed set, unbounded set, closed unbounded set, club set
This is version 1 of club, born on 2002-07-29.
planetmath.org /encyclopedia/Club.html   (66 words)

 In a few words...
A set that can't be split into a union of two sets each of which is both open and closed.
Any set that may be brought into 1-1 correspondence with the set of real numbers.
A set is a topological space if some of its subsets are declared to be open subject to these conditions.
www.cut-the-knot.com /do_you_know/few_words.shtml   (3748 words)

 PlanetMath: closed set
is open (for it being union of two open sets).
is not closed under the standard topology on
This is version 2 of closed set, born on 2002-03-02, modified 2002-03-09.
planetmath.org /encyclopedia/ClosedSet.html   (74 words)

 Infinitude of Primes - A Topological Proof
A finite union of closed sets is closed.
One may start with neighborhoods and deduce from their properties properties of open and closed sets or those of the operation of closure.
It's also possible to start with Statements 1 and 2 and the stipulation that the empty set and the whole space are closed, and define neighborhoods and open sets with all the expected properties of the latter.
www.cut-the-knot.org /proofs/Furstenberg.shtml   (601 words)

 Math Forum - Ask Dr. Math   (Site not responding. Last check: 2007-10-06)
Date: 07/01/2001 at 12:14:19 From: Doctor Ian Subject: Re: Algebra Hi Craig, A set of elements is closed under an operation if, when you apply the operation to elements of the set, you always get another element of the set.
For example, the whole numbers are closed under addition, because if you add two whole numbers, you always get another whole number - there is no way to get anything else.
So the statement that 'the complex numbers are closed under addition' means that if you add two complex numbers together, you are guaranteed to get a complex number as the sum.
mathforum.org /library/drmath/view/52452.html   (166 words)

 Closed set : Closed subset   (Site not responding. Last check: 2007-10-06)
Note that thisn'tion depends on the concept of "outside", the surrounding space with respect to which the complement is taken.
For instance, the unit interval [0,1] is closed in the real numbers, and the set [0,1] ∩ Q of rational numbers between 0 and 1 (inclusive) is closed in the space of rational numbers, but [0,1] ∩ Q isn't closed in the real numbers.
Am I not mirrored in those eyes of while she is dressed in the glories of leaf and flower, and draw the the poverty of winter?'.
www.termsdefined.net /cl/closed-subset.html   (718 words)

 Open and Closed Sets in Metric Spaces
To construct this set, start with the closed interval [0,1] and recursively remove the open middle-third of each of the remaining closed intervals...
This finite union of closed intervals is closed.
The Cantor set is the intersection of this (decreasing or nested) sequence of sets and so is also closed.
www.math.sc.edu /~sharpley/math555/Lectures/MetricSpaceTopol.html   (232 words)

The concept of an H-bounded set was defined by Lambrinos in [1] as a weakening of the notion of a bounded set which he defined in [2].
In particular a new characterization of H-bounded sets is given and the structure of H-bounded sets is explored.
Since every subset of a compact set is H-bounded, H-bounded sets need not be closed (which is not the case for compact sets, H-closed sets, or H-sets).
at.yorku.ca /b/a/a/g/08.l2h   (600 words)

 Lecture Notes 3 - Math 3210
Def: A supporting line for a two dimensional set with interior points is a line through at least one boundary point of the set such that all points of the set are in the same closed half-plane determined by the line.
Thm 3.8 : A simple closed curve S is the boundary of a 2-dim convex body K if and only if each closed closed polygon determined by successive points on S is the boundary of a convex polygonal region.
Def: A point A of convex set K is called an extreme point of K iff it is not an interior point of any line segment contained in K. Thm 3.17 : A convex polygonal region S is the convex hull K of its extreme points.
www-math.cudenver.edu /~wcherowi/courses/m3210/hg3lc3.html   (764 words)

 Closed sets in   (Site not responding. Last check: 2007-10-06)
Closed sets are more difficult to characterize directly: we describe the concepts of isolated and cluster points for a set
Closed sets can be can be very bizarre: a famous example is the Cantor set: this is the set of all real numbers in the interval
Now iterate: at each iteration, the number of intervals doubles and the interval length is multiplied by a factor of one third.
www.math.pitt.edu /~sparling/23014/23014notes6/node18.html   (146 words)

 Open and Closed Sets
It is fairly clear that when combining two open sets (either via union or intersection) the resulting set is again open, and the same statement should be true for closed sets.
The basic open (or closed) sets in the real line are the intervals, and they are certainly not complicated.
The worst-case scenario for the open sets, in fact, will be given in the next result, and we will concentrate on closed sets for much of the rest of this chapter.
pirate.shu.edu /projects/reals/topo/open.html   (733 words)

 Monotype Imaging: Closed Caption Fonts
Monotype Imaging’s closed caption font set supports the Federal Communications Commission’s requirements for closed captioning display on digital and analog television sets.
Requirements for closed captioning have existed since the 1990 passage of the Television Decoder Circuitry Act.
Devices affected by the rules include digital sets with wide-screen displays measuring at least 7.8 inches vertically, digital TVs with conventional displays measuring at least 13 inches vertically, and stand-alone digital TV tuners, whether or not they are marketed with display screens.
www.agfamonotype.com /ce/closed_caption.asp   (242 words)

 Proposition 6.4.1: Continuity and Topology   (Site not responding. Last check: 2007-10-06)
If D is closed, then the inverse image of every closed set under f is again closed.
The inverse image of every open set under f is the intersection of D with an open set.
The inverse image of every closed set under f is the intersection of D with a closed set.
www.shu.edu /projects/reals/cont/proofs/conttop.html   (317 words)

 This is Jordon Cooper's weblog: Open and Closed Set Thinking in the Church   (Site not responding. Last check: 2007-10-06)
Applying this to the Church, 'closed set' believers have a 'territorial' concept of God's kingdom, enclosed within a boundary.
For 'open set' believers, membership of the Kingdom is defined by movement towards or away from Christ as centre.
Which set theory one adheres to has nothing to do with whether one's positions on specific doctrines are liberal or conservative.
jordoncooper.com /2003/08/open-and-closed-set-thinking-in-church.html   (812 words)

 Proof of the main result
The implications from (2) to (3) and (5) to (6) are straightforward by considering the countable set
The implications from (3) to (4) and (6) to (7) are straightforward by considering the countable set
(as C is a code for a weak-*-closed set), whereas `n is in the range of f' is
www.math.psu.edu /simpson/papers/convex-l/node5.html   (1044 words)

 5aSC20 Closed-set response formats: An evaluation within the framework of   (Site not responding. Last check: 2007-10-06)
Preliminary results indicate that, relative to the open-set condition, when the difficulty of alternatives is controlled, identification within the closed-set format is improved for hard, but not for easy, words.
It is hypothesized that the improved word recognition for hard words in closed-set formats is the result of reducing the viable candidate set.
In contrast, the already small candidate set of easy words minimizes the benefits of reducing the number of viable candidate items.
www.auditory.org /asamtgs/asa95stl/5aSC/5aSC20.html   (230 words)

 A Fourier analytical characterization of the Hausdorff dimension of a closed set and of related Lebesgue spaces ...   (Site not responding. Last check: 2007-10-06)
Let \Gamma be a closed set in R n with the Lebesgue measure j\Gammaj = 0.
92 The geometry of fractal sets (context) - Falconer - 1985 ACM
1 The dimension of a closed subset of R n and related function..
citeseer.ist.psu.edu /98365.html   (451 words)

 Closed hospital set to reopen - The Clarion-Ledger
Closed hospital set to reopen - The Clarion-Ledger
A north Jackson hospital, closed by court order in 1999 after competitors challenged its need, will reopen this fall as a long-term, rehabilitation facility.
The Ridgewood Road hospital, known as the CMMC North Campus, closed after losing an appeal to the state Supreme Court, as well as appeals to the governor and Legislature, to remain in operation.
www.clarionledger.com /apps/pbcs.dll/article?AID=/20050818/NEWS01/508180371/1002   (520 words)

 Current Issues   (Site not responding. Last check: 2007-10-06)
Fisheries Minister Pete Hodgson is banning set nets from inshore waters of the North Island's upper west coast to protect the critically endangered Maui's dolphin.
All commercial set netting will be banned within four nautical miles of the coast (excluding most harbour areas) from Maunganui Bluff (north of Dargaville) to Pariokariwa Point (north of New Plymouth).
Recreational set netters will also be affected, particularly by the ban on netting in the Manukau Harbour entrance.
www.fish.govt.nz /current/press/pr240103.htm   (558 words)

 closed set - OneLook Dictionary Search
Closed set : A Glossary of Mathematical Terms [home, info]
Closed Set : Eric Weisstein's World of Mathematics [home, info]
Phrases that include closed set: closed set in a compact space is compact, gcd closed set, union closed set
www.onelook.com /cgi-bin/cgiwrap/bware/dofind.cgi?word=closed+set   (154 words)

 closedc1   (Site not responding. Last check: 2007-10-06)
Closed Economy Set C: In this set we continue to explore models which include a government sector.
Consider the following set of equations which depict an imaginary economy.
Solve this model for the equilibrium level of Y. Assume that output is demand determined and that all income is distributed to households.
www.economics.unimelb.edu.au /rdixon/101/multiplier/closedc1.html   (93 words)

 Closed set
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An alternative characterization of closed sets is available via sequences and netss.
Visit NewsDaily for the latest headlines from around the world.
www.sciencedaily.com /encyclopedia/closed_set   (578 words)

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