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Topic: Disjoint union (topology)

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disjoint union

Disjoint sets

Almost disjoint sets

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	Encyclopedia :: encyclopedia : Topology (Site not responding. Last check: 2007-10-29)
		Topology (Greek topos, place and logos, study) is a branch of mathematics concerned with spatial properties preserved under bicontinuous deformation (stretching without tearing or gluing); these are the topological invariants.
		Topology has sometimes been called rubber-sheet geometry, because it does not distinguish between a circle and a square (a circle made out of a rubber band can be stretched into a square) but does distinguish between a circle and a figure eight (you cannot stretch a figure eight into a circle without tearing).
		In pointless topology one considers instead the lattice of open sets as the basic notion of the theory, while Grothendieck topologies are certain structures defined on arbitrary categories which allow the definition of sheaves on those categories, and with that the definition of quite general cohomology theories.
	www.hallencyclopedia.com /topic/Topology.html (1776 words)

	Disjoint union - Wikipedia, the free encyclopedia
		In set theory, a disjoint union (or discriminated union) is a union of a collection of sets whose members are pairwise disjoint.
		This notation is meant to be suggestive of the fact that the cardinality of the disjoint union is the sum of the cardinalities of the terms in the family.
		In the language of category theory, the disjoint union is the coproduct in the category of sets.
	en.wikipedia.org /wiki/Disjoint_union (348 words)

	[No title]
		Topology Glossary Mainly extracted from (a) UC Davis Math:Profile Glossary (http://www.math.ucdavis.edu/profiles/glossary.html) by Greg Kuperberg (http://www.math.ucdavis.edu/profiles/kuperberg.html), and (b) Topology Atlas Glossary (http://www.achilles.net/~mtalbot/TopoGloss.html).
		An early result in topology states that every closed 3-manifold (closed meaning that the manifold is finite and connected but has no boundary) has a Heegaard splitting and a resulting description in terms of a Heegaard diagram, which describes how the two handlebodies are glued together.
		Given a vector space of functions of a parameter or functions on a manifold, an operator may have a kernel or matrix whose rows and columns are indexed by the parameter or by points on the manifold.
	www.ornl.gov /sci/ortep/topology/defs.txt (5717 words)

	PlanetMath: disjoint union
		This is also often called being pairwise disjoint and is a much stronger condition than that the intersection of all the images is empty.
		Of course, there are many categories where this usage is unnatural: In the category of pointed sets, the coproduct will be the disjoint union with the distinguished points identified.
		This is version 4 of disjoint union, born on 2004-03-01, modified 2004-03-23.
	planetmath.org /encyclopedia/DisjointUnion.html (217 words)

	Topology glossary
		This is a glossary of some terms used in the branch of mathematics known as topology.
		A set of open sets is a sub-base for a topology if every open set is a union of finite intersections of sets in the sub-base.
		A space X is connected if it is not the union of a pair of disjoint nonempty open sets.
	www.ebroadcast.com.au /lookup/encyclopedia/lo/Local_base.html (1004 words)

	Foundations of Point Set Topology
		The other division of the subject is algebraic topology Point-set topology involves reducing the relationship among points to its most general level and formulating axioms in order to prove theorems of greatest generality.
		The discrete topology for a set S is the collection of all subsets of S. The indiscrete topology for S is the collection consisting of only the whole set S and the null set ∅.
		In the discrete topology no point is the limit point of any subset because for any point p the set {p} is open but does not contain any point of any subset X. The closure of a set Q is the union of the set with its limit points.
	www.sjsu.edu /faculty/watkins/topology2.htm (2151 words)

	Algebraic Topology: Topology
		A topological space is a set X together with a collection of subsets OS the members of which are called open, with the property that (i) the union of an arbitrary collection of open sets is open, and (ii) the intersection of a finite collection of open sets is open.
		The topology on A defined by F is the weakest topology (i.e., the smallest collection OA) for which all these functions become continuous.
		The topology on B defined by F is the strongest topology (i.e., the largest collection OB) for which all these functions become continuous.
	www.win.tue.nl /~aeb/at/algtop-2.html (1509 words)

	Algebraic Topology: Two-dimensional manifolds
		This is a covering of the space with finitely many triangles, homeomorphic images of an ordinary plane triangle, where two triangles either are disjoint, or have a point or an edge in common, each edge is in precisely two triangles, and the residue at a point is a polygon.
		By induction we see that the subspace on the union of an initial part of this sequence of triangles is homeomorphic to a plane regular polygon of which some pairs of edges are identified.
		Indeed, the boundary will be a union of disjoint cycles, and if we glue a disk (a `cap') in the hole bounded by a cycle, we fill the hole.
	www.win.tue.nl /~aeb/at/algtop-4.html (985 words)

	Storage and Topology in the Redesigned MAF/TIGER
		A distinct advantage of using topology derives from the relationships that are maintained between the "topological primitives", i.e., nodes, edges, and faces.
		Topology is utilized to a significant degree by many of the applications that comprise the Census Bureau's Geographic Support System.
		Alternatively, if topology is used to organize spatial data, the coordinates for a section of road that also bounds geographic areas are only stored once, and the correction becomes more straightforward, although it is still subject to business rules that govern feature consistency.
	www.census.gov /geo/mtep_obj2/topo_and_data_stor.html (4867 words)

	Disjoint union (topology) - Wikipedia, the free encyclopedia
		In topology and related areas of mathematics, the disjoint union (also called the direct sum, free union, or coproduct) of a family of topological spaces is a space formed by equipping the disjoint union of the underlying sets with a natural topology called the disjoint union topology.
		The disjoint union topology on X is defined as the finest topology on X for which the canonical injections are continuous (i.e.
		This shows that the disjoint union is the coproduct in the category of topological spaces.
	en.wikipedia.org /wiki/Disjoint_union_(topology) (370 words)

	Masnet Formation Control
		This topic of switching communication topologies is addressed after the rendezvous case study with fixed topologies.
		The union of the five different communication trees is a well connected communication tree.
		However, as the communication topologies are cycled through, the union of the trees eventually brings the whole group together in final position close to the fixed topology case.
	www.engineering.usu.edu /ece/faculty/wren/masnet/Masnet-consensus.htm (3194 words)

	Product Space
		The topology of p is defined by a base, where a base set in p is given by the cross product of open sets in the component spaces.
		Once again, verify that the base for the weak topology is valid, and the topology of p includes all finite cross products of open sets in the component spaces, where the remaining components are unconstrained.
		The coproduct of disjoint spaces is their union, where a set is open iff its restriction to each subspace is open.
	www.mathreference.com /top,prod.html (1822 words)

	Connectedness
		R with its usual topology is not connected since the sets [0, 1] and [2, 3] are both open in the subspace topology.
		The spaces [0, 1] and (0, 1) (both with the subspace topology as subsets of R) are not homeomorphic.
		[2, 3] with the subspace topology inherited from R, are the subspaces [0, 1] and [2, 3].
	www-groups.dcs.st-and.ac.uk /~john/MT4522/Lectures/L19.html (619 words)

	MAT246: Outline of Point Set Topology
		A topological space is a set together with a collection of subsets, called the open sets, such that the union of any number of open sets is open, the intersection of a finite number of open sets is open, and the empty set and the entire space are both open.
		Definition: If S is a subset of a topological space, then the relative topology on S is the topology in which a subset of S is open if and only if it is the intersection of S and an open subset of the entire space.
		Problem #6: Prove that there is a topology on the plane under which it is homeomorphic to the line with its usual topology.
	www.math.toronto.edu /jkorman/Math246Y/top.htm (766 words)

	AdjunctionSpace.htm
		We consider the unlying sets to be "implicitly" disjoint, in particular we consider
		is defined to be the disjoint union of the underlying sets.
		We can also consider the quotient topology to be the "largest" topology (most open sets) for which
	www.umsl.edu /~siegel/Topology/AdjunctionSpace.htm (105 words)

	Oz's crib sheet: basic set theory (Site not responding. Last check: 2007-10-29)
		Union: If C is a collection of sets, then the set of all of C's elements' elements is UC, the union of C.
		Exercise that links with the topology crib sheet: Recall that the formal definition of topology that Michael gave consisted of a set X along with a collection T of subsets of X, such that T satisfied 3 conditions.
		Note that, because the union and intersection of a family of sets doesn't depend on how often the sets appear, or the order they appear in, or so on, I could define union and intersection of simple collections of sets earlier.
	math.ucr.edu /~toby/Oz/sets (3103 words)

	Topology MAT 530
		This is the largest (finest, strongest) topology such that the canonical projection (from the space to the quotient-space) is continuous.
		A counterexample is the set of all rational numbers with the topology induced from the reals (which is the same as the order topology) --- all rationals are separate connected components, but they are not open.
		The Urysohn lemma states that for a normal topological space X and two disjoint closed subsets A and B of it, there exists a continuous function from X to [0,1] that is 0 on A and 1 on B.
	www.math.sunysb.edu /~azinger/mat530/fall04/index.htm (2907 words)

	Topology Course Lecture Notes
		We learnt that, for metric spaces, sequential convergence was adequate to describe the topology of such spaces (in the sense that the basic primitives of `open set', `neighbourhood', `closure' etc. could be fully characterised in terms of sequential convergence).
		This topology is 'just right' in the sense that it is barely fine enough to guarantee the continuity of the coordinate projection functions while being just course enough allow the important result of Theorem.
		A basic formal distinction between algebra and topology is that although the inverse of a one-one, onto group homomorphism [etc!] is automatically a homomorphism again, the inverse of a one-one, onto continuous map can fail to be continuous.
	at.yorku.ca /i/a/a/b/23.dir/index.htm (8277 words)

	A point-set theorem
		The thing is, I'm not 100 percent convinced the closures of G and H must be disjoint.
		A set X is not connected iff there exist disjoint, relatively open (to X), nonempty subsets G and H of X such that (G union H) = X. That doesn't seem to require that the closures of G and H are disjoint.
		A set X is not connected iff there exist disjoint, relatively open (to X), nonempty subsets G and H of X such that (G union H) = X. As it turns out, this statement also appears to hold true if we replace "relatively open" with "relatively closed".
	www.physicsforums.com /showthread.php?t=65973 (1409 words)

	Topology - Abstract Shape (Site not responding. Last check: 2007-10-29)
		Any set X has two obvious topologies: the trivial topology with only two open subsets, the null set and the whole set X; the discreet topology where every subset is open.
		Notice that if a space can be decomposed into the disjoint union of two closed subsets, then, since closed subsets include all their limit points, this intuitively means the space is composed of two pieces "which do not touch".
		In this case, f~(U) and f~(V) are two disjoint, open subsets of X whose union is all of X- which is impossible since X is connected.
	ourworld.cs.com /jamessfreeman16/Topology.htm (2436 words)

	Topology Seminar (Site not responding. Last check: 2007-10-29)
		We will cover the basics of point-set topology, including simple applications in calculus, as well as some rudimentary aspects of differentiable topology, leading up to a brief treatment of the recent solution of the Poincare conjecture.
		A topology on set X is a collection tau of subsets of X such that: (i) tau contains the empty set and X (X is required to be nonempty) (ii) the intersection of any two members of tau is in tau (iii) the union of any subfamily of tau is in tau.
		#2 Show that the collection of all subsets of X forms a topology for any nonempty set X. Let the cofinite topology kappa on some infinite set X be defined to be the collection of all subsets of X which have a finite complement, together with the empty set.
	www.georgetown.edu /faculty/kainen/topsem.html (623 words)

	Urysonh's Metrization Criteria
		Each isolated point is open, and is the union of base sets, hence there is a base open set for each point, which contradicts second countable.
		In general, the topology induced by a function such as f is the weakest topology needed to make f continuous.
		The weakest topology that makes f continuous has to preserve the open sets of t, as determined by the topology of p.
	www.mathreference.com /top-ms,umc.html (1002 words)

	Remarks on Proving The Fundamental Theorem of Algebra
		In summary, it is impossible for the union of two disjoint open intervals to form an interval.
		We have shown that the union of U and V fails to include the point c, which lies between a and b.
		In the language of "general topology", a set which cannot be decomposed into two disjoint, non-empty open sets is called "connected", and we have shown that Bolzano's theorem is in essence equivalent to the theorem that a closed interval is a connected set.
	www.cut-the-knot.org /fta/brodie.shtml (1369 words)

	Elliptic Curves and Modular Functions
		For instance, the set of all rotations of the plane about the origin is a group, and the orbit of any particular point in the plane is a circle whose radius is the distance of the point from the origin.
		If the set which is acted upon has a topology, then the set of orbits also has a topology which is called the "quotient topology", and the resulting topological space is called the "quotient space".
		Since a Riemann surface has a topology, a group of analytic transformations acting upon it defines a quotient space, which is also a Riemann surface.
	www.mbay.net /~cgd/flt/flt05.htm (2994 words)

	[No title] (Site not responding. Last check: 2007-10-29)
		I have tried to "attach" a picture illustrating two disjoint open subsets of the Moore plane whose union contains the entire horizontal axis.
		Ramsay [2] provided a Hausdorff topology on the upper half of the complex plane in which the topology induced on the real axis is discrete (about as disconnected as they come), but no pair of disjoint open sets can be found that both meet the real axis and cover it with their union.
		take X to be the deleted Tychonoff plank (example 87 in Steen and Seebach's Counterexamples in Topology) and stick a unit interval between each successive ordinal in omega_1 and omega.
	www.math.niu.edu /~rusin/known-math/98/connected (352 words)

	Manifolds
		An example of a compact, disconnected one dimensional manifold is a union of two disjoint circles.
		One of the goals in topology is to capture the nature of certain subsets terms of topological properties.
		As a rule, in topology, a situation is simpler in the compact case compared to the non-compact case.
	www.math.uiowa.edu /~roseman/tom/tom/node3.html (4066 words)

	disjoint union - OneLook Dictionary Search
		Tip: Click on the first link on a line below to go directly to a page where "disjoint union" is defined.
		Disjoint Union : Eric Weisstein's World of Mathematics [home, info]
		Phrases that include disjoint union: disjoint union topology
	www.onelook.com /cgi-bin/cgiwrap/bware/dofind.cgi?word=disjoint+union (115 words)

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