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Topic: Finite intersection property

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	Finite intersection property - Wikipedia, the free encyclopedia
		In topology, the finite intersection property is a property of a collection of subsets of a set X.
		This is trivially satisfied if the intersection over the entire collection is nonempty (in particular, if the collection itself is empty), and it is also trivially satisfied if the collection is nested, meaning that for any finite subcollection, a particular element of the subcollection is contained in all the other elements of the subcollection, e.g.
		The finite intersection property is useful in formulating an alternative definition of compactness: a space is compact if and only if every collection of closed sets satisfying the finite intersection property has nonempty intersection itself.
	en.wikipedia.org /wiki/Finite_intersection_property (250 words)

	Tychonoff's theorem - Wikipedia, the free encyclopedia
		To actually prove Tychonoff's theorem, we use the definition of compactness based on the FIP, by taking an FIP collection A of sets, and showing that the intersection over closures of elements of A is nonempty.
		But then these basis elements intersect every element of D, and so x is a limit point of each element of D, and so is in the closure of each element of D.
		Another proof uses the Alexander subbase theorem, and yet another proof follows trivially from the properties of nets on product spaces, in particular that a net converges in a product space iff each coordinate converges and the fact that compactness can be expressed in terms of nets.
	en.wikipedia.org /wiki/Tychonoff's_theorem (1201 words)

	Properties
		Properties, by contrast, do not seem to have spatial parts; indeed, they are sometimes said to be wholly-present in each of their instances.
		Properties and relations measured on an interval scale are invariant under positive linear transformations, but this isn't true of all properties and relations measured on ordinal scales.
		The fact that properties confer causal powers on their instances is also naturally understood as the claim that the instances of a property have those powers in all possible worlds in which that property exists.
	plato.stanford.edu /entries/properties (18888 words)

	The Tikhonov Product Theorem
		Let f be any finite subcollection of m, and note that the intersection of f and u and v is the intersection of f with w.
		Let f be a finite subcollection, and let g be their nonempty intersection.
		Embed t in a collection m of (not necessarily closed) sets, that is maximal with respect to the finite intersection property.
	www.mathreference.com /top-cs,tpt.html (1013 words)

	Set Theory Papers of Andreas R. Blass
		We prove that the number of near-coherence classes of non-principal ultrafilters on the natural numbers is either finite or at least the larger of the dominating number and the minimum number of generators for such an ultrafilter.
		We study some weakenings of the finite intersection property for families of subsets of the natural numbers.
		The weakenings involve (1) requiring intersections for only a fixed number of sets from the family and (2) requiring the sets to have elements near each other rather than actually intersecting.
	www.math.lsa.umich.edu /~ablass/set.html (2059 words)

	Compact_space
		Then the intersection of all the U(x) and the union of all the V(a) are the required neighbourhoods of x and A.
		An equivalent definition of compact spaces, sometimes useful, is based on the finite intersection property.
		There are a number of topological properties which are equivalent to compactness in metric spaces, but are inequivalent in general topological spaces.
	www.freecaviar.com /search.php?title=Compact_space (1413 words)

	Difference of two Cantor sets
		First, we'll see that any number with a finite ternary expansion can be written as the difference of two elements from the Cantor set.
		A sequence of closed sets is said to possess a finite intersection property, if any finite subsequence has a non-empty intersection.
		A metric space is compact iff the intersection of every sequence of closed sets with the finite intersection property is not empty.
	www.cut-the-knot.org /do_you_know/cantor3.shtml (944 words)

	Citations: Topology: A First Course - Munkres (ResearchIndex) (Site not responding. Last check: 2007-10-22)
		A finite complete K ary tree in which all of the leaves have the same depth is termed balanced.
		Recall that a topology on a set is abstractly defined as a family of subsets of that contains and and is closed under finite intersections and arbitrary unions.
		Hence from the finite intersection property we deduce that 2 Gamma C 6= Let t be a truth assignment in 2 Gamma C.
	citeseer.ist.psu.edu /context/79191/0 (1858 words)

	Xerox Finite State Tool
		give information about the content and properties of the network on the top of the stack, the language or relation it represents, the state of the system, and the available commands.
		Property list commands modify or display the attribute-value pairs on the top network's property list.
		If an attribute to be added already exists on the property list of the network, 'add properties' will reset it to a new value.
	www.cis.upenn.edu /~cis639/docs/xfst.html (7033 words)

	Journey through Intersection Graph County (Site not responding. Last check: 2007-10-22)
		Although a large amount of work has been done for several intersection graph models, there are still only very few basic concepts and ideas which work for general intersection graphs, and even these ideas are not as widely known as they maybe should.
		In application it is often important to find certain large substructures in the intersection graphs occuring, see this or this or this example.
		By duality, intersection graphs should be viewed as the union of certain complete subgraphs---if we have the Helly-property, these complete subgraphs are really (maximal) cliques.
	www.math.uni-hamburg.de /spag/gd/mitarbeiter/prisner/Pris/Rahmen.html (662 words)

	Finite Rings (Site not responding. Last check: 2007-10-22)
		Hence the intersection of all prime ideals is also the ideal consisting of all nilpotent elements of a ring; this ideal is called the nil radical of the ring.
		When the ring under consideration is Noetherian, the nil radical is finitely generated so there is an upper bound k on the index of nilpotency of the generators.
		The Jacobson radical of an Artinian ring is the product of its (finite collection of) maximal ideals and is a nilpotent ideal; each prime ideal is maximal and consists of zero-divisors; the complement of the union of all maximal ideals consists of units.
	www.imsc.ernet.in /~kapil/geometry/caag/finite.html (2519 words)

	NTU Info Centre: Hypergraph (Site not responding. Last check: 2007-10-22)
		It is also possible to define a hypergraph in a more general way as a pair of sets (X,E) where E is some arbitrary subset of the power set of X.
		A property of graphs that encourages their study, is that it is easy to draw a representation of them on a piece of paper.
		This is not so for hypergraphs and so their study tends to be conducted using the nomenclature of set theory rather than the more pictoral descriptions (for instance 'trees','forests' and 'cycles') of graph theory.
	www.nowtryus.com /article:Hypergraph (448 words)

	wikien.info: Main_Page (Site not responding. Last check: 2007-10-22)
		In topology, the finite intersection property is a property of a collection of subsets of a set
		The finite intersection property is useful in formulating an alternative definition of compactness.
		In particular, a space is compact if every collection of closed sets satisfying the finite intersection property has nonempty intersection itself.
	www.hostingciamca.com /index.php?title=Finite_intersection_property (243 words)

	All finite distributive lattices occur as intervals between Hausdorff topologies
		forms a lattice under inclusion, in which the meet of two topologies is their intersection, while the join is the topology with their union as a sub-basis.
		In this paper we are concerned with the local structure of this lattice.
		Hence a finite lattice can be realized as such an interval if and only if it is distributive.
	pear.math.pitt.edu /mathzilla/Examples/lattices.xml (830 words)

	[No title]
		A property of numbers which states that the operations of addition and multiplication are associative.
		A property of numbers which states that the operations of addition and multiplication are commutative.
		It may also be expressed as the intersection of E with the closure of its complement.
	www.mathacademy.com /platonic_realms/encyclop/main.txt (8404 words)

	Topology MAT 530
		The Baire theorem states that in a complete metric space, the intersection of countably many open dense sets is dense ("dense" means that the closure is the whole space).
		We say that "almost all points" of a complete metric space (in the sense of the Baire category) satisfy a certain property, if the set of points with this property is the intersection of countably many open dense sets.
		If a collection of closed sets has a finite intersection property, then its intersection is nonempty.
	www.math.sunysb.edu /~timorin/mat530.html (2904 words)

	World Intellectual Property Organization (Site not responding. Last check: 2007-10-22)
		The distribution of Compton scattered photons from a particular point in space can be described as the sum of a set of annula (rings) centered at the intersection of the second detection plane with the vector originating at the scattering point and continuing along the direction of the incoming photon, as shown in Figures 4C-4E.
		The axis of incidence intersects the second detector layer at a displacement y".
		The first and second detector layers (666 and 668) are shown in a second position with radius vector 676, chosen for rebinning purposes to be perpendicular to this incident path (for the purpose of equivalent parallel collimation).
	www.wipo.int /ipdl/IPDL-CIMAGES/view/pct/getbykey5?KEY=01/88493.011122&ELEMENT_SET=DECL (7572 words)

	G3M08 FINITE AUTOMATA (Site not responding. Last check: 2007-10-22)
		This theorem can be regarded as providing a precise description of the computational power of any machine or organism endowed with a finite memory.
		In the first, we introduce the two main concepts on which the course is based: finite state automata and formal languages.
		The complement of a recognisable language is recognisable, the intersection of recognisable languages is recognisable, and the union of recognisable languages is recognisable.
	www.informatics.bangor.ac.uk /public/math/teaching/ugrads/h201/g3m08.html (295 words)

	Glossary (Site not responding. Last check: 2007-10-22)
		Topology is a branch of mathematics concerned with those properties of geometric configurations (as sets of points) which are unaltered by elastic deformations (as a stretching or a twisting) that are homeomorphisms.
		The centroid of a triangle is the intersection of the medians.
		a tree is a graph with the property that there is a unique path from any vertex to any other vertex traveling along the edges.
	mcraeclan.com /MathHelp/BasicSetTopologyGlossary.htm (2727 words)

	Finite Solid Primitives (Site not responding. Last check: 2007-10-22)
		This formula has the nice property that it is exactly equal to the strength parameter at the center of the component and drops off to exactly 0 at a distance from the center of the component that is equal to the radius value.
		Both are 4-D generalizations of the complex numbers but neither satisfies all the field properties (all the properties of real and complex numbers that many of us slept through in high school).
		The intersection test with a SOR object involves solving a cubic polynomial while the test with a lathe object requires to solve a 6th order polynomial (you need a cubic spline for the same smoothness).
	www.blastwave.org /docs/povray-3.50c/povdoc_187.html (6624 words)

	[No title]
		No other finite group has a universal bundle which is so easily pictured; it is th* *is case which motivated some of our terminology.
		Then the fibres of the map fip are 4 orbits under the action of G. In particular, the map fip is at most n-to-one.
		Remark: Properties intermediate between H-antipolarity and H-incompleteness may be useful if Theorem 4 is to be extended to a larger class of groups.
	hopf.math.purdue.edu /Feldman-Wilce/fibdegen.txt (5513 words)

	Metamath Proof Explorer - fiint (Site not responding. Last check: 2007-10-22)
		Description: Equivalent ways of stating the finite intersection property.
		This theorem is applicable to a topology, which (among other axioms) is closed under finite intersections.
		Some texts use the left-hand version of this axiom and others the right-hand version, but as our proof here shows, their "intuitively obvious" equivalence can be non-trivial to establish formally.
	us.metamath.org /mpegif/fiint.html (71 words)

	NTU Info Centre: Compact space (Site not responding. Last check: 2007-10-22)
		Some authors (including Bourbaki) use the term quasicompact instead and reserve the term compact for compact Hausdorff spaces, but this encyclopedia follows the usual current practice of allowing compact spaces to be non-Hausdorff.
		The spectrum of any continuous linear operator on a Hilbert space is a compact subset of '''C'''.
		At one time, when primarily metric spaces were studied, compact was taken to mean the weaker sequentially compact (every sequence has a convergent subsequence).
	www.nowtryus.com /article:Compact_space (1409 words)

	11 Finite State (Site not responding. Last check: 2007-10-22)
		The idea is to achieve a finite closure after some number of states have been added.
		Regular expressions are finite symbol strings, but the sets they denote can be finite or infinite.
		The reason for this step is to isolate the properties of being initial and accepting so that we can more easily apply the transformations in the second step.
	www.cs.hmc.edu /claremont/keller/webBook/ch12 (10653 words)

	Compact (Site not responding. Last check: 2007-10-22)
		Of a design, describes the valuable property that it can all be apprehended at once in one's head.
		For example, (0,1] is not compact, since the sequence (0,1/n] of closed sets (in (0,1]) is nested, and so clearly has the finite intersection property, but has empty intersection.
		All intellectual property rights in and to the game are owned in the U.S.A and Canada by Hasbro Inc., and throughout the rest of the world by J.W. Spear & Sons Limited of Maidenhead, Berkshire, England, a subsidiary of Mattel Inc. Mattel and Spear are not affiliated with Hasbro.
	www.websters-online-dictionary.org /co/compact.html (3617 words)

	[No title]
		The completeness axiom - bounds, lub=supremum, glb=infimum, max/min; completeness axiom; real number system; - Archimedian property; density of Q in R; 13.
		partition, lower/upper sums and integrals, Riemann integral - Properties of lower/upper sums and integrals; - Criterion for integrability 30.
		Properties of Riemann integral - Classes of Integrable functions: monotone, continuous; f int.
	www.ilstu.edu /~lmiones/247rvf03.doc (703 words)

	[No title]
		The statement is in fact true for infinite collections of arbitrary size; in this case it depends heavily on the peculiar definition of the
		finite intersection property (FIP), there is a maximal set
		But we know these spaces are compact, and so we can choose a point in each space from the intersection of that space's projected
	en-cyclopedia.com /wiki/Tychonoff's_theorem (886 words)

	World Intellectual Property Organization (Site not responding. Last check: 2007-10-22)
		The section concludes with simulation results on the convergence properties, and the efficiency trade-offs.
		As we will presently show, this property is preserved by the PSP rule in the more general case of sharing an arbitrarily divisible resource, and this leads to stability (Nash equilibrium).
		The key property of PSP is that, for a given opponent profile, a player cannot do much better than simply tell the truth, which in this setting means bidding at a price equal to the marginal valuation, i.
	www.wipo.int /ipdl/IPDL-CIMAGES/view/pct/getbykey5?KEY=01/88811.011122&ELEMENT_SET=DECL (9602 words)

	The Complete Nontrivial-Intersection Theorem For Systems Of Finite Sets - Ahlswede, Khachatrian (ResearchIndex)
		Abstract: Recently we proved in [4] a complete intersection theorem for systems of finite sets.
		8 Intersection theorems for systems of finite sets (context) - os, Ko et al.
		1 Intersection theorems for finite sets and geometric applicat..
	citeseer.ist.psu.edu /ahlswede96complete.html (513 words)

	Semicontinuous
		Hint, an open interval is the intersection of two rays.
		The intersection is an open set, containing x, whose image under h is below u.
		Choose a finite subcover, and the image of s is bounded.
	www.mathreference.com /top-cs,semi.html (675 words)

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