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Topic: Closed subsets

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	PlanetMath: closed set
		Closed subsets can also be characterized as follows:
		Cross-references: cluster points, lower limit topology, open sets, union, standard topology, open, complement, subset, topological space
		This is version 7 of closed set, born on 2002-03-02, modified 2007-02-06.
	planetmath.org /encyclopedia/ClosedSet.html (78 words)

	PlanetMath: closed subsets of a compact set are compact
		"closed subsets of a compact set are compact" is owned by Wkbj79.
		See Also: closed set in a compact space is compact, a compact set in a Hausdorff space is closed
		This is version 13 of closed subsets of a compact set are compact, born on 2003-09-04, modified 2007-05-30.
	www.planetmath.org /encyclopedia/ClosedSubsetsOfACompactSetAreCompact.html (179 words)

	[No title]
		The irreducible closed subsets which occur in this uniquely determined decomposition are called the irreducible components of X.
		This implies that S is a union of finitely many irreducible closed subsets, thus contradicting the hypothesis that S is nonempty and thereby proving the first part of the theorem.
		Therefore, the same irreducible closed subsets occur in the two decompositions.
	www.math.umn.edu /~roberts/math8203/irred_var.html (839 words)

	Topology MAT 530
		All connected subsets of the real line are open intervals (that may be empty and may be infinite) with, possibly, some of the ends attached.
		Closed subsets of compact spaces are compact in the subspace topology, the product of two compact spaces is also compact.
		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 /~timorin/mat530.html (2896 words)

	NationMaster - Encyclopedia: Closed subset (Site not responding. Last check: )
		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 is not closed in the real numbers.
		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.
		Any intersection of arbitrarily many closed sets is closed, and any union of finitely many closed sets is closed.
	www.nationmaster.com /encyclopedia/Closed-subset (723 words)

	Locally compact space
		All open or closed subsets of a locally compact Hausdorff space are locally compact in the subspace topology.
		This provides several examples of locally compact subsets of Euclidean spaces, such as the unit disc (either the open or closed version).
		The first two examples show that a subset of a locally compact space need not be locally compact, which contrasts with the open and closed subsets in the previous section.
	www.ebroadcast.com.au /lookup/encyclopedia/lo/Locally_compact_space.html (1265 words)

	PlanetMath: topological space
		Can a topology be defined as a subset of an arbitrary complete (and complemented) lattice, instead of a power set?
		Ok, without a complement operator, it will be hard to define what a closed element should be...
		it might be interesting to note that you can equivalently define a topology in terms of it's closed sets, by demorgan's set laws.
	planetmath.org /encyclopedia/TopologicalSpace.html (302 words)

	PlanetMath: alternative characterizations of Noetherian topological spaces, proof of
		satisfies the descending chain condition for closed subsets.
		Cross-references: unions, subspace topology, stationary, subcover, finite, open cover, sequence, proper subset, strictly, infinite, bijective, map, compact, subset, maximal element, minimal element, open subsets, closed subsets, topological space
		This is version 9 of proof of alternative characterizations of Noetherian topological spaces, born on 2005-07-27, modified 2006-09-14.
	www.planetmath.org /encyclopedia/ProofOfAlternateCharacterizationsOfTheNoetherianCondition2.html (304 words)

	Springer Online Reference Works (Site not responding. Last check: )
		The concept of a compact topological space is fundamental in topology and modern functional analysis; certain fundamental properties of compact spaces (with numerous applications) are already considered in mathematical analysis, e.g.
		-compacta, so too the class of compact spaces is invariant with respect to transition to the space of closed subsets (taken with the Vietoris topology); moreover, the weight of the space does not increase.
		A normal space is said to be perfectly normal if each closed set in it is the intersection of countably many open sets.
	eom.springer.de /c/c023530.htm (2017 words)

	Solution 5
		In the usual topology finite sets are closed, but there are some closed subsets that are not finite.
		Hence every finite subset is closed and so every finite set is open in the subspace topology which is therefore discrete.
		In the cofinite topology on R, the subset 2Z is not closed and hence it is not closed in the subspace topology on Z.
	www-groups.dcs.st-and.ac.uk /~john/MT4522/Solutions/S5.html (315 words)

	Topology
		Theorem: If A is a connected subset of X wrt t and f:X \to Y is a continuous map of (X,t) to (Y,s), then f(A) is a connected subset of Y wrt s.
		Theorem: If A is a compact subset of X wrt t and f:X \to Y is a continuous map of (X,t) to (Y,s), then f(A) is a compact subset of Y wrt s.
		Hence, if A is any nonempty subset of X, the intersection of all closed subsets of X which contain A must also be closed, and we call this set the _closure_ of A, denoted cl(A), or cl(A,t) if it is necessary to indicate the topology t on X with respect to which closure is taken.
	www.georgetown.edu /faculty/kainen/topology.html (1132 words)

	Topology - Abstract Shape (Site not responding. Last check: )
		Since subsets of a topological space are themselves topological spaces under their relative topology, any definition or discussion involving subsets applies to the whole space.
		From the definition of open and closed subsets, it is obvious that the null set and the whole space X are both open and closed.
		The closed interval [0,2] of the rationals is not compact - the sequence of finite decimals which converges to the square root of two is an infinite subset without a limit point.
	ourworld.cs.com /jamessfreeman16/Topology.htm (2436 words)

	Trimethoric (and trisynaptic) polyhedra
		As used here the term 'polyhedron' denotes a set of plane polygons forming a single closed surface, 'face' denotes the entire area within a bounding polygon and 'region' denotes a part of a face bounded by edges or by lines of intersection with other faces and traversed by none.
		Although some closed subsets may be found for the trimethoric polyhedra described, in no case do all faces fall into such subsets.
		The ability of a conventional polyhedron's surface to close in a finite way around a defined internal space is associated with the ability to trace a circuit around any vertex figure.
	www.steelpillow.com /polyhedra/trimethoric/trimethoric.htm (1325 words)

	Compact Sets (Site not responding. Last check: )
		A set S is compact if every collection of closed subsets of S with the finite intersection property has a nonempty intersection.
		A closed set in S comes from a closed set in T, whose complement is open in T. The open sets form a cover iff the closed sets in S have no intersection.
		Any closed subset Y of a compact set S is compact.
	www.mathreference.com /top-cs,intro.html (636 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.
		A Hausdorff space X is normal if and only if for each pair of disjoint closed sets A and B there exists a map f from X to the unit interval I that is identically 0 on A and identically 1 on B.
		In particular, an image of the closed unit interval [0,1] (sometimes called an arc or a path) is connected.
	www.win.tue.nl /~aeb/at/algtop-2.html (1509 words)

	Limit points and closed sets in metric spaces
		A subset A is said to be a closed subset of X if it contains all its limit points.
		The subset X is a closed subset of itself.
		The closed interval [0, 1] is closed subset of R with its usual metric.
	www-history.mcs.st-and.ac.uk /~john/MT4522/Lectures/L9.html (463 words)

	physics - Relatively compact subspace
		In mathematics, a relatively compact subspace (or relatively compact subset) Y of a topological space X is a subset whose closure is compact.
		Since closed subsets of compact spaces are compact, every set in a compact space is relatively compact.
		In the case of a metric topology, or more generally when sequences may be used to test for compactness, the criterion for relative compactness becomes that any sequence in Y has a subsequence convergent in X.
	www.physicsdaily.com /physics/Relatively_compact (190 words)

	Kids.Net.Au - Encyclopedia > Second category
		The interior of every union of countably many nowhere dense sets is empty.
		Whenever the union of countably many closed subsets of X has an interior point, then one of the closed subsets must have an interior point.
		every space that is homeomorphic to an open subset of a complete pseudometric space is a Baire space (this includes the irrational numbers)
	www.kids.net.au /encyclopedia-wiki/se/Second_category (276 words)

	Compact Set
		It says that K is compact iff given ANY collection of closed subsets of K, then if any finite subcollection of those subsets has nonempty intersection, then the whole collection has nonempty intersection.
		I'm not entirely sure what you mean by "K is compact iff any collection of closed subsets of K with the finite intersection property has nonempty intersection" since "a collection of sets is said to have the finite intersection property iff any finite subecollection has nonempty intersection".
		Suppose that if C is a collection of closed subsets of K with the finite intersection property also has nonempty intersection.
	www.physicsforums.com /showthread.php?p=959862 (1206 words)

	XML-Data
		It implies that every valid instance of the subset is a valid instance of the superset.
		Ranges and other constraints are cummulative, that is, all apply (though the exact effect of this depends on the semantics of the constraint type).
		To indicate that the content model of the subset should inherit the content model of a superset, we use a particular kind of superType called "genus" of which only one is allowed per ElementType.
	www.w3.org /TR/1998/NOTE-XML-data-0105 (6061 words)

	The Unapologetic Mathematician
		In some sense these will be “closeness preserving”, but as we’ll see they work a little differently than the various algebraic categories we’re used to.
		Topology is, roughly speaking, the study of spaces where we have an idea of what it means for points to be “close” to each other, and functions which “preserve closeness”.
		are called the closed subsets of the topological space, and their complements are called the open subsets.
	unapologetic.wordpress.com (2458 words)

	Math Forum Discussions - Regular Closed Subsets and Subspaces
		If U is a regular open subset of a regular open subspace A, then U is regular open subset of the overspace S. U is regular open when U = int cl U = interior of closure of U
		If K is a regular closed subset of a regular closed
		The Math Forum is a research and educational enterprise of the Drexel School of Education.
	www.mathforum.com /kb/thread.jspa?forumID=253&threadID=1131622&messageID=3710850 (350 words)

	Math Forum Discussions
		So the power of the set of closed sets in R is also c.
		C is in fact closed, so it is trivially F(sigma).
		Because of these two facts we can deduce that there are 2^c subsets of C, and they are all measure zero.
	mathforum.org /kb/thread.jspa?messageID=511716&tstart=0 (515 words)

	Noetherian topological space (Site not responding. Last check: )
		In mathematics, a Noetherian topological space is a topological space in which closed subsets satisfy the descending chain condition.
		Equivalently, we could say that the open subsets satisfy the ascending chain condition, since they are the complements of the closed subsets.
		It can also be shown to be equivalent that every open subset of such a space is compact, and in fact the seemingly stronger statement that every subset is compact.
	en.askmore.net /Noetherian_topological_space.htm (342 words)

	Compactness of the Bounded Closed Subsets of $\cal E^2_\rm T$
		The compactness of the bounded closed subset of ${\cal E}^2_{\rm T}$ is proven.
		Families of subsets, subspaces and mappings in topological spaces.
		The Jordan's property for certain subsets of the plane.
	www.cs.ualberta.ca /~piotr/Mizar/mirror/http/JFM/Vol11/topreal6.html (232 words)

	[No title]
		To finagle the concept of "closed" into here, recall that if Z is a complete metric space, a subset X of Z is closed in Z iff it is complete in its own right.
		Thus for subsets of a _complete_ metric space, X is compact iff it is closed and _totally_ bounded.
		So, to answer your question, a cosed and bounded subset of a metric space will be compact provided: 1) The space is complete (closed subsets of a complete space are complete).
	www.math.niu.edu /~rusin/known-math/95/compact.nss (1036 words)

	[No title]
		In the next stage there are 4 disjoint subsets in which 1/3 are leaf nodes, totaling 4/27 of paths terminated at this level.
		That is not a different summation, but it arises from a different viewpoint from that which first resulted from a count of the number of leaf nodes descendent from the extensions.
		Furthermore, the various fractions of the Fibonacci series which appear for a given value of b in that table of cardinalities have closed sums because the sum(F(i)/2^(i+1),i=1..infinity) is 1, where F(i) are the Fibonacci numbers.
	www-personal.ksu.edu /~kconrow/counting.html (2316 words)

	Borel algebra
		The Borel algebra may alternatively and equivalently defined as the smallest σ-algebra which contains all the closed subsets of T.
		A subset of T is a Borel set if and only if it can be gotten from open sets by using a countable series of the set operations union, intersection and complement.
		A particularly important example is the Borel algebra on the set of real numbers.
	www.ebroadcast.com.au /lookup/encyclopedia/bo/Borel_algebra.html (124 words)

	Math 323 Answer to Homework 6 (Site not responding. Last check: )
		The whole argument for the case of open subsets almost works for the case of closed subsets (with the help of a similar version of Exercise 5.39, in which open are changed to closed).
		For a counterexample for the case of inifinite union of closed subsets, consider any map f: R → R, where both R have the usual topology.
		Suppose A×B is closed, and b ∈ B (i.e., B is not empty).
	www.math.ust.hk /~mamyan/ma323/hwk06.shtml (2413 words)

	Topology history
		He called a simple closed curve on a surface which does not intersect itself an irreducible circuit if it cannot be continuously transformed into a point.
		This is not as straightforward as it might appear since even in three dimensions it is possible to have a surface that cannot be reduced to a point yet closed curves on the surface can be reduced to a point.
		He also defined closed subsets of the real line as subsets containing their first derived set.
	www-groups.dcs.st-and.ac.uk /~history/HistTopics/Topology_in_mathematics.html (1456 words)

	Two-Arc Closed Subsets of Graphs (ResearchIndex)
		Abstract: A subset of vertices of a graph is said to be 2-arc closed if it contains every vertex that is adjacent to at least two vertices in the subset.
		In this paper, 2-arc closed subsets generated by pairs of vertices at distance at most 2 are studied.
		Several questions are posed about the structure of such subsets and the relationships between two such subsets, and examples are given from the class of partition graphs.
	citeseer.ist.psu.edu /416004.html (373 words)

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