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Topic: Dense topology

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	Statement of Research
		A dense topology may induce high interference, which, in turn, reduces the effective network capacity due to limited spatial reuse and may cause unnecessarily high energy consumption.
		Topology control for ad hoc networks aims to maintain a specified topology, such as requiring that the network be connected.
		The desired effect of topology control is to reduce energy consumption, reduce MAC layer interference between adjacent nodes, and to increase the effective network capacity.
	www.eecis.udel.edu /~zhaol/publications/StatementOfResearch.html (1399 words)

	My Topology Questions
		Eg in cofinite topology, since only finite sets are closed, either X, if X is finite, or any countably infinite set is dense hence X is separable.
		In real life you only ever need the standard metric topology or some other topology of continuity of functions that you pick a posteri to make things work (eg topology of pointwise convergence, topology of uniform convergence), or the zariski topology, or the compact open topology.
		By example show that the zariski topology on R^2 is not the product topology from the zariski topology on two copies of R. It's not hard but is an (the only?) interesting example of a case where the product topology is not what you think it might be.
	www.physicsforums.com /showthread.php?t=106672 (2886 words)

	Dense set - Wikipedia, the free encyclopedia
		In topology and related areas of mathematics, a subset A of a topological space X is called dense (in X) if, intuitively, any point in X can be "well-approximated" by points in A.
		The real numbers with the usual topology have the rational numbers and the irrational numbers as dense subsets.
		A metric space M is dense in its completion γM.
	en.wikipedia.org /wiki/Dense_set (206 words)

	Topological Preliminaries
		Topology is one of (quite a few) mathematical theories that permeate other branches of Mathematics connecting them into one coherent whole.
		Most of the examples will be drawn on the 2-dimensional plane but, given the definitions of the distance and neighborhood could be carried over to the 1- and many dimensional cases.
		The set Q of all rational numbers is dense in R, thick, neither open nor closed, and without internal points.
	www.cut-the-knot.org /do_you_know/topology.shtml (759 words)

	Dense MANET Configuration
		The primary idea is not to maintain a centralized, global, and always up-to-date vision of all the nodes and of their network topology, but to design a simple, lightweight, and completely decentralized protocol where any node autonomously determines whether it belongs to the dense MANET.
		Notwithstanding this introduced random delay, the time needed to complete the dense MANET identification protocol is limited and largely acceptable; in fact, it shows a linear dependence on the diameter of the dense region, not on the number of its participants.
		Dense MANET nodes periodically (every Check Hello Period) check whether their table entries are still valid; if an entry has expired, the node removes it from the table, and verifies whether the condition to belong to the dense MANET still holds.
	lia.deis.unibo.it /Research/Redman/Pages/dmc.html (1028 words)

	PlanetMath: Linearly Ordered Spaces in ZF
		Without imposing the algebraic structure upon an ordered set we can investigate into the characteristics of the order topologies of the sets.
		Proof: As the topology is first countable, there is a countable, nested local bace
		We claim that this local base has got the properties of the assumption portion of Lemma2.2.
	planetmath.org /encyclopedia/LinearlyOrderedSpacesInZFC2.html (640 words)

	ZIVCG.COM - tutorials
		Have a dense mesh where you want lots of details, but be cheap on polygons where you don't.
		When your model already has too much of a dense topology in a certain area and you are looking for a quick way to flatten / soften it, using relax + soft selection can work very well sometimes (see example 3B).
		The topology is the base on which the details are modeled upon.
	zivcg.com /creatutorial.html (1668 words)

	Cartan's Corner : Point Set Topology
		Remarks: In the definition of a topology when the number of elements of the set in not finite, the logical intersection of open sets is restricted to any pair, and the logical union of closed sets to restricted to any pair.
		When the closure of a subset is the whole set X, the subset is said to be dense in X relative to the specified topology.
		Relative to the topology T4(open), the interior of (ab) is the singleton, (a):
	www22.pair.com /csdc/car/carfre64.htm (2727 words)

	Topology
		The family t is called a topology (for X) when it satisfies these axioms and its elements are called _open sets_ (open wrt the topology).
		The reader should now check that continuity in the sense of calculus of a function from R to R is equivalent to continuity as a map of topological spaces, with respect to the topology m.
		Call a topology t _stronger_ than the topology t' (both for the same set X) if t is contained in t'.
	www.georgetown.edu /faculty/kainen/topology.html (1132 words)

	Closure (topology) - Wikipedia, the free encyclopedia
		If one considers on R the topology in which the only open (closed) sets are the empty set and R itself, then cl((0, 1)) = R.
		In any indiscrete space X, since the only open (closed) sets are the empty set and X itself, we have that the closure of the empty set is the empty set, and for every non-empty subset A of X, cl(A) = X.
		In particular, S is dense in A iff A is a subset of Cl
	en.wikipedia.org /wiki/Closure_(topology) (1175 words)

	General Topology - NoiseFactory Science Archives (http://noisefactory.co.uk) (Site not responding. Last check: 2007-10-13)
		If we assign P the discrete topology, in which every subset is open, these will include all the inverse images of open sets in the various factor spaces.
		The standard topologies on N, Z, Q, and R are all (defined to be) their order topologies.
		The topology on C is not an order topology - in fact, there is no possible ordering of the complex field which generates the standard topology.
	noisefactory.co.uk /maths/topology.html (4788 words)

	RFC2702
		The virtual topology is constructed from virtual circuits which appear as physical links to the IGP routing protocols.
		For Traffic Engineering in large dense networks, it is desirable to equip MPLS with a level of functionality at least commensurate with current overlay models.
		Resource attributes are part of the topology state parameters, which are used to constrain the routing of traffic trunks through specific resources.
	www.unix.org.ua /rfc/rfc2702.html (7652 words)

	Dense, Separable (Site not responding. Last check: 2007-10-13)
		A set s is dense in t if it intersects every nonempty open set in t.
		A space is separable if it contains a countable dense set.
		The reals are separable, since the rationals are dense in the reals, and the rationals are countable.
	www.mathreference.com /top,dense.html (44 words)

	Exercises 5
		A subset A of a topological space X is said to be dense in X if the closure of A is X.
		Show that the subspace topology on any finite subset of R is the discrete topology.
		Show that the subspace topology on the subset Z is not discrete.
	www-groups.dcs.st-and.ac.uk /~john/MT4522/Tutorials/T5.html (187 words)

	Dense subgroups may or may not produce the same character group by W. W. Comfort, S. U. Raczkowski and F. Javier ... (Site not responding. Last check: 2007-10-13)
		Dense subgroups may or may not produce the same character group
		Denote by [^G] the character group of G, i.e., the group of characters of G with operation defined pointwise and equipped with the compact-open topology.
		The authors have granted their consent to include this document in Topology Atlas.
	at.yorku.ca /v/a/a/a/74.htm (227 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 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).
	www.math.sunysb.edu /~azinger/mat530/fall04/index.htm (2907 words)

	pub (Site not responding. Last check: 2007-10-13)
		Perfect compacta and basis problems in topology, to appear in Open Problems in Topology II (with J.T. Moore).
		Products of Frechet spaces, workshop notes written as a survey and submitted to the proceedings of the 2005 Summer Topology Conference at Denison U. Quotients of countably based spaces are not closed under sobrification, Math.
		Base-paracompactness and base-normality of GO-spaces, Q & A in General Topology, to appear.
	www.auburn.edu /~gruengf/preprints.html (132 words)

	Mathematics Topology Homework Help
		dense subsets of the unit circle in the complex plane
		to find theta, and prove that {e^{in theta} : n nonnegative integer} is a dense subset of the unit circle.
		This is one of the basic courses for students beginning study towards the Ph.D. degree in mathematics.
	www.brainmass.com /homeworkhelp/math/topology/pg4 (398 words)

	Recent Progress in the Topology of Generalized Ordered Spaces by Harold R Bennett and David J. Lutzer
		there is a dense-in-itself LOTS Y that does not have a \sigma-closed-discrete dense set, and yet each nowhere dense subspace of Y does have a \sigma-closed-discrete dense subset (in its relative topology).
		We asked whether a GO-space X must be quasi-developable provided each subspace of X has a \sigma-minimal base for its relative topology, and whether a compact LOTS Y must be metrizable if each of its subspaces has a \sigma-minimal base.
		If the topology coincides with the open interval topology of the order, then X is a linearly ordered topological space (LOTS).
	at.yorku.ca /t/a/i/c/36.htm (1375 words)

	Zariski Topology
		The intersection of any finite collection of open sets is open.
		The Zariski topology is a different kind of topology.
		This confirms once again that the Zariski topology is much coarser than the analytic topology.)
	mathcircle.berkeley.edu /BMC3/alg-geom/node3.html (244 words)

	dense - OneLook Dictionary Search
		Dense : Online Plain Text English Dictionary [home, info]
		Phrases that include dense: dense blazing star, dense leaved elodea, dense bodies, dense fog advisory, dense set, more...
		Words similar to dense: thick, compact, obtuse, densely, denseness, denser, densest, dim, dull, dumb, heavy, impenetrable, slow, crowded, dimwitted, foggy, stupid, thickheaded, thickset, more...
	www.onelook.com /?loc=pub&w=dense (320 words)

	mp_arc 95-193 (Site not responding. Last check: 2007-10-13)
		We prove that in the weak topology of measure preserving transformations, a dense $G_{\delta}$ has purely singular continuous spectrum in the orthocomplement of the constant functions.
		In the uniform topology, a dense $G_{\delta}$ of aperiodic transformations has singular continuous spectrum.
		We show that a dense $G_{\delta}$ of shift-invariant measures has purely singular continuous spectrum.
	www.ma.utexas.edu /mp_arc-bin/mpa?yn=95-193 (94 words)

	Concerning the dual group of a dense subgroup by W. W. Comfort, S. U. Raczkowski and F. Javier Trigos-Arrieta (Site not responding. Last check: 2007-10-13)
		Concerning the dual group of a dense subgroup
		Throughout this Abstract, G is a topological Abelian group and [^G] is the space of continuous homomorphisms from G into T in the compact-open topology.
		Bohr compactification, Bohr topology, character, character group, Au{\ss}enhofer-Chasco Theorem, compact-open topology, dense subgroup, determined group, duality, metrizable group, reflexive group, reflective group.
	www.univie.ac.at /EMIS/proceedings/TopoSym2001/04.htm (210 words)

	RFC 2702 - Requirements for Traffic Engineering Over MPLS
		6.0 Resource Attributes Resource attributes are part of the topology state parameters, which are used to constrain the routing of traffic trunks through specific resources.
		Informational [Page 22] RFC 2702 MPLS Traffic Engineering September 1999 This document uses the term "constraint-based routing" however, because it better captures the functionality envisioned, which generally encompasses QoS routing as a subset.
		Informational [Page 24] RFC 2702 MPLS Traffic Engineering September 1999 There are many important details associated with implementing constraint-based routing on Layer 3 devices which we do not discuss here.
	members.tripod.com /rfc_archive/rfc_2702.html (7990 words)

	RFC 2702
		This document uses the term "constraint-based routing" however, because it better captures the functionality envisioned, which generally encompasses QoS routing as a subset.
		For routers that use topology driven hop by hop IGPs, constraint- based routing can be incorporated in at least one of two ways:
		There are many important details associated with implementing constraint-based routing on Layer 3 devices which we do not discuss here.
	library.n0i.net /rfc/html/rfc2702.html (7409 words)

	[No title]
		and exponentially drops around this value (Poisson distribution)¡2$ çÿK bSparse topology
		£ð 40 (in AS hops Þð hundreds of IP hops!)¡Ê¸ çÿPçÿª,6Q-ó$¨EK&K path length increase for dense topologies is intuitively expected ÒArea organization on a sparse topology
		®ð 1 There are remote points¡jj+ çÿ çÿ ª,1$ ÄArea organization on a dense topology
	www.caida.org /~dima/pub/klein-msrw.ppt (791 words)

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