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Topic: Sierpinski space

	Space-Filling Curves (Site not responding. Last check: 2007-10-13)
		A space-filling curve is typically defined as the limit of a sequence of curves.
		The Sierpinski group of curves use isosceles right-angled triangles as cells.
		The Sierpinski_C curve and the Sierpinski curve are, in a sense, complements of each other: the shape of the areas excluded by one curve is the shape included by the other.
	www.cs.utexas.edu /users/vbb/misc/sfc/Oindex.html (511 words)

	Sierpinski Tautology Map (Site not responding. Last check: 2007-10-13)
		In the case of 'and' the persistent image of the Sierpinski gasket appears in the upper left as a value pattern for contradiction; in the case of 'or' it appears in the lower right as a value pattern for tautology.
		The initially startling appearance of the Sierpinski gasket as a map of tautologies under the Sheffer stroke can thus be understood in terms of (i) what a binary representation of values means and (ii) a corresponding rendering of the familiar 'doubling the distance from the nearest vertex' route to the Sierpinski in terms of binaries.
		It is well known that the points constitutive of the Sierpinski gasket within a continuous unit square are infinitely many, but nonetheless 'very few' in the sense that a random selection of points has a probability approaching zero of hitting such a point.
	www.sunysb.edu /philosophy/fractal/Sierpins.html (2671 words)

	Chu Spaces (Site not responding. Last check: 2007-10-13)
		A Chu space (X,r,A) consists of just three things: a set X of points, individuals, or subjects, a set A of states or predicates, and a lookup table or matrix r: X×A → K which specifies for every subject x and predicate a the value a(x) of that predicate for that subject.
		All finite topological spaces are trivially Alexandrov spaces.
		Chu spaces are important to the foundations of mathematics because they demonstrate that when one has stepped back to view the mathematical landscape from a sufficient distance, a global symmetry appears, duality, that is not apparent when standing inside any particular category.
	chu.stanford.edu (5453 words)

	PlanetMath: Hausdorff space
		space is also known as a Hausdorff space.
		Important examples of Hausdorff spaces are metric spaces, manifolds, and topological vector spaces.
		This is version 17 of Hausdorff space, born on 2002-02-08, modified 2006-04-15.
	planetmath.org /encyclopedia/T2Space.html (292 words)

	Locally Compact Spaces in Abstract Stone Duality (Site not responding. Last check: 2007-10-13)
		The idea of a sober space or datatype is that it has exactly the points that are required by its open subspaces or computable predicates.
		This recovers the lattice of open subspace of a space as a comma square involving those of an open subspace and its closed complement, together with a map that encodes the way in which these fit together (§ D 1).
		Kuratowski-finite subspace of an overt discrete space is overt and compact, so it is represented by its pair ([], < >) of modal operators, on which it is easy to express the naïve set-theoretic operations (§§ E 3 and 6).
	www.cs.man.ac.uk /~pt/ASD/loccpct.html (3364 words)

	Hyperspace Travel Guide
		Of course, to view the three-dimensional projection of a four-dimensional object on a computer monitor (or a piece of paper, or whatever means of viewing pictures is available to you), you must project it down another dimension, onto a two-dimensional plane.
		The Sierpinski gasket is a fractal object similar to the Menger sponge, but with the tetrahedron as the base form.
		A Sierpinski gasket is by no means a two-dimensional object, it is a fractal object in three-space whose fractal dimension happens to be exactly 2.
	www.hyperspace-travel.de /travel_guide.html (1701 words)

	Sierpinski space
		In topology, Sierpiński; space S is the simplest example of a topological space that does not satisfy the T
		Then T = {{},{1},{0,1}} is a topology on S, and the resulting topological space is called Sierpinski space.
		If X is a topological space with topology T, then the weak topology on X generated by C(X,S) coincides with T.
	publicliterature.org /en/wikipedia/s/si/sierpinski_space.html (169 words)

	Normal space information information - Search.com (Site not responding. Last check: 2007-10-13)
		X is a completely normal space or a hereditarily normal space if every subspace of X is normal.
		A locally normal space is a topological space where every point has an open neighbourhood that is normal.
		A classical example of a completely regular locally normal space that is not normal is the Niemitzky plane.
	c10-ss-1-lb.cnet.com /reference/Normal_space (1024 words)

	A Universal Separable Metric Space Based on the Triangular Sierpinski Curve (Site not responding. Last check: 2007-10-13)
		Sierpinski has constructed two significant curves at the beginning of this century.
		A year later he constructed his rectangular curve and proved that is universal for planar one-dimensional spaces.
		The subset of this product which consists of points having at least one irrational coordinate is a universal space for the class of all separable metrizable spaces of dimension less than or equal n.
	www.pmf.ukim.edu.mk /mathematics/ivansic.htm (186 words)

	Digital Doodles - the Sierpinski Gasket
		P happens to be near the middle of the largest white space (forbidden zone), but it could be anywhere on the figure (even outside of the triangle).
		Once the enclosing white space decreases to the smallest size that can be resolved on the computer screen, however, the point will be indistinguishable from points which are not in white spaces (such as the corners of the given white space).
		The point might even pop into a white space, but it would be a white space much, much smaller than the resolution of the screen, with the result again being the continuous avoidance of the larger (resolvable) white spaces.
	www.nmsr.org /digdudle.htm (1908 words)

	Sierpinski Gasket and Tower of Hanoi
		To represent a puzzle graphically, associate various (legitimate) configurations of the puzzle with dots in some space (practically, it's a piece of paper or a computer screen we'll be working with).
		In the Tower of Hanoi puzzle, disks stacked on one peg are to be moved to another with, perhaps an intermediate stop at a third, auxiliary peg.
		Convergence is established in the framework of metric spaces with the Hausdorff distance.
	www.cut-the-knot.org /triangle/Hanoi.shtml (1058 words)

	Normal space - Wikipedia, the free encyclopedia
		Most spaces encountered in mathematical analysis are normal Hausdorff spaces, or at least normal regular spaces:
		All metric spaces (and hence all metrizable spaces) are perfectly normal Hausdorff;
		All pseudometric spaces (and hence all pseudometrisable spaces) are perfectly normal regular, although not in general Hausdorff;
	en.wikipedia.org /wiki/Normal_space (1053 words)

	Compact space
		An equivalent definition of compact spaces, sometimes useful, is based on the finite intersection property.
		Every topological space X is a dense subspace of a compact space which has at most one point more than X.
		At one time, when primarily metric spaces were studied, compact was taken to mean the weaker sequentially compact, that every sequence has a convergent subsequence.
	www.fact-index.com /c/co/compact_space.html (1294 words)

	Zoltan Developer's Guide: Refinement Tree
		The method generates a space filling curve which is cut into K appropriately-sized pieces to define contiguous partitions, where the size of a piece is the sum of the weights of the elements in that piece.
		Unless the user provides the order through which to traverse the elements of the initial grid, a path is determined through the initial elements along with the "in" vertex and "out" vertex of each element, i.e., the vertices through which the path passes to move from one element to the next.
		This path can be determined by a Hilbert space filling curve, Sierpinski space filling curve (triangles only), or an algorithm that attempts to make connected partitions (connectivity is guaranteed for triangles and tetrahedra).
	www.cs.sandia.gov /~kddevin/Zoltan_html/dev_html/dev_reftree.html (885 words)

	Compact space (Site not responding. Last check: 2007-10-13)
		In mathematics, a compact space is a space that resembles a closed and bounded subset of Euclidean space R
		That is, any collection of open sets whose union is the whole space has a finite subcollection whose union is still the whole space.
		Suppose X is a Hausdorff space, and we have a point x in X and a finite subset A of X not containing x.
	compact-space.iqnaut.net (1408 words)

	Recent Papers of Robert L. Devaney
		We also show that these Sierpinski holes are always homeomorphic to disks and that there is a unique parameter in each hole for which the critical orbit lands on infinity at iteration k+2.
		These Sierpinski curves have complementary domains that consist not only of the basin of attraction of infinity, but also of the basins of other finite attracting cycles.
		These Julia sets are generalized Sierpinski gaskets and are produced when the critical orbits for these maps land on repelling periodic points in the basin of infinity.
	math.bu.edu /people/bob/papers.html (2846 words)

	KS5
		Space filling curve prefractals and their affine images.
		As we mentioned before, an interesting application of space filling curves in computer graphics is to halftoning and dithering of the images (see [4], [15]).
		The different shading of the cube faces in the picture on the right is obtained by thicker or lighter filling of the space, corresponding to preattractors at different positions in the Hutchinson orbits corresponding to the single faces.
	www.mi.sanu.ac.yu /vismath/kocic1/KS5.html (330 words)

	Math Forum: Lee: Lesson 5 - Leaky Pyramids and Rubik's Cubes, part 2 (Site not responding. Last check: 2007-10-13)
		It is said a line drawn on paper is one-dimensional, a flat object in plane is two-dimensional, and a solid object in space is three-dimensional.
		We now extend the Sierpinski function to a three-dimensional object of 3-sided pyramid, also known as the tetrahedron, which means an object of four faces.
		That is, the Sierpinski pyramid function breaks up the input pyramid into half-size pyramids and throws away all but the four corner ones, as shown in figure 3.
	mathforum.org /~lisab/fifth_lesson/part_2.htm (345 words)

	[No title] (Site not responding. Last check: 2007-10-13)
		The underlying set functor on Top\op takes a space to the set of all pairs (U,A) where U is open and A is an arbitrary subset of U. The frame operations are the usual, while (U,A)' = (U,U - A).
		The first condition is sufficient to ensure that, for each topological space X, the canonical counit map X -----> B^Top(X,B) is an injection, and the second ensures that it is the embedding of a subspace.
		This particular algebraic theory to which you refer is perhaps exactly that arising from the space B having three points, one of which is closed, with open complement (containing the other two points) as the only non-trivial open.
	www.mta.ca /~cat-dist/catlist/1999/topop (385 words)

	Topological space - Wikipedia, the free encyclopedia
		Sierpiński space is the simplest non-trivial, non-discrete topological space.
		Every subset of a topological space can be given the subspace topology in which the open sets are the intersections of the open sets of the larger space with the subset.
		A quotient space is defined as follows: if X is a topological space and Y is a set, and if f : X → Y is a surjective function, then the quotient topology on Y is the collection of subsets of Y that have open inverse images under f.
	en.wikipedia.org /wiki/Topological_space (1829 words)

	Yale Topology Seminar (Site not responding. Last check: 2007-10-13)
		Convex submanifolds of hyperbolic space, Sierpinski carpets, and minimal volume.
		Discrete groups which act cocompactly on (infinite volume) convex proper submanifolds of hyperbolic 3-space usually admit nontrivial deformations.
		In the case of acylindrical groups, there is a connection between the convex submanifold of minimal co-volume, the geometry of the boundary of the convex submanifold, and the geometry of the limit set in the Riemann sphere (which is topologically a Sierpinski carpet).
	www.math.yale.edu /~yhm3/topsem/spr04/storm_abs.html (84 words)

	Many ways to form the Sierpinski gasket
		The most common construction of the Sierpinski gasket is by splitting a triangle into four by the midlines and removing the middle triangle and then applying the same procedure recursively to the three remaining triangles.
		Being a uniform limit of curves (for it's defined by an L-System), the Sierpinski gasket is known to be the image of a continuous map from [0,1].
		Similar to, but different from, the discussion below, the Sierpinski gasket could be obtained as a result of a step by step automation taken modulo 2.
	www.cut-the-knot.org /ctk/Sierpinski.shtml (1149 words)

	Wacław Sierpiński - Wikipedia, the free encyclopedia
		Three well-known fractals are named after him (the Sierpinski triangle, the Sierpinski carpet and the Sierpinski curve), as are Sierpinski numbers and the associated Sierpiński problem.
		Sierpinski began to study set theory and, in 1909, he gave the first ever lecture course devoted entirely to the subject.
		To avoid the persecution that was all too common for Polish foreigners, Sierpinski spent the rest of the war years in Moscow working with Luzin.
	en.wikipedia.org /wiki/Waclaw_Sierpinski (875 words)

	SPACE.com -- Beyond the Two Cultures: Artists and Scientists to Gather in Paris, Composing Messages to ET.
		Sierpinski Gasket: This fractal is constructed by starting with an triangle having all sides of equal length.
		In the middle of this triangle, a smaller triangle is cut out, leaving three other triangles at the corners of the original triangle.
		Because of space limitations, the Workshop will be closed to the public,and there will be very limited room for the press, by invitation only.
	www.space.com /searchforlife/seti_vakoch_020311.html (1187 words)

	The Original "Studio 102" Hat
		Unless otherwise stated, each stitch is worked in the next available space (in the case of hdc) or around the next available post (in the case of all the fp stitches).
		Also, the number of repetitions of the pattern, and the spacings used, can be modified for the best fit on your particular hat-sizing.
		Since the triangles themselves use 17 stitches per pair, the number of stitches used is r(17+2s), where s is the number of spacing stitches, and r is the number of motif repetitions.
	math.ucsd.edu /~dwildstr/crochet/studio90.html (1735 words)

	Sober spaces and continuations (Site not responding. Last check: 2007-10-13)
		A topological space is sober if it has exactly the points that are dictated by its open sets.
		A new definition of sobriety is formulated in terms of lambda calculus and elementary category theory, with no reference to lattice structure, but, for topological spaces, this coincides with the standard lattice-theoretic definition.
		Nor is this ``denotational semantics of continuations using sober spaces'', though that could easily be derived.
	www.tac.mta.ca /tac/volumes/10/12/10-12abs.html (252 words)

	Glossary
		a topological space is compact if every collection of open sets that covers the space has a finite subset that also covers the space.
		Two points lie in the same path component of a space X iff there is a path in X from one point to the other.
		a separable space is one that has a countable dense subset, that is a countable subset whose closure is the whole space.
	mcraefamily.com /MathHelp/BasicSetTopologyGlossary.htm (2717 words)

	brownian motion on fractals (Site not responding. Last check: 2007-10-13)
		The initial motivation for this area came from physicists working in the theory of condensed matter, who consider fractals to be good models of some objects (such as polymers) which arise there, and who were interested in `transport properties' of these objects, such as conduction of heat or transmission of vibrations.
		It was therefore natural to wish to study equations such as the heat equation on fractal spaces, and to do this one has to define and study a `Laplacian operators' on fractal spaces.
		In [] a very detailed study was made of Brownian motion on the simplest non-trivial fractal space, the Sierpinski* gasket.
	www.nonserviam.com /apeiron/fractals/diffusion.html (302 words)

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