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	Hausdorff space - Wikipedia, the free encyclopedia
		Hausdorff spaces are named for Felix Hausdorff, one of the founders of topology.
		Pseudometric spaces typically are not Hausdorff, but they are preregular, and their use in analysis is usually only in the construction of Hausdorff gauge spaces.
		The terms "Hausdorff", "separated", and "preregular" can also be applied to such variants on topological spaces as uniform spaces, Cauchy spaces, and convergence spaces.
	en.wikipedia.org /wiki/Hausdorff_space (1262 words)

	Hausdorff dimension - Wikipedia, the free encyclopedia
		In mathematics, the Hausdorff dimension is an extended non-negative real number (that is a number in the closed infinite interval [0, ∞]) associated to any metric space.
		The Hausdorff dimension gives an accurate way to measure the dimension of an arbitrary metric space; this includes complicated sets such as fractals.
		Fractals often are spaces whose Hausdorff dimension strictly exceeds the topological dimension.
	en.wikipedia.org /wiki/Hausdorff_dimension (1904 words)

	Hausdorff (Site not responding. Last check: 2007-11-07)
		Hausdorff studied at Leipzig University under Heinrich Bruns and Adolph Mayer, graduating in 1891 with a doctorate in applications of mathematics to astronomy.
		Hausdorff married Charlotte Sara Goldschmidt in Leipzig in 1899.
		Hausdorff returned to Bonn in 1921, by this time an emminent mathematician, and he worked there until 1935 when he was forced to retire by the Nazi regime.
	www-groups.dcs.st-and.ac.uk /~history/Mathematicians/Hausdorff.html (1823 words)

	Felix Hausdorff - Wikipedia, the free encyclopedia
		Felix Hausdorff (November 8, 1868 – January 26, 1942) was a German mathematician who is considered to be one of the founders of modern topology and who contributed significantly to set theory and functional analysis.
		He defined and studied partially ordered sets, Hausdorff spaces, and the Hausdorff dimension, proved the Hausdorff maximality theorem, solved what is now called the Hausdorff moment problem, and published philosophical and literary works under the pseudonym "Paul Mongré".
		When the Nazis came to power, Hausdorff, who was Jewish, felt that as a respected university professor he would be spared from persecution.
	en.wikipedia.org /wiki/Felix_Hausdorff (213 words)

	Encyclopedia: Hausdorff distance (Site not responding. Last check: 2007-11-07)
		Hausdorff distance measures how far two compact subsets of a metric space are from each other.
		Hausdorff distance can be defined the same way for closed non-compact subsets of M, but in this case the distance may take infinite value and the topology of F(M) starts to depend on particular metric on M (not only on its topology).
		Hausdorff distance can be defined the same way for closed closed set is a set whose complement is open.
	www.nationmaster.com /encyclopedia/Hausdorff-distance (609 words)

	Hausdorff, Felix (1868-1942)
		Among several concepts named after him is the Hausdorff dimension, which gives a way of assigning a fractional dimension to a curve or shape.
		Hausdorff also published philosophical and literary works under the pseudonym "Paul Mongré." He studied at Leipzig and taught mathematics there until 1910, when he became professor of mathematics at Bonn.
		When the Nazis came to power, Hausdorff, a Jew, felt that as a respected university professor he would be safe from persecution.
	www.daviddarling.info /encyclopedia/H/Hausdorff_Felix.html (236 words)

	Hausdorff space
		In topology and related branches of mathematics, Hausdorff spaces and preregular spaces are particularly nice kinds of topological spaces.
		Hausdorff spaces are named after Felix Hausdorff, who helped originate general topology.
		In fact, Hausdorff's original definition of topological space required all topological spaces to be Hausdorff (a requirement that is not made today).
	www.gamesinathens.com /olympics/h/ha/hausdorff_space.shtml (841 words)

	Hausdorff-Based Matching (Site not responding. Last check: 2007-11-07)
		The generalized Hausdorff measure provides a means of determining the resemblance of one point set to another, by examining the fraction of points in one set that lie near points in the other set (and perhaps vice versa).
		A C implementation of Hausdorff matching (for matching with translation or with translation and scaling) is available as a tar file via ftp.
		We have used the generalized Hausdorff measure to search for a two-dimensional model (a point set represented as a bitmap) in a bitmap image (usually the intensity edges from some image) under various transformations.
	www.cs.cornell.edu /vision/hausdorff/hausmatch.html (410 words)

	Hausdorff dimension
		The Hausdorff dimension, named after Felix Hausdorff, coincides with the more familiar notion of dimension in the case of well-behaved sets.
		For example a straight line or an ordinary curve, such as a circle, has a Hausdorff dimension of 1; any countable set has a Hausdorff dimension of 0; and an n-dimensional Euclidean space has a Hausdorff dimension of n.
		But a Hausdorff dimension is not always a natural number.
	www.daviddarling.info /encyclopedia/H/Hausdorff_dimension.html (248 words)

	Hausdorff - Wikipedia, the free encyclopedia
		A Hausdorff space, when used as an adjective, as in "the real line is Hausdorff."
		Felix Hausdorff, the German mathematician that Hausdorff spaces are named after.
		Hausdorff dimension, a measure theoretic concept of dimension.
	en.wikipedia.org /wiki/Hausdorff (115 words)

	PlanetMath: Hausdorff dimension (Site not responding. Last check: 2007-11-07)
		Hausdorff dimension is easy to calculate for simple objects like the Sierpinski gasket or a Koch curve.
		Each of these may be covered with a collection of scaled-down copies of itself.
		This is version 9 of Hausdorff dimension, born on 2002-05-31, modified 2004-07-28.
	planetmath.org /encyclopedia/HausdorffDimension.html (163 words)

	Felix Hausdorff
		Hausdorff worked at Bonn until 1935 when he was forced to retire by the Nazi regime.
		Hausdorff's main work was in topology and set theory.
		In 1919 he introduced the notion of Hausdorff dimension, which was a real number lying between the topological dimension of an object and 3.
	umm.kou.edu.tr /math/felix_hausdorff.htm (304 words)

	Felix Hausdorff (Site not responding. Last check: 2007-11-07)
		Felix Hausdorff (November 8, 1868 - January 26, 1942) was aGerman mathematician who is considered to be one of the founders ofmodern topology and who contributed significantly to set theory and functional analysis.
		Whenthe Nazis came to power, Hausdorff, who was Jewish, felt that as a respected universityprofessor he would be spared from persecution.
		When in 1942 he could nolonger avoid being sent to a concentration camp, Hausdorff committed suicide togetherwith his wife and sister-in-law.
	www.therfcc.org /felix-hausdorff-87025.html (163 words)

	Hausdorff space (Site not responding. Last check: 2007-11-07)
		Pseudometric spaces typically are not Hausdorff but they preregular and their use in analysis is only in the construction of Hausdorff gauge Indeed when analysts run across a non-Hausdorff it is still probably at least preregular then they simply replace it with its quotient which is Hausdorff.
		In contrast non-preregular spaces are encountered much frequently in abstract algebra and algebraic geometry in particular as the Zariski topology on an algebraic variety or the spectrum of a ring.
		All regular spaces are preregular as are all Hausdorff There are many results for topological spaces hold for both regular and Hausdorff spaces.
	www.freeglossary.com /Hausdorff_space (966 words)

	Hausdorff dimension (Site not responding. Last check: 2007-11-07)
		The Hausdorff dimension agrees with the (topological) dimension on "well-behaved sets" but it applicable to many more sets and is always a natural number.
		The Hausdorff dimension should not be with the (similar) box-counting dimension.
		The Hausdorff dimension is a well-defined extended real number for any set E and we always have 0 ≤ d (E) ≤ ∞.
	www.freeglossary.com /Fractal_dimension (987 words)

	Hausdorff, Felix (Site not responding. Last check: 2007-11-07)
		Hausdorff was born in Breslau, Germany (now Wroclaw, Poland), and studied at Berlin, Freiburg, and Leipzig.
		Hausdorff formulated a theory of topological and metric spaces, proposing that such spaces be regarded as sets of points and sets of relations among the points, and introduced the principle of duality.
		A topological space is understood to be a set E of elements x and certain subsets Sx of E which are known as neighbourhoods of x.
	www.cartage.org.lb /en/themes/Biographies/MainBiographies/H/Hausdorff/1.html (207 words)

	Felix Hausdorff (Site not responding. Last check: 2007-11-07)
		When the Nazis came to power Hausdorff who was felt that as a respected university professor would be spared from persecution.
		He sent his daughter to Great Britain but stayed with his wife in When in 1942 he could no longer being sent to a concentration camp Hausdorff suicide together with his wife and sister-in-law.
		Hausdorff was a leader in the second wave of set theory (post Cantor) and a founder of point-set topology.
	www.freeglossary.com /Felix_Hausdorff (376 words)

	Hausdorff dimension - Term Explanation on IndexSuche.Com
		''Hausdorff outer measure'' of dimension ''s'', denoted ''H's'' is the outer measure corresponding to the function ''p''''s'' on ''C''.
		Finally, if ''H's''(''E'')=∞ for all positive ''s'', then ''E'' has Hausdorff dimension ∞ The Hausdorff dimension is a well-defined extended real_number for any set ''E'' and we always have 0 ≤ ''d''(''E'') ≤ ∞.
		For example, the Cantor_set (a zerodimensional topological space) is a union of two copies of itself, each copy shrunk by a factor 1/3; this fact can be used to prove that its Hausdorff dimension is ln(2)/ln(3) (see natural_logarithm).
	www.indexsuche.com /Hausdorff_dimension.html (454 words)

	Online Encyclopedia and Dictionary - Hausdorff space
		The Hausdorff condition is one in a series of separation axioms that can be imposed on a topological space, however it is the one that is most frequently used and discussed.
		X is a Hausdorff space if any two distinct points of X can be separated by neighborhoods which is the reason Hausdorff spaces are also called T
		Let f : X → Y be a quotient map with X a compact Hausdorff space.
	www.fact-archive.com /encyclopedia/Preregular_space (1238 words)

	Hausdorff-Based Matching
		The most successful methods that we have develoepd are based on using the generalized Hausdorff measure to compare portions of one image with another.
		This measure determines the resemblance of one point set to another, by examining the fraction of points in one set that are near points in the other (and perhaps vice versa).
		This distance measure differs from correspondence-based techniques such as point matching methods and binary correlation, in that there is no pairing of points in the two sets being compared.
	www.cs.cornell.edu /~dph/hausdorff/hausdorff1.html (423 words)

	Hausdorff Edition (Site not responding. Last check: 2007-11-07)
		Hausdorff hat bis zu seinem Tod wissenschaftlich gearbeitet, konnte aber in Deutschland nicht mehr publizieren.
		Felix Hausdorff pflegte Separata seiner Arbeiten an einen sehr weiten Kreis von Mathematikern zu versenden.
		Wenn Sie Zugang zu Nachlassbeständen eines der unten genannten Korrespondenten Hausdorffs haben, sind wir für jeden Hinweis auf mögliche Korrespondenz mit Hausdorff dankbar.
	www.aic.uni-wuppertal.de /fb7/hausdorff (583 words)

	Hausdorff Distance Image Comparison
		The function h(A,B) is called the directed Hausdorff `distance' from A to B (this function is not symmetric and thus is not a true distance).
		Thus the Hausdorff distance, H(A,B), measures the degree of mismatch between two sets, as it reflects the distance of the point of A that is farthest from any point of B and vice versa.
		This verification, or reverse Hausdorff fraction, ensures that a given portion of the points in the image (covered by the model array) are actually near points of M+ t.
	www.cs.cornell.edu /vision/hausdorff/hausdist.html (830 words)

	Hausdorff Distance
		Felix Hausdorff (1868 -1942) devised a metric function between subsets of a metric space.
		two sets are within Hausdorff distance r from each other iff any point of one set is within distance r from some point of the other set.
		The Hausdorff metric h(A,B) is defined in terms of the neighborhoods.
	www.cut-the-knot.org /do_you_know/Hausdorff.shtml (687 words)

	Ars Mathematica » Blog Archive » Hausdorff Surprises
		The Hausdorff dimension is used to define the dimension of fractals, for example, the dimension of the Sierpinski triangle is log(3)/log(2).
		For integer dimensions, the Hausdorff measure is equivalent to the Lebesgue measure.
		The Hausdorff dimension of a set is the point where the d-dimensional Hausdorff measure changes from infinity to zero, i.e.
	www.arsmathematica.net /archives/2005/05/16/hausdorff-surprises (285 words)

	Encyclopedia: Hausdorff dimension (Site not responding. Last check: 2007-11-07)
		This article is in need of attention from an expert on the subject.
		For example box-counting dimension, generalises the idea of counting the squares of graph paper in which a point of X can be found, as the size of the squares is made smaller and smaller.
		Dimension (from Latin "measured out") is, in essence, the number of degrees of freedom available for movement in a space.
	www.nationmaster.com /encyclopedia/Hausdorff-dimension (440 words)

	Hausdorff Distance Image Comparison
		Intuitively, if the Hausdorff distance is d, then every point of A must be within a distance d of some point of B and vice versa.
		It is well known that the Hausdorff distance is a metric over the set of all closed, bounded sets - it obeys the properties of identity, symmetry and triangle inequality.
		In general, we are interested in using the Hausdorff distance to identify instances of some ``model'' bitmap M in some image bitmap I.
	www.cs.cornell.edu /Info/People/dph/hausdorff/hausdorff.html (830 words)

	Hausdorff distance
		The two distances h(A, B) and h(B, A) are sometimes termed as forward and backward Hausdorff distances of A to B. Although the terminology is not stable yet among authors, eq.
		Hausdorff distance gives an interesting measure of their mutual proximity, by indicating the maximal distance between any point of one polygon to the other polygon.
		One of the main application of the Hausdorff distance is image matching, used for instance in image analysis, visual navigation of robots, computer-assisted surgery, etc. Basically, the Hausdorff metric will serve to check if a template image is present in a test image ; the lower the distance value, the best the match.
	cgm.cs.mcgill.ca /~godfried/teaching/cg-projects/98/normand/main.html (1810 words)

	OUP: Hausdorff on Ordered Sets: Plotkin (Site not responding. Last check: 2007-11-07)
		Hausdorff eschewed foundations and developed set theory as a branch of mathematics worthy of study in its own right, capable of supporting both general topology and measure theory.
		Hausdorff published seven articles in set theory during the period 1901-1909, mostly about ordered sets.
		Also available from the AMS by Felix Hausdorff are the classic work, Grundzuge der Mengenlehre, and its English translation, Set Theory, as Volume 69 and Volume 119 in the AMS Chelsea Publishing series.
	www.oup.co.uk /isbn/0-8218-3788-5 (610 words)

	Compact and Hausdorff
		An important corollary is that a continuous map of a compact space into a hausdorff space is bicontinuous.
		We already showed that a map from a compact space onto a hausdorff space is a homeomorphism, and that isn't the case here; hence the domain of f is not compact.
		Since the domain is compact and the range hausdorff, f is a homeomorphism.
	www.mathreference.com /top-cs,haus.html (693 words)

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