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# Topic: Cantor space

###### In the News (Wed 17 Jul 19)

 Cantor biography Cantor was promoted to Extraordinary Professor at Halle in 1872 and in that year he began a friendship with Dedekind who he had met while on holiday in Switzerland. Cantor continued to correspond with Dedekind, sharing his ideas and seeking Dedekind's opinions, and he wrote to Dedekind in 1877 proving that there was a 1-1 correspondence of points on the interval [0, 1] and points in p-dimensional space. Cantor also discussed the concept of dimension and stressed the fact that his correspondence between the interval [0, 1] and the unit square was not a continuous map. www-groups.dcs.st-and.ac.uk /~history/Biographies/Cantor.html   (3038 words)

 Space-filling curve - Wikipedia, the free encyclopedia Peano's purpose was to construct a continuous mapping from the unit interval to the unit square, in order to demonstrate Georg Cantor's earlier counterintuitive result that the infinite number of points in a unit interval is the same cardinality as the infinite number of points in any finite-dimensional manifold, such as the unit square. The composition f of H and g is a continuous function mapping the Cantor set onto the entire unit square. In one direction a compact Hausdorff space is a normal space and, by the Urysohn metrization theorem, second-countable then implies metrizable. en.wikipedia.org /wiki/Space-filling_curve   (779 words)

 Cantor space - Wikipedia, the free encyclopedia In mathematics, the term Cantor space is sometimes used to denote the topological abstraction of the classical Cantor set: A topological space is a Cantor space if it is homeomorphic to the Cantor set. A topological characterization of Cantor spaces is given by Brouwer's theorem: As a corollary, we see that separable, completely metrizable spaces satisfy the Continuum hypothesis: Every such space is either countable or has the cardinality of the continuum. en.wikipedia.org /wiki/Cantor_space   (524 words)

 PlanetMath: Cantor space It is a compact subspace of the Baire space, the set of all infinite sequences of integers (again with the natural product topology). Cross-references: integers, Baire space, subspace, compact, Polish space, perfect, product topology, sequences, binary, infinite This is version 11 of Cantor space, born on 2003-07-10, modified 2005-03-18. planetmath.org /encyclopedia/CantorSpace.html   (80 words)

 Georg Ferdinand Ludwig Philipp Cantor   (Site not responding. Last check: 2007-10-11) Cantor was promoted to Extraordinary Professor in 1872, and in that year he began a friendship with Dedekind. Cantor was surprised at his own discovery and wrote: "I see it, but I don't believe it!" Of course this had implications for geometry and the notion of dimension of a space. Cantor published a rather strange paper in 1894 which listed the way that all even numbers up to 1000 could be written as the sum of two primes. www.stetson.edu /~efriedma/periodictable/html/Ca.html   (608 words)

 Wikinfo | Cantor set Since the Cantor set is the complement of a union of open sets, it itself is a closed subset of the reals, and therefore a complete metric space. Thus the Cantor set is nowhere dense in the unit interval and totally disconnected. The Cantor set is also homeomorphic to the p-adic integers, and, if one point is removed from it, to the p-adic numbers. www.wikinfo.org /wiki.php?title=Cantor_set   (1128 words)

 Against the Black Crusade Cantor recoiled in horror; the daemon was barely coherent on the plane of the Real, but on the psychic plane it was like a thing of nightmare, shrieking and slashing at its target and jibbering like a madthing. Cantor squeezed his eyes closed tightly, knowing the act would cause him to miss whole streams of data across the holoscreens...but praying silently to the Emperor that the doing of it would restore to him enough concentration to focus, when he opened them. For a space marine to reach such a state was almost unheard of--but so much had been set in motion, the moment Abaddon had fallen on the surface of Bray; and as the chief eyewitness, he had taken transcribing the defeat of this Black Crusade as his personal Ordeal. members.aol.com /VoidPhantoms/Against_the_Black_Crusade.htm   (6168 words)

 Cantor & Nissel - Animations and Special Effects Lenses Cantor and Nissel Limited technical artists can work to any artwork or dimensions within the range quoted. Manufactured from resin Cantor and Nissel manufacture a range of spheres from ¼ to ¾ or total sphere, diameters from 12.00mm to 152.00mm. Cantor and Nissel produce a complete range of artificial eyes that can be used in Waxworks / mannequins / heads. www.cantor-nissel.co.uk /animations.html   (625 words)

 Dynamic Topological Logic   (Site not responding. Last check: 2007-10-11) Let a dynamic topological system be a topological space X together with a continuous function f. The function f can be thought of in temporal terms, moving the points of the topological space X from one moment to the next. space is a topological space where the arbitrary intersection of open sets is open: these are the topological analogues to Kripke frames); and the DTL of measure-preserving homeomorphisms on the unit ball of dimension individual.utoronto.ca /philipkremer/DynamicTopologicalLogic.html   (577 words)

 :: EricCantor.com :: Until very recently the information age was defined by mass communication, now the information age is defined by mass empowerment. Solutions Factory aims to serve as a place for genuine political dialogue between citizens from all walks of life about the issues that most affect their everyday lives and the lives of their children. Cantor asks for investigation of VA hospital WDBJ7 www.cantorforcongress.com   (145 words)

 Ellen Cantor Interior Design Home Office   (Site not responding. Last check: 2007-10-11) has the knowledge and expertise to turn your home office space, whether it be a home-based business, an after-hours office or a home-management office, into an efficient, pleasing work environment. Research shows that employees (including the manager) who work in a well-designed space produce better results and are more successful. It may be as simple as moving the furniture around, adding a few key pieces of storage or improving the lighting. www.palosverdes.com /ellencantor/home_office.cfm   (247 words)

 Subfaculteit Wiskunde   (Site not responding. Last check: 2007-10-11) We suggest a different intuitionistic notion of determinacy and prove that every subset of Cantor space is determinate in the proposed sense. In Cantor space both player I and player II have two alternative possibilities for each move. More generally, every subset of a space where player II has, for each one of his moves, no more than a finite number of alternative possibilities, is determinate according to our definition. www.math.ru.nl /colloquium/101104-dut.shtml   (333 words)

 CiteULike: Propositional logic of continuous transformations in Cantor space   (Site not responding. Last check: 2007-10-11) CiteULike: Propositional logic of continuous transformations in Cantor space Propositional logic of continuous transformations in Cantor space Note: You or your institution must have access rights to this article. www.citeulike.org /user/greg_restall/article/312134   (52 words)

 Titles & Abstracts Abstract While there is, up to homeomorphism, only one Cantor space, i.e., one zero-dimensional, perfect, compact, nonempty metric space, there are many measures on Cantor Space which are not topologically equivalent. In particular, we indicate the usefulness of a handle decomposition of a manifold in proving the genericity of (periodic) shadowing and strong tolerance stability. We present a result relating basin cells for a basin to prime ends of type 3, and results relating a finite number of prime ends of either type 2 or 4 to situations when an accessible periodic saddle on the basin boundary has some specified tangencies of its stable and unstable manifolds. www.math.udel.edu /GeomDyn/talks.html   (1970 words)

 AMCA: Good measures on Cantor space by Ethan Akin   (Site not responding. Last check: 2007-10-11) We call a probability measure mu on a Cantor space X "good" if for all clopen subsets U,V of X, mu(U) < mu(V) implies there exists a clopen subset W of V such that mu(U) = mu(W). This mild-seeming condition is quite strong and is nicely related to measure conditions on orderings of the space. The author(s) of this document and the organizers of the conference have granted their consent to include this abstract in Atlas Mathematical Conference Abstracts. at.yorku.ca /c/a/k/b/50.htm   (175 words)

 [No title] An {\em open neighbourhood} in this metric space is the set of all the signals which have the same prefix. We will show that if the system is strictly causal then the feedback configuration is guaranteed to have at most one behaviour, i.e., it can have either a unique behaviour or no behaviour at all. We will first show that the cantor metric is indeed a metric. ptolemy.eecs.berkeley.edu /~eal/ee290n/lec6/26820_lec6   (623 words)

 ☆ http://theory.stanford.edu/~tingz/papers/s4c.html ☆ S4C Paper A topological space X often acquires more interesting structures when it is the domain of a dynamical topological system, that is, a pair &lang X,T &rang where X is the topological space and T is a continuous transformation on X. Dynamical topological logic studies dynamical topological systems by logical means. These spaces do not satisfy topological separability axioms and are not very natural mathematically. In this paper we prove completeness of S4C for Cantor space, a space that is very popular in the theory of dynamical systems. theory.stanford.edu /~tingz/papers/s4c.html   (334 words)

 Abstract   (Site not responding. Last check: 2007-10-11) Type 2 computational complexity of functions on Cantor's space. Continuity and computability on Cantor's space C has turned out to be a very natural basis for a Type 2 theory of effectivity (TTE). Finally, we demonstrate that an optimization of the input-lookahead used by a machine may result in a non-polynomial increase of computation time. www.cs.cornell.edu /home/kreitz/Abstracts/91tcs-Type2.html   (204 words)

 Topology of data types   (Site not responding. Last check: 2007-10-11) It is known from various lines of attack (intuitionistic and constructive mathematics, recursion theory, domain theory, programming language semantics, type-two theory of effectivity) that domains of computation, or data types, are topological spaces, and that computable maps are continuous. To give a simple example, it follows from the compactness of the Cantor space that equality of integer-valued predicates on the Cantor space is decidable. For example: (1) Compactness of the Cantor space is proved by constructing a program for universally quantifying over it. www.cs.nott.ac.uk /~txa/mgs/TOP.html   (275 words)

 Melvin Henriksen, Jorge Mart\'{\i }nez, R. Grant Woods   (Site not responding. Last check: 2007-10-11) This article is devoted to a systematic study of these spaces, which are an obvious generalization of $P$-spaces. The compact quasi $P$-spaces are characterized as the compact spaces which are scattered and of Cantor-Bendixson index no greater than 2. If $X$ is normal and $X$ and $Y$ are cozero-complemented spaces and $f:X\longrightarrow Y$ is a closed continuous surjection which has the property that $f^{-1}(Z)$ is nowhere dense for each nowhere dense zeroset $Z$, then if $X$ is quasi $P$, so is $Y$. www.univie.ac.at /EMIS/journals/CMUC/cmuc0302/abs/henrimar.htm   (304 words)

 natural theology > notes > 12 november 2000   (Site not responding. Last check: 2007-10-11) Now the space of all genomes is very big but not infinite. Genes create an organism with a large number of degrees of freedom which must learn how to navigate the space it is born into. We understand this in function space, that is mapping space that is permutation space, that is meaning space. www.naturaltheology.net /Notes/Notes00/notesM11D12.html   (2461 words)

 [No title]   (Site not responding. Last check: 2007-10-11) 1 The interest of Space When thinking about the physical world, modal logicians have taken Time as their main interest, because it fits so well with an interest in the flow of information and computation. Today Space remains intriguing — both for mathematical reasons, and given the amount of work in CS and AI on visual reasoning and image processing. Recent work includes a nice completeness proof by Mints for Cantor Space (the binary tree with its natural topology), which is like the real interval [0, 1] but not quite. www.stanford.edu /~sarenac/Stanford_Space.doc   (1249 words)

 [No title]   (Site not responding. Last check: 2007-10-11) Logics of space form a scattered area of research, unlike temporal logic. This is a bit surprising, since so much fundamental work in logic was inspired by the foundations of geometry. A completeness for S4 with respect to the real line R. This work extends an earlier proof for Cantor space of G. Mints. www.stanford.edu /~sarenac/LogicsOfSpace.htm   (326 words)

 The Hierarchy of Borel Universal Sets Each section is further divided in two, with the first part dealing with general spaces, and the second the special case of compact spaces. In particular, a space is separable metrisable if and only if parametrised by a separable metrisable space. All spaces are regular Hausdorff topological spaces unless stated otherwise. pear.math.pitt.edu /mathzilla/Examples/DynHieruni.xml   (1264 words)

 EE290N Lecture 4 Notes An open neighbourhood in this metric space is the set of all the signals which have the same prefix. be the set of one sided discrete event signals and the cantor metric is defined. , i.e., the cantor metric is an ultra metric. ptolemy.eecs.berkeley.edu /~eal/ee290n/lec6/lec6.html   (519 words)

 13operator for a topological space is a collection of sets that generates the open sets of the space when closed under arbitrary union. Elements of Baire space may be thought of as "data streams" involving outcomes through time drawn from a countably infinite set. Unlike Baire and Cantor space, it may be impossible to "empirically decide" whether the function is defined in a given place. www.andrew.cmu.edu /user/kk3n/recursionclass/14top.htm   (2260 words)

 natural theology > notes > 7 january 2001   (Site not responding. Last check: 2007-10-11) In shaping, first we consider the plasticity available (the space) and then we consider the design. Solving mindspace and physical space problems with bullets is aesthetically repulsive. The structure of god is driven by beauty and necessity tempered by the transfinite space of possibility. www.naturaltheology.net /Notes/Notes01/notesM01D07.html   (648 words)

 Discrete Mathematics -- Spring 2003 The set of all languages consisting of finite words over a finite alphabet equipped with the standard language metric is homeomorphic to the Cantor space. We characterize the continuous morphisms on this space and discuss morphic properties of forbidding-enforcing families. We give necessary and sufficient conditions for a finite set of code words to generate (through concatenation) an infinite set of code words with the same properties. www.math.usf.edu /Research/spring03/seminars/discrete.html   (542 words)

 Atlas: $\omega$-Languages and the Borel Hierarchy in Cantor Space by Ludwig Staiger   (Site not responding. Last check: 2007-10-11) Atlas: $\omega$-Languages and the Borel Hierarchy in Cantor Space by Ludwig Staiger \omega-Languages and the Borel Hierarchy in Cantor Space Here the fundamentals were established in the sixties by the work of Büchi, Trakhtenbrot, McNaughton and Rabin who investigated the behaviour of automata on infinite sequences. atlas-conferences.com /cgi-bin/abstract/cait-66   (714 words)

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