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# Topic: Homeomorphism

###### In the News (Fri 19 Jul 19)

 Homeomorphism - Wikipedia, the free encyclopedia Thus a square and a circle are homeomorphic. The traditional joke is that the topologist can't tell the coffee cup she is drinking out of from the donut she is eating, since a sufficiently pliable donut could be reshaped to the form of a coffee cup by creating a dimple and progressively enlarging it, while shrinking the hole into a handle. Intuitively a homeomorphism maps points in the first object that are "close together" to points in the second object that are close together, and points in the first object that are not close together to points in the second object that are not close together. en.wikipedia.org /wiki/Homeomorphism   (789 words)

 Homeomorphism - Wikipedia, the free encyclopedia In the mathematical field of topology a homeomorphism or topological isomorphism (from the Greek words homeos = identical and morphe = shape) is a special isomorphism between topological spaces which respects topological properties. Roughly speaking a topological space is a geometric object and the homeomorphism is a continuous stretching and bending of the object into a new shape. This characterization of a homeomorphism often leads to confusion with the concept of homotopy, which is actually defined as a continuous deformation, but from one function to another, rather than one space to another. en.wikipedia.org /wiki/Homeomorphic   (789 words)

 Station Information - Homeomorphism If two objects are homeomorphic, one can find a continuous function which maps points from the first object to corresponding points of the second object, and vice versa. Such a function is called a homeomorphism; intuitively, it maps points in the first object that are "close together" to points in the second object that are close together, and points in the first object that are not close together to points in the second object that are not close together. As such, the composition of two homeomorphisms is again a homeomorphism, and the set of all homeomorphisms X → X forms a group. www.stationinformation.com /encyclopedia/h/ho/homeomorphism.html   (633 words)

 Encyclopedia: Homeomorphism   (Site not responding. Last check: 2007-10-22) In topology, a local homeomorphism is a map from one topological space to another that respects locally the topological structure of the two spaces. Homeomorphisms are the ((biology) similarity or identity of form or shape or structure) isomorphisms in the (Click link for more info and facts about category of topological spaces) category of topological spaces. As such, the composition of two homeomorphisms is again a homeomorphism, and the set of all self-homeomorphisms X → X forms a ((chemistry) two or more atoms bound together as a single unit and forming part of a molecule) group, called the homeomorphism group of X, often denoted Homeo(X). www.nationmaster.com /encyclopedia/Homeomorphism   (621 words)

 Encyclopedia: Local homeomorphism   (Site not responding. Last check: 2007-10-22) It is shown in complex analysis that a complex analytic function f gives a local homeomorphism precisely when the derivative f′(z) is non-zero for all z in the domain of f. All covering maps are local homeomorphisms; in particular, the universal cover p : C → X of a space X is a local homeomorphism. A bijective local homeomorphism is therefore a homeomorphism. www.nationmaster.com /encyclopedia/Local-homeomorphism   (647 words)

 HOMEOMORPHISM   (Site not responding. Last check: 2007-10-22) In topology, two geometrical objects are called homeomorphic if, roughly speaking, the first can be deformed into the second by stretching and bending; cutting is also allowed, but only if the two parts are later glued back together along exactly the same cut. A hollow sphere containing a smaller solid ball is homeomorphic to a hollow sphere with a solid ball outside of it. It is an isotopy between the identity map on X and the homeomorphism from X to Y. www.yotor.org /wiki/en/ho/Homeomorphism.htm   (625 words)

 PlanetMath: homeomorphism We also say that two spaces are homeomorphic if such a map exists. If two topological spaces are homeomorphic, they are topologically equivalent -- using the techinques of topology, there is no way of distinguishing one space from the other. This is version 8 of homeomorphism, born on 2001-11-16, modified 2004-11-16. planetmath.org /encyclopedia/Homeomorphism.html   (84 words)

 Search Results for homeomorphism - Encyclopædia Britannica A homeomorphism can be defined as a one-to-one correspondence between the points of two... Two topological spaces are called topologically equivalent, or homeomorphic, if there is a one-to-one correspondence between them that is continuous both ways. It was shown in 1960 by Morton Brown of the United States that an (n 1)-sphere in is tame under certain conditions. www.britannica.com /search?query=homeomorphism&submit=Find&source=MWTEXT   (323 words)

 Responses to "Creative homeomorphism" April 19th, 2005 You can probably see an example of architectural homeomorphism from where you sit--window panes are generally divided from one another by equidistant mullions. Homeomorphism is one way to beguile the eye. Homeomorphism is a device I use because it gives a naïve, almost other-worldly presence to objects. www.painterskeys.com /clickbacks/homeomorphism.asp   (3777 words)

 Questions on homeomorphism groups of chainable and homogeneous continua by Beverly L. Brechner In [2], we began a study of the topological structure, in particular dimension properties, of homeomorphism groups of various continua. The homeomorphism groups of those two continua are totally disconnected, and it is still an open question to determine what the dimension is. Wayne Lewis, Stable homeomorphisms of the pseudo-arc, Canad. at.yorku.ca /i/a/a/z/01.htm   (503 words)

 PlanetMath: homeomorphism Adapted with permission of the author from Modern Graph Theory by Béla Bollobás, published by Springer-Verlag New York, Inc., 1998. This is version 3 of homeomorphism, born on 2002-03-07, modified 2002-07-17. Object id is 2773, canonical name is Homeomorphic. planetmath.org /encyclopedia/Homeomorphic.html   (39 words)

 Homeomorphism   (Site not responding. Last check: 2007-10-22) A criterion for homeomorphism between closed Haken manifolds... Homeomorphism between [:\$cal E^i_rm T, cal E^j_rm T\$:] and \$cal E^i+j_rm T... Homeomorphism and the Equivalence of Logical Systems, Stephen Pollard... www.scienceoxygen.com /math/338.html   (141 words)

 Homeomorphism The reader who has had abstract algebra will note that homeomorphism is the analogy in the setting of topological spaces and continuous functions to the notion of isomorphism in the setting of groups (or rings) and homomorphisms, and to that of linear isomorphism in the context of vector spaces and linear maps. In general, it may be quite difficult to demonstrate that two spaces are homeomorphic (unless a homeomorphism is obvious or can easily be discovered). is homeomorphic to (0,1) with its induced metric topology, it is necessary to demonstrate, for instance, that at.yorku.ca /course/atlas1/node17.html   (255 words)

 [No title] Clearly, any homeomorphism will have to fix the point of attachment, but more is true: the points of the interval have the property that X-{x} is not connected, but the points of the circle don't; yet all these points have the same kind of neighborhoods. It's easy to find a homeomorphism of the unit ball carrying the origin to any other spot, and in fact to do so without moving points on the boundary of the ball. I think it may be possible to find other examples based on, say, the Zariski topology, since typically an algebraic variety has only a finite number of self-isomorphisms (and thus in particular the orbit of a point under this family of maps cannot be the whole space). www.math.niu.edu /~rusin/known-math/95/homogen   (1110 words)

 Publication - Abstract Page   (Site not responding. Last check: 2007-10-22) We prove the existence of a homeomorphism between a set of mutually non-intersecting simple composite curves and the corresponding heap of linear approximating segments which do not have inappropriate intersections. The objective of this project is to generate numerically robust and topologically reliable unstructured triangular mesh of surfaces which are the faces of complex solid models. The overall objective of this project is to lay the framework for a new generation of robust solid modelers for representing and interrogating solid models with curved boundaries in the context of imprecise (discrete) computer arithmetic. czms.mit.edu /cho/publication/abstract.html   (1713 words)

 Local homeomorphism   (Site not responding. Last check: 2007-10-22) More precisely, the function f : X → Y is a local homeomorphism if for every point x of X there exists an open neighbourhood U of x and an open set V in Y such that f yields a homeomorphism between U and V. Every homeomorphism is of course also a local homeomorphism, but this is boring. Then this is a local homeomorphism for all non-zero n, but a homeomorphism only in the cases where it is bijective, i.e. www.portaljuice.com /local_homeomorphism.html   (460 words)

 Homeomorphism (graph theory) - Wikipedia, the free encyclopedia In graph theory, a homeomorphism exists between two graphs G and G′ if there exists a graph H such that both G and G′ are subdivisions of that graph. If the edges of a graph are thought of as lines drawn from one vertex to another (as they are usually depicted in illustrations), then two graphs are homeomorphic to each other in the graph-theoretic sense precisely if they are homeomorphic in the sense in which the term is used in topology. these two graphs are homeomorphic, since given the graph en.wikipedia.org /wiki/Homeomorphism_(graph_theory)   (211 words)

 On Local Homeomorphism of Mapping ...   (Site not responding. Last check: 2007-10-22) On local homeomorphism of mapping with bounded distortion with the coefficient of distortion close to identity In 1971 V. Goldshtein has proved a theorem about local homeomorphism of mappings with bounded distortion with the coefficient of distortion close to identity. The problem is that the structure of 1-quasiregular mappings is unknown on an arbitrary Carnot group. www.mat.utfsm.cl /publicaciones/preprints2002/11-Local   (139 words)

 What exactly IS a homeomorphism?   (Site not responding. Last check: 2007-10-22) I've already used two homeomorphisms; the topologies of A and B are the usual ones. A is homeomorphic to X/~1; B is homeomorphic to X/~2. To say that there's a homeomorphism between two spaces is to say that they have exactly the same topological structure; only the 'names' of the points are different, and the map tells you how the points in one space correspond to those in the other. www.thehelparchive.com /new-2359222-279.html   (1756 words)

 Homeomorphism: Definition and Links by Encyclopedian.com - All about Homeomorphism Homeomorphism: Definition and Links by Encyclopedian.com - All about Homeomorphism If two objects are homeomorphic, one can find a function which maps points from the first object to corresponding points of the second object, and vice versa. The intuitive criterion of stretching, bending, cutting and glueing back together takes a certain amount of practice to apply correctly--it is not obvious from the above description that deforming a line segment to a point is impermissable, for instance. www.encyclopedian.com /ho/Homeomorphism.html   (644 words)

 Atlas: Continua whose homeomorphism groups are generated by small neighborhoods of the identity by Wayne Lewis   (Site not responding. Last check: 2007-10-22) A compactum with the property that its homeomorphism group is generated by arbitrarily small neighborhoods of the identity is said to have a micro-generated homeomorphism group. A homeomorphism of a compactum which is a composition of homeomorphisms in an arbitrarily small neighborhood of the identity is said to be micro-generated. We discuss results on homeomorphisms of continua which are micro-generated and on continua which have micro-generated homeomorphism groups. atlas-conferences.com /cgi-bin/abstract/caem-07   (185 words)

 Finding a homeomorphism from circle to square with rotation A circle and a square are homeomorphic (isomorphic?). Is there some proceedure for finding the equivalent transformations or operations on the image of a set by a homeomorphism. That depends on the particular homeomorphism you're using. www.groupsrv.com /science/post-680193.html   (990 words)

 homeomorphism vs isomorphism homeomorphism is an invertible mapping continuous in both directions. properties and are not to be confused with Homeomorphisms. I guess it was Poincaré who coined the term "homeomorphism" in topology. sci4um.com /post-33324.html   (1017 words)

 The Ends of a Homotopy That's hardly a homeomorphism; there isn't even a well defined function from s to t. is hausdorff, and a homotopy between the two would imply a homeomorphism, yet the sphere and torus are not homeomorphic, as we showed in an earlier section. Put two spheres in disjoint spaces, and they are obviously homeomorphic, even though there is no homotopy that carries one sphere onto the other. www.mathreference.com /at,ends.html   (662 words)

 Homeomorphism of 2-complexes is equivalent to graph isomorphism   (Site not responding. Last check: 2007-10-22) Homeomorphism of 2-complexes is equivalent to graph isomorphism We show that graph isomorphism can be reduced efficiently to 2-complex homeomorphism, and that Whittlesey's criterion can be reduced efficiently to graph isomorphism. Therefore graph isomorphism and 2-complex homeomorphism are polynomial-time equivalent. www.maths.tcd.ie /~odunlain/abstracts/2complex.html   (61 words)

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