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Topic: Simply connected space


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In the News (Tue 19 Mar 19)

  
  Simply connected space - Wikipedia, the free encyclopedia
For instance, a doughnut (with hole) is not simply connected, but a ball (even a hollow one) is. A circle is not simply connected but a disk and a line are.
An equivalent formulation is this: X is simply connected if and only if it is path connected, and whenever p : [0,1] → X and q : [0,1] → X are two paths (i.e.: continuous maps) with the same start and endpoint (p(0) = q(0) and p(1) = q(1)), then p and q are homotopic relative {0,1}.
If a space X is not simply connected, one can often rectify this defect by using its universal cover, a simply connected space which maps to X in a particularly nice way.
en.wikipedia.org /wiki/Simply_connected_space   (931 words)

  
 Connected space - Wikipedia, the free encyclopedia
A space X is said to be arc-connected if any two distinct points can be joined by an arc, that is a path f which is a homeomorphism between the unit interval [0,1] and its image f([0,1]).
The connected components of a space are disjoint unions of the path-connected components.
The closure of a connected subset is connected.
en.wikipedia.org /wiki/Connected_space   (942 words)

  
 Talk:Simply connected space - Wikipedia, the free encyclopedia
It is possible to construct a bounded simply connected subset of the plane with disconnected complement.
If "simply connected" is meant to express the notion "completely filled up", isn't it better to define it as a connected subspace of which the complement's interior is also connected, or something like that?
The idea of "simply connected" is that there is a path from every point to every other point (this is path-connectedness) and that there is essentially only one such path (thus "simply" connected), in the sense that any two paths from A to B can be deformed into one another.
en.wikipedia.org /wiki/Talk:Simply_connected   (836 words)

  
 PlanetMath: example of a semilocally simply connected space which is not locally simply connected
"example of a semilocally simply connected space which is not locally simply connected" is owned by antonio.
This is version 2 of example of a semilocally simply connected space which is not locally simply connected, born on 2003-02-05, modified 2003-02-05.
The Hawaiian rings are defined in "example of a space which is not semilocally simply connected." I added a request for "hawaiian rings" and "hawaiian earrings" (both common names for that space) to be added to that entry as synonyms.
planetmath.org /encyclopedia/ExampleOfASemilocallySimplyConnectedSpaceWhichIsNotLocallySimplyConnected.html   (244 words)

  
 Encyclopedia: Semilocally simply connected
In mathematics, in particular topology, a topological space X is called semi-locally simply connected if every point x in X has a neighborhood U such that the homomorphism from the fundamental group of U to the fundamental group of X, induced by the inclusion map of U into X, is trivial.
Evidently, a space that is locally simply connected is semi-locally simply connected.
An example of a space that is not semi-locally simply connected is the Hawaiian earring: the union of the circles in the Euclidean plane with centers (1/n, 0) and radii 1/n, for n a natural number.
www.nationmaster.com /encyclopedia/Semilocally-simply-connected   (255 words)

  
 PlanetMath: example of a space that is not semilocally simply connected
In contrast, the same set endowed with the CW topology is just a bouquet of countably many circles and (as any CW complex) it is semilocaly simply connected.
"example of a space that is not semilocally simply connected" is owned by mathcam.
This is version 12 of example of a space that is not semilocally simply connected, born on 2003-02-04, modified 2004-09-23.
planetmath.org /encyclopedia/HawaiianRings.html   (194 words)

  
 Simply connected space -- Facts, Info, and Encyclopedia article   (Site not responding. Last check: 2007-10-21)
A third way to express the same: X is simply connected if and only if X is path-connected and the (Click link for more info and facts about fundamental group) fundamental group of X is trivial, i.e.
A surface (two-dimensional topological (A pipe that has several lateral outlets to or from other pipes) manifold) is simply connected if and only if it is connected and its ((biology) taxonomic group containing one or more species) genus is 0.
If a space X is not simply connected, one can often rectify this defect by using its (Click link for more info and facts about universal cover) universal cover, a simply connected space which maps to X in a particularly nice way.
www.absoluteastronomy.com /encyclopedia/s/si/simply_connected_space.htm   (1060 words)

  
 Topology - Wikipedia, the free encyclopedia
In 1914, Felix Hausdorff, generalizing the notion of metric space, coined the term "topological space" and gave the definition for what is now called Hausdorff space.
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.
The continuous image of a connected space is connected.
en.wikipedia.org /wiki/Topology   (1548 words)

  
 Simply connected: Definition and Links by Encyclopedian.com - All about Simply connected
A geometrical object is called simply connected if it consists of one piece and doesn't have any "holes" or "handles".
For instance, a doughnut is not simply connected, but a ball (even a hollow one) is. A circle is not simply connected but a disk and a line is.
The Riemann mapping theorem states that any two such non-empty open simply connected subsets of C can be conformally and bijectively mapped to the unit disk.
www.encyclopedian.com /si/Simply-connected.html   (479 words)

  
 Encyclopedia: Simply connected space
Topological spaces are structures that allow one to formalize concepts such as convergence, connectedness and continuity.
In mathematics and astronomy, Euclidean space is a generalization of the 2- and 3-dimensional spaces studied by Euclid.
The notion of simply connectedness is important in complex analysis because of the following facts: Complex analysis is the branch of mathematics investigating holomorphic functions, i.
www.nationmaster.com /encyclopedia/Simply-connected-space   (1919 words)

  
 NTU Info Centre: Topology glossary   (Site not responding. Last check: 2007-10-21)
Each connected component is closed, and the set of connected components of a space is a partition of that space.
A partition of unity of a space X is a set of continuous functions from X to [0, 1] such that any point has a neighbourhood where all but a finite number of the functions are identically zero, and the sum of all the functions on the entire space is identically 1.
A space X is semilocally simply connected if, for every point x in X, there is a neighbourhood U of x such that every loop at x in U is homotopic in X to the constant loop x.
www.nowtryus.com /article:Partition_of_unity   (4072 words)

  
 Fundamental group - Wikipedia
It is a group associated with every point of a topological space and conveying information about the 1-dimensional structure of the space.
Take some space, and some point in it, and consider all the loops at this point -- paths which start at this point, wander around as much they like and eventually return to the starting point.
An example of a space with a non-Abelian fundamental group is the complement of a trefoil knot in R
nostalgia.wikipedia.org /wiki/Fundamental_group   (600 words)

  
 Semi locally simply connected   (Site not responding. Last check: 2007-10-21)
In mathematics, in particular topology, a topological space X is calledsemi-locally simply connected if every point x in Xhas a neighborhood U such that the homomorphism from the fundamental group of U to the fundamental group of X, induced by the inclusion map of U into X, is trivial.
An example of a space that is notsemi-locally simply connected is the Hawaiian earring : the union of the circles in the Euclidean plane with centers(1/n, 0) and radii 1/n, for n a natural number.
In the theory of covering spaces, a space has a universal cover if and only if it is path-connected, locallypath-connected, and semi-locally simply connected.
www.therfcc.org /semi-locally-simply-connected-35904.html   (212 words)

  
 Topology_of_the_universe   (Site not responding. Last check: 2007-10-21)
One aspect of local geometry to emerge from General Relativity and the FLRW model is that the critical density is related to the curvature of the geometry of space.
If the global geometry is non-simply connected then at some points in the geometry, near a the junction of a "handle", paths of light may reach an observer by two routes, a "closed path" - a path through the main body and a path via the handle.
In a hyperbolic local geometry, a non-simply connected space is unlikely to be detected unless the observer is near a closed path.
www.freecaviar.com /search.php?title=Topology_of_the_universe   (1342 words)

  
 Simply Whispers   (Site not responding. Last check: 2007-10-21)
An example of a space that is not semi-locally simply connected is the Hawaiian earring : the unio 6: simply connected, but it is clearly not locally simply connected.
Simply connected space 4: but a disk and a line are.
The opposite is '''non-simply connected''' or, in a somewhat old-fashioned term 10: s the loop caught in the object, then it '''is''' simply connected.
www.elusiveeye.com /side3671-simply-whispers.html   (635 words)

  
 Simply Connected   (Site not responding. Last check: 2007-10-21)
A topological space is simply connected if it is path connected, and it has no holes.
If spheres must shrink continuously to points, then 3 space without the origin is no longer simply connected, as the unit sphere about the origin cannot shrink to a point.
Unless otherwise stated, simply connected refers to paths and circles, regardless of the dimension of the containing space.
www.mathreference.com /top,sconnect.html   (213 words)

  
 Semi-locally simply connected -- Facts, Info, and Encyclopedia article   (Site not responding. Last check: 2007-10-21)
This is true of the 'best' spaces such as (A pipe that has several lateral outlets to or from other pipes) manifolds and (Click link for more info and facts about simplicial complex) simplicial complexes.
The property of semi-locally simple connectivity is weaker than that of local simple connectivity.
It is (Click link for more info and facts about contractible) contractible and therefore semi-locally simply connected, but it is clearly not locally simply connected.
www.absoluteastronomy.com /encyclopedia/S/Se/Semi-locally_simply_connected.htm   (171 words)

  
 Universal Path Spaces   (Site not responding. Last check: 2007-10-21)
The latter hypothesis that each point in the base space has a relatively simply connected open neighborhood---necessary and sufficient for the existence of a simply connected covering space---is abandoned, thus admitting as base even those spaces that contain arbitrarily small essential loops at wild points.
When the base space is a wild metric 2-complex, the universal path space is simply connected if and only if the fundamental group is an omega-group--a group whose elements acquire non-negative real weights and form countable products of all order-type whenever their weights vanish as their appearance in the order-type deepens.
The standard features of covering space theory are thus engulfed by the richer features of universal path space theory for a variety of base spaces whose wild local topology prevents the application of traditional covering space theory.
oregonstate.edu /~bogleyw/research/upsAbs.html   (245 words)

  
 Multiply Connected Spacetimes
Periodic configuration spaces are of considerable importance in a wide range of topics, particularly those relating to unified models and condensed matter physics.
However, for the purpose of investigating gauge symmetries on such background spaces, the idealization is an reasonable one and results in a significant simplification.
Essentially, the non-simple connectivity arises by demanding that the physics of a system of identical particles be invariant under permutations of their labels.
www.iu.hio.no /~mark/physics/thesis/node4.html   (575 words)

  
 HOW THE PETRI-NETS ARE DERIVED FROM TREE-GENERATED ORDER . CONCEQUENCES TO THEIR IMPLEMENTATION .
simply connected if it is path-connected space (that is  any two points can be connected with a continuous path) and if  p
A sufficient condition for the existence of universal covering space  is given by the next theorem  Munkress, (1975).
The relation is simply the precedence of  vertices from the next vertices (in direction from a root to the leaves of the tree if it is finite).
www.softlab.ntua.gr /~kyritsis/PapersInComputerScience/Hrm98Stf.htm   (1683 words)

  
 The fundamental group
Another problem is that spaces with interesting connectivity may be declared as simply connected.
A stronger concept than simply connected for a space is that its homotopy groups of all orders are equal to the identity group.
In the plane, the notions of contractible and simply connected are equivalent; however, in higher-dimensional spaces, such as those arising in motion planning, the term contractible should be used to indicate that the space has no interior obstacles (holes).
msl.cs.uiuc.edu /planning/node155.html   (907 words)

  
 Heriot-Watt Maths Research Report HWM99-23   (Site not responding. Last check: 2007-10-21)
We introduce the homotopy surface category of a space which generalizes the 1+1-dimensional cobordism category of circles and surfaces to the situation where one introduces a background space.
We explain how for a simply connected background space, monoidal functors from this category to vector spaces can be interpreted in terms of Frobenius algebras with additional structure.
P R Turner, M Brightwell, Representations of the homotopy surface category of a simply connected space, Journal of Knot Theory and its Ramifications, 9 No. 1, 855-864, (2000).
www.ma.hw.ac.uk /maths/deptreps/HWM99-23.html   (95 words)

  
 v
Given a topological space X, a path in X from x to y is a continuous map f:[a,b] ®X of some closed real interval into X,such that f(a)=x and f(b)=y.A space X is said to be path connected if every pair of points of X can be joined by a path in X.
A topological space B is said to be locally path connected at x if for every neighborhood U of x,there is a path-connected neighborhood V of x contained in U.If B is locally path connected at each of its points,then it is said to be locally path connected.
It is not difficult to prove that real(G) is path connected, locally path connected and semilocally simply connected topological space.
www.softlab.ntua.gr /~kyritsis/PapersInComputerScience/6BARCEL9.html   (3782 words)

  
 Topics: Connectedness
Idea: A space which is "all in one piece." Of course, this depends crucially on the topology imposed on the set; every discrete topological space is "totally" disconnected.
Totally disconnected space: One in which each connected component is a single point; The only perfect, totally disconnected metric topological space is the Cantor set, a fractal.
Relationships: Local arcwise connectedness implies local connectedness; There are topological spaces which are simply connected, but not locally pathwise connected, or not locally connected (think of comb spaces).
www.phy.olemiss.edu /~luca/Topics/c/connected.html   (303 words)

  
 Simply connected   (Site not responding. Last check: 2007-10-21)
For instance, a doughnut isn't simply connected, but a ball (even a hollow one) is. A circle isn't simply connected but a disk and a line is.
The special orthogonal group SO(n,R) isn't simply connected for n≥2; the special unitary group SU(n) is simply connected.
The long line L is simply connected, but its compactification, the extended long line L* isn't (since it isn't even path connected).
www.explainthis.info /si/simply-connected.html   (624 words)

  
 [No title]   (Site not responding. Last check: 2007-10-21)
Cosheaf spaces have a topological characterization that is almost identical to Fox’s notion of a complete spread [R. Fox, Covering spaces with singularities, Algebraic Geometry and Topology: A Symposium in Honor of S. Lefschetz, R. Fox et al., editors, pp.
A covering space of a locally connected space is an unramified map, but over a locally path-connected and semi-locally simply connected space we establish the converse, i.e., we show that an unramified map is a covering space.
We provide an example of a connected unramified map over a connected, locally path-connected space that is not a covering space.
www.pphmj.com /abstracts/jpgt/vol1issue3/ab-3.htm   (209 words)

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