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Topic: Trivial topology

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  Trivial topology - Wikipedia, the free encyclopedia
In topology, a topological space with the trivial topology is one where the only open sets are the empty set and the entire space.
The trivial topology is the topology with the least possible number of open sets, since the definition of a topology requires these two sets to be open.
The trivial topology belongs to a pseudometric space in which the distance between any two points is zero, and to a uniform space in which the whole cartesian product X × X is the only entourage.
en.wikipedia.org /wiki/Trivial_topology   (538 words)

 Encyclopedia: List of general topology topics   (Site not responding. Last check: 2007-10-21)
In topology and related areas of mathematics a topological property or topological invariant is a property of a topological space which is invariant under homeomorphisms.
In topology and related areas of mathematics, the disjoint union (also called the direct sum, free union, or coproduct) of a family of topological spaces is a space formed by equipping the disjoint union of the underlying sets with a natural topology called the disjoint union topology.
In topology and related fields of mathematics, a discrete space is a particularly simple example of a topological space or similar structure, one in which the points are isolated from each other in a certain sense.
www.nationmaster.com /encyclopedia/List-of-general-topology-topics   (4593 words)

 Topology glossary - Wikipedia, the free encyclopedia
The topology T is the smallest topology on X containing B and is said to be generated by B.
The trivial topology (or indiscrete topology) on a set X consists of precisely the empty set and the entire space X.
The weak topology on a set, with respect to a collection of functions from that set into topological spaces, is the coarsest topology on the set which makes all the functions continuous.
en.wikipedia.org /wiki/Topology_glossary   (4512 words)

 Trivial topology -- Facts, Info, and Encyclopedia article   (Site not responding. Last check: 2007-10-21)
The trivial topology is the topology with the least possible number of (Click link for more info and facts about open set) open sets, since the definition of a topology requires these two sets to be open.
Two topological spaces carrying the trivial topology are (Click link for more info and facts about homeomorphic) homeomorphic (Click link for more info and facts about iff) iff they have the same (Click link for more info and facts about cardinality) cardinality.
In some sense the opposite of the trivial topology is the (Click link for more info and facts about discrete topology) discrete topology, in which every subset is open.
www.absoluteastronomy.com /encyclopedia/t/tr/trivial_topology.htm   (821 words)

 Topology glossary - Wikipedia, the free encyclopedia
This is a glossary of some terms used in the branch of mathematics known as topology.
If T is a topology on a space X, and if A is a subset of X, then the subspace topology on A induced by T consists of all intersections of open sets in T with A.
Algebraic topology is the study of topologically invariant abstract algebra constructions on topological spaces.
en.wikipedia.org /wiki/Topology_Glossary   (4512 words)

 Station Information - Trivial topology
Because of these, it is not an order topology, and it is not metrizable.
Arbitrary productss of trivial topology spaces, with either the product topology or box topology, have the trivial topology.
If F : Top → Set is the functor that assigns to each topological space its underlying set (the so-called forgetful functor), and G : Set → Top is the functor that puts the trivial topology on a given set, then G is right adjoint to F.
www.stationinformation.com /encyclopedia/t/tr/trivial_topology.html   (433 words)

 PlanetMath: discrete space
The discrete topology is the finest topology one can give to a set.
Note that any bijection between discrete spaces is trivially a homeomorphism.
The product of an infinite number of discrete spaces is discrete under the box topology, but if an infinite number of the spaces have more than one element, it is not discrete under the product topology.
planetmath.org /encyclopedia/Discrete.html   (231 words)

 Topology glossary   (Site not responding. Last check: 2007-10-21)
Although there is no clear distinction between different areas of topology, this glossary focuses primarily on general topology and on definitions that are fundamental to a broad range of areas.
A collection of open sets is a subbase (or subbasis) for a topology if every open set in the topology is a union of finite intersections of sets in the subbase.
If B is any collection of subsets of a set X, the topology on X generated by B is the smallest topology containing B; this topology consists of all unions of finite intersections of elements of B.
www.sciencedaily.com /encyclopedia/topology_glossary   (3712 words)

 Topology Reading Course
Students are expected to become familiar with the basic concepts and methodology of point-set topology: separation properties, connectedness, and compactness, as well as subspaces, quotient spaces, and the properties of continuous mappings.
That is, excluding the trivial smallest topology (consisting of only the empty set and the entire set, called the ``co-discrete'' topology) and the trivial largest topology (which consists of _all_ subsets and is called the ``discrete'' topology), there are 27 nontrivial topologies.
I mentioned in class that this is trivial if S is finite; in that case, the cofinite topology is the discrete topology.
www.georgetown.edu /faculty/kainen/topol-02.html   (1223 words)

 Closure (topology)   (Site not responding. Last check: 2007-10-21)
In topology and mathematical analysis, the closure of a subset
In the trivial topology, the closure of any non-empty set is the whole space.
In the discrete topology, the closure of any set is that set itself.
www.sciencedaily.com /encyclopedia/closure__topology_   (416 words)

 Comparison of topologies - Enpsychlopedia   (Site not responding. Last check: 2007-10-21)
All possible polar topologies on a dual pair are finer than the weak topology and coarser than the strong topology.
The join, however, is not generally the union of those topologies (the union of two topologies need not be a topology) but rather the topology generated by the union.
In the case of topologies, the greatest element is the discrete topology and the least element is the trivial topology.
psychcentral.com /wiki/Coarser_topology   (549 words)

 Topological space Article, Topologicalspace Information   (Site not responding. Last check: 2007-10-21)
The Zariski topology is a purely algebraically defined topologyon the spectrum of a ring or an algebraic variety.
Many sets of operators in functional analysis are endowed with topologies that are defined by specifying when a particularsequence of functions converges to the zero function.
Any set can be given the trivial topology in which only theempty set and the whole space are open.
www.anoca.org /spaces/set/topological_space.html   (2012 words)

 IPN (Iranian Physics News)   (Site not responding. Last check: 2007-10-21)
The non trivial topologies, can not be foliated by a family of surfaces of constant time.
The trivial topology, periodically identified anti de Sitter space, fills in the torus, but so also do non-trivial topologies, the best known of which is Schwarzschild anti de Sitter.
The simplest topology that fits inside that boundary, is the trivial topology, S1 cross D3, the three disk.
ipn.persianblog.com /1383_5_ipn_archive.html   (2933 words)

 Citebase - Hyperbolic Universes with a Horned Topology and the CMB Anisotropy   (Site not responding. Last check: 2007-10-21)
Citebase - Hyperbolic Universes with a Horned Topology and the CMB Anisotropy
Hyperbolic Universes with a Horned Topology and the CMB Anisotropy
On the one hand, we show that the horned topology does not lead to a flat spot in the CMB sky maps in the direction of the horn as stated in the literature.
citebase.eprints.org /cgi-bin/citations?id=oai:arXiv.org:astro-ph/0403597   (1573 words)

 Cleveland Cosmology-Topology Workshop
The majority of Thurston's talk was devoted to 3-manifolds, where both topology and geometry find their optimal balance between flexibility and rigidity.
Concepts such as the prime decomposition of 3-manifolds were made accessible to the physics audience by relating the underlying ball-gluing construction to wormholes.
Since the best window on topology is provided by the cosmic microwave background radiation, David Spergel provided a review of CBMR physics and observations.
www.phys.lsu.edu /mog/mog11/node20.html   (848 words)

 ONT Topology   (Site not responding. Last check: 2007-10-21)
A 'topology' is a family !T! of sets which satisfies the two conditions:
The space X of the topology is always open, and the void set is
topology, but it occurs frequently enough to deserve a name; it is
0-suo.ieee.org.csulib.ctstateu.edu /ontology/msg03863.html   (299 words)

 Encyclopedia article on Topological space [EncycloZine]   (Site not responding. Last check: 2007-10-21)
A topology is completely determined if for every net in X the set of its accumulation points is specified.
The Zariski topology is a purely algebraically defined topology on the spectrum of a ring or an algebraic variety.
For any such structure which is not finite, we often have a natural topology which is compatible with the algebraic operations in the sense that the algebraic operations are still continuous.
encyclozine.com /Topological_space   (2350 words)

 Is the Universe like a Klein Bottle?   (Site not responding. Last check: 2007-10-21)
The topology corresponding to the torus and rectangle) analogies is only one possibility of a "non-trivial" topology.
The other topologies can be thought of in each case by starting with a (3-D) polyhedron of some sort, and by identifying faces (not necessarily opposite) with each other in certain ways.
The consequence of a non-trivial topology to what we see through telescopes would be that we see the same objects (galaxies, or quasars) in different directions and at different times in the past.
www2.iap.fr /users/roukema/top/top_easyE.html   (464 words)

topology generated by a basis (=basis (topology)) owned by rspuzio
topology induced by uniform structure owned by n3o
topology of the complex plane owned by matte
planetmath.org /encyclopedia/T   (1425 words)

 Science Fair Projects - Talk:Limit point
To give an example, I could write articles on Sierpinski space or the upper- or lower-limit topologies on the reals, or other articles, and in all of these, I might at some time want to talk about "limit points" and "accumulation points" as separate definitions.
General (point-set, descriptive) topology is a sort of guilty pleasure hobby for me, so I've explored more than a bit "life without T2" or some of the bizarre spaces one can get from considering various cardinalities or even different axiom systems.
If a space has more than one point, then its topology is trivial if and only if every point is a limit point of every nonempty subset.
www.all-science-fair-projects.com /science_fair_projects_encyclopedia/Talk:Limit_point   (1683 words)

Something that is trivial (and triviality etc.) is something that anyone can grasp, understand, and explain to others, as opposed to something sublime, transcendental etc.
The word trivial comes from Latin, and is originally a word for the kind of things discussed in a trivium, a crossroads where three roads meet.
In the Roman empire a trivium would often have a tavern (Latin: taverna) or similar, where trivial things could be discussed, as opposed to the things discussed in other locations.
www.fact-library.com /trivial.html   (192 words)

 Vincent Moncrief
The interests of this group cover a broad range of topics ranging from the perturbation theory of cosmological models with non-trivial topologies to the study of the Cosmic Censorship and Hoop conjectures for Einstein's theory of General Relativity to the treatment of sufficiently small but fully non-linear perturbations of certain background cosmological spacetimes.
This research involves the expansion of the perturbations in suitably defined tensor harmonics, the derivation of the associated perturbation equations and the reformulation of these equations in terms of naturally defined gauge invariant quantities.
The aim of these projects is to study the asymptotic properties, for expanding universes, of the spacetime metric and matter perturbations and to determine how these properties are related to the underlying spatial topology.
www.yale.edu /physics/research/moncrief.html   (539 words)

 Physics Help and Math Help - Physics Forums - topology question
A topology on a set is a collection of subsets that satisfy certain axioms (which I'll omit).
There may be more then one topology defined on the same set A-- in fact there are as many topologies on A as there are subsets of P(A) that satisfy the topology axioms.
Consider two topological spaces X and Y with topologies S and T. There are many set-maps between X and Y. We have a simple criterion that allows us to decide if such a set map is continuous or not.
www.physicsforums.com /printthread.php?t=14028&pp=40   (5136 words)

 Solution 8
V are open in the subspace topology on A and separate x, y.
In fact, if you take the identity map from R with the usual topology to R with the trivial topology, you can see that it doesn't.
In fact the identity map from R with the usual topology to R with the trivial topology (which is the example of Exercises 7 Question 4 : R
www-groups.dcs.st-and.ac.uk /~john/MT4522/Solutions/S8.html   (484 words)

 Non trivial Topologies of FLRW Universes   (Site not responding. Last check: 2007-10-21)
According to the Value of "k" in the FL equation, the spatial isotropic and homogenous part of the Universe can be flat, have positive curvature or negative curvature.
Aside from trivial topology, we may have hyper-torus S1xS1xS1 topology for flat Universe for instance.
My question is about positive curvature topology: aside S3 solution, which is hypersphere (and some exotic non orientable manifold RP3?), I do not see some topology corresponding to "Hyper-ellipsoide ".
www.lns.cornell.edu /spr/2003-03/msg0049736.html   (131 words)

 A shift-invariant metric on S Z inducing a non-trivial topology - Cattaneo, Formenti, Mazoyer, Margara (ResearchIndex)   (Site not responding. Last check: 2007-10-21)
The metric topology induced by this metric is perfect but not compact.
Moreover we prove that the new space is "suitable" for the study of the dynamical behavior of CA.
In this context sensitivity assumes a stronger meaning than before (usually S Z is given the product topology).
sherry.ifi.unizh.ch /cattaneo97shiftinvariant.html   (547 words)

 Stony Brook Math Calendar
These fl holes can be of non-trivial topology with, for example, handles behind the horizon.
We describe an analytic continuation procedure that sends a fl hole spacetime into a hyperbolic 3-manifold having the topology of a handlebody.
Physical (thermodynamical) properties of the fl hole are encoded in the conformal geometry of the boundary Riemann surface.
www.math.sunysb.edu /~calendar/day.php?LocationID=1&Date=2003-02-05   (94 words)

 ► » Topology of the universe   (Site not responding. Last check: 2007-10-21)
a given local curvature can exist in more than one global topology.
Dodecahedral space topology as an explanation for weak wide-angle
finite, but also that it has a specific, rather rigid topology.
www.science-one.org /Topology-of-the-universe-3479677.html   (545 words)

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