Where results make sense

About us   |   Why use us?   |   Reviews   |   PR   |   Contact us

Wikinfo | Topology In mathematics, topology is a branch concerned with the study of topological spaces. Topology is also concerned with the study of the so-called topological properties of figures, that is to say properties that do not change under bicontinuous one-to-one transformations (called homeomorphisms). In pointless topology one considers instead the lattice of open sets as the basic notion of the theory, while Grothendieck topologies are certain structures defined on arbitrary categories which allow the definition of sheaves on those categories, and with that the definition of quite general cohomology theories. www.wikinfo.org /wiki.php?title=Topology   (1228 words)

Normal space You'll also find terms like normal regular space and normal Hausdorff space; these simply mean that the space both is normal and satisfies the other condition mentioned. An important example of a non-normal topology is given by the Zariski topology on an algebraic variety or on the spectrum of a ring, which is used in algebraic geometry. A non-normal space of some relevance to analysis is the topological vector space of all functions from the real line R to itself, with the topology of pointwise convergence[?]. www.ebroadcast.com.au /lookup/encyclopedia/no/Normal_regular_space.html   (888 words)

[No title] Turned the other way ("a Hausdorff topology cannot be properly contained in a compact topology") it says Hausdorff topologies have to have kind of a lot of open sets. Taken together ("no two compact, Hausdorff topologies are comparable") this makes it clear that compact+Hausdorff is quite a restrictive condition, and thus one from which many nice results can be expected to follow (indeed, Bourbaki _defines_ compact to include the Hausdorff axiom). Undoubtedly there are non-trivial topologies on spaces X for which this Ibokor compactification would turn out to be the extreme case (X, coarse). www.math.niu.edu /~rusin/known-math/95/bijection   (909 words)

Topology - Wikipedia, the free encyclopedia Topology (Greek topos, place and logos, study) is a branch of mathematics, which is an extension of geometry. Topology begins with a consideration of the nature of space, investigating both its fine structure and its global structure. The most basic division within topology is into point-set topology, which investigates such concepts as compactness, connectedness, and countability, and algebraic topology, which investigates such concepts as homotopy, homology, and knot theory. en.wikipedia.org /wiki/Topology   (1817 words)

Wikinfo | Topological space The Zariski topology is a purely algebraically defined topology on 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 particular sequence of functions converges to the zero function. A space carries the trivial topology if all points are "lumped together" in the sense that there are only two open sets, the empty set and the whole space. www.wikinfo.org /wiki.php?title=Topological_space   (2014 words)

Tychonoff space In topology and related brances of mathematics, Tychonoff spaces and completely regular spaces are particularly nice kinds of topological spaces. (The phrase "completely regular Hausdorff", however, is unambiguous, and always means a Tychonoff space.) For more on this issue, see History of the separation axioms[?]. Tychonoff spaces are precisely those topological spaces which can be embedded in a compact Hausdorff space. www.ebroadcast.com.au /lookup/encyclopedia/ty/Tychonov_space.html   (401 words)

Tychonoff space - ExampleProblems.com In topology and related branches of mathematics, Tychonoff spaces and completely regular spaces are particularly nice kinds of topological spaces. More precisely, for every Tychonoff space X, there exists a compact Hausdorff space K and an injective continuous map j from X to K such that the inverse of j is also continuous. It is characterised by the universal property that, given a continuous map f from X to any other compact Hausdorff space Y, there is a unique continuous map g from βX to Y that extends f in the sense that f is the composition of g and j. www.exampleproblems.com /wiki/index.php/Tychonoff_space   (583 words)

Springer Online Reference Works The upper bound of a family of topologies is also known as the inductive limit of a family of topologies. This interpretation of the upper bound of a family of topologies makes it possible to understand that the upper bound of any family of Hausdorff topologies is a Hausdorff topology, and the upper bound of any family of (completely) regular topologies is a (completely) regular topology. However, the upper bound of a countable family of metrizable topologies (with a countable base) is a metrizable topology (with a countable base). eom.springer.de /U/u095840.htm   (306 words)

Research Beside the Hausdorff metric topology, the best known and probably the most important weak hyperspace topology is the Wijsman topology, introduced in the context of convex analysis and then studied in abstract by a number of authors (e.g. The interest for weak topologies arises among others from the fact that under some natural conditions they are measurably compatible, which in turn allows to express the multifunction measurability as ordinary measurability of an associated single-valued function. The so-called generalized compact-open topology on the space of partial maps with domains that are closed in a topological space X has been studied in connection with problems arising in differential equations, in mathematical economics, in convergence of dynamic programming models and other fields. www.uncp.edu /home/laszlo/research.html   (1016 words)

Algebraic Topology: Topology The topology on A defined by F is the weakest topology (i.e., the smallest collection OA) for which all these functions become continuous. The topology on B defined by F is the strongest topology (i.e., the largest collection OB) for which all these functions become continuous. A Hausdorff space X is normal if and only if for each pair of disjoint closed sets A and B there exists a map f from X to the unit interval I that is identically 0 on A and identically 1 on B. www.win.tue.nl /~aeb/at/algtop-2.html   (1509 words)

Tenntop Abstracts The hull-kernel topology (also known as the Stone topology or the Zariski topology) on a family of prime ideals in a ring is an important tool in algebra and analysis. After briefly reviewing one construction of the h-k topology and noting that the derivation has little to do with rings, I will use some simple notions from the theory of ordered sets to investigate h-k and other topologies on families of subsets of a given set. Thus the non-commutativity of A and the asymmetry of topologies on the ideal spaces of A are intimately intertwined. math.tntech.edu /tenntop/abstracts.html   (4721 words)

Topology MAT 530 Some other examples of topological spaces: the 3 essentially different topologies on a 2-point set, the order topology of a linearly ordered set, we also know how to define topology on a partially ordered set such that any pair of elements admits a lower bound. This is the largest (finest, strongest) topology such that the canonical projection (from the space to the quotient-space) is continuous. A counterexample is the set of all rational numbers with the topology induced from the reals (which is the same as the order topology) --- all rationals are separate connected components, but they are not open. www.math.sunysb.edu /~azinger/mat530/fall04/index.htm   (2907 words)

PlanetMath: Linearly Ordered Spaces in ZF Without imposing the algebraic structure upon an ordered set we can investigate into the characteristics of the order topologies of the sets. Proof: As the topology is first countable, there is a countable, nested local bace We claim that this local base has got the properties of the assumption portion of Lemma2.2. planetmath.org /encyclopedia/LinearlyOrderedSpacesInZFC2.html   (640 words)

Lorentzian Wormholes The "test particle limit" in which spacetime is fixed and Hausdorff and we look at the quantum mechanics of a particle on a closed timelike world line. The particular "i epsilon" rule of Feynman for the contour in the complex energy plane around the mass shell poles of the propagator use both retarded waves of positive frequency and advanced waves from the future of negative frequency. Topology change can cause violations of the strong equivalence principle if one wants to hang on to causality at all costs -- the way I read Theorem 11 on p. www.zamandayolculuk.com /cetinbal/LorentzianWormholes.htm   (5849 words)

[No title] This implies in particular that the topology induced by W is the same as the topology induced by W+. The right topology has as a basis the intervals of the form [x; y), the 13 left topology the intervals (y; x], and the bilateral topology is the discrete topology. Now the right topology has as a basis the intervals of the form (\Gamma 1; x), dually for the left topology, and the the bilateral topology coincides with the usual topology in R. The onesided topologies are not Hausdorff. www.math.rutgers.edu /~sontag/dissipation.html   (5969 words)

Stone-Cech Compactification - NoiseFactory Science Archives (http://noisefactory.co.uk)   (Site not responding. Last check: 2007-09-10) Topologies can be very general indeed, so we need to impose some restrictions if we're to be able to derive useful results. Being Hausdorff essentially means that there are enough open sets to stop the points in X getting 'tangled up' with one another. If X is one of the standard spaces used in complicated mathematics it probably isn't compact, and that means it may be quite hard to reason about its properties. noisefactory.co.uk /maths/stone-cech.html   (1649 words)

A Mathematician’s Scratchpad » Blog Archive » Topology, Separation and finite combinatorics Another reason why I thought Hausdorff spaces were really the appropriate setting for what I was doing (and while I still maintain that Hausdorff is part of the definition of topological group). One thing that seems to be the case is that the theories of Hausdorff and non-Hausdorff topological spaces seem to be very different - definitions important in one either don’t work or become essentially trivial in the other. Almost everything you’d normally file under ‘point-set topology’ is a bit uninteresting or downright wrong when you start weakening separation conditions. david.efnet-math.org /?p=6   (1153 words)

Springer Online Reference Works Conversely, the topology of any locally convex space can be defined by some set of semi-norms — for example, by the set of gauge functions (Minkowski functionals) of an arbitrary subbase of neighbourhoods of zero consisting of balanced convex sets. An inductive topology is a projective topology, being the supremum of a collection of topologies. The topology of any locally convex space can be considered as the topology of convergence on some set of subsets of the dual space. eom.springer.de /t/t093180.htm   (2856 words)

Good Math, Bad Math : Meet the Manifolds Also, the sets that are "in the topology" are the ones we're going to call "open." So the Cantor set not being in the topology means it's not open. *Ahem* -- not all non-Hausdorff topologies are pathological. From my perspective this 'Auditory Space' may be a stereoscopic dual coordinate topology possibly employing a vector operator algebra similar to that of Borcherds for a 3D-space with string and time dimensions. scienceblogs.com /goodmath/2006/10/meet_the_manifolds.php   (2990 words)

ZENO\'s paradox - Advanced Physics Forums Let 0 represent the state of the lamp being off and 1 the state of the lamp being on. For a unique limit (limits in non-Hausdorff topologies aren\'t usually unique) of f(x) as x->2 we require f to be a continuous function. This means imposing the indiscrete topology on {0,1}. www.advancedphysics.org /forum/showthread.php?t=938   (636 words)

Springer Online Reference Works A number of general properties of locally convex spaces follows immediately from the corresponding properties of locally convex topologies; in particular, subspaces and Hausdorff quotient spaces of a locally convex space, and also products of families of locally convex spaces, are themselves locally convex spaces. The fundamental results of duality theory include the bipolar theorem (a form of the Hahn–Banach theorem), the Alaoğlu–Bourbaki theorem (on equicontinuous sets in the dual) and the Mackey–Arens theorem (characterizing the topologies which are compatible with a given dual pair). There is an abstract duality between topology and bornology, and equicontinuous sets provide an important example of compactology. eom.springer.de /L/l060360.htm   (1540 words)

Well-behaved - Wikipedia, the free encyclopedia Continuous functions are better-behaved than Riemann-integrable functions on compact sets in calculus. Continuous functions are better-behaved than discontinuous ones in topology. Hausdorff topologies are better-behaved than those in arbitrary general topology. en.wikipedia.org /wiki/Well-behaved   (326 words)

Abstract Stone Duality   (Site not responding. Last check: 2007-09-10) Abstract Stone Duality (ASD) is a type theory in which the topology on a space is an exponential with a lambda-calculus, not an infinitary lattice. However, point-set topology is highly non-constructive - even at a student level, it often uses the axiom of choice. For the troublesome purely infinitary directed joins, we turn the idea of the Scott topology on its head: this is defined using directed joins, so the idea is that, as all functions (including homomorphisms) are to be ``continuous'', they will preserve whatever directed joins are actually needed. www.cs.man.ac.uk /~pt/ASD/manifesto.html   (2776 words)

All finite distributive lattices occur as intervals between Hausdorff topologies forms a lattice under inclusion, in which the meet of two topologies is their intersection, while the join is the topology with their union as a sub-basis. Two obvious directions for possible improvements on this result are to ask for more separation on the topologies or to ask for smaller sets. topologies on some set without assuming the existence of a measurable cardinal. pear.math.pitt.edu /mathzilla/Examples/svg.xml   (830 words)

BSc Course Outline This course includes the basic operations of set theory (union, intersection etc.); mappings including subjective, injective, injective and inverse; countable and uncountable sets including the diagonal process and the unaccountability of R and countable unions of countable sets. Ideas of continuity, neighbourhood and open set; standard topologies on R and Rn and discrete, indiscrete and cofinite topologies on a general set; other topologies on R; closed sets; metrics in R and Rn, convergence of sequences in a metric space, Cauchy sequences and Completeness. The formalism that will be developed provides a unifying thread which is crucial to most areas of mathematics including geometry, analysis, and topology and number theory. web.lums.edu.pk /~webdev/BSc_Course_outline/BSc_courseoutline8.htm   (1036 words)

"visualizing" topological spaces (give an example of a topological space where points are closed that is not hausdorff; and give an example of a compact topological space which is not hausdorff are examples of these kinds of questions). So in a Hausdorff spaces points are closed, but there are spaces where points are closed that are not Hausdorff (R with the finite complement topology for example). If R is Hausdorff, maybe a coaser topology won't be, but maybe finite point sets will still be closed. www.physicsforums.com /showthread.php?t=94974   (550 words)

HJM, Vol. 31, No. 2, 2005 We study the Hausdorff dimension of the intersection between stable manifolds and basic sets for an Axiom A holomorphic endomorphism on the complex projective space of dimension 2. Weaker connected Hausdorff topologies on spaces with a σ-locally finite base, pp. We show that if (X,t) is a disconnected Hausdorff space with a sigma locally finite base, then there is a weaker connected Hausdorff topology on X if and only if X is not H-closed. www.math.uh.edu /~hjm/Vol31-2.html   (1654 words)

Van Zandt: Other Research Abstract: A topological tension arises in optimization in general choice spaces because the topology on the choice space should be weak enough for the choice set to be compact and strong enough for the preference relation to be continuous. Upper semicontinuity of the objective function and upper hemicontinuity of the constraint correspondence are assumed with respect to one topology, and lower semicontinuity and lower hemicontinuity are assumed with respect to the other topology. Abstract: Using a result of Landers and Rogge, it is shown that the Hausdorff metric of sigma-fields is uniformly equivalent to the metric induced by the Hausdorff distance between sets of measurable functions. faculty.insead.edu /vanzandt/research-other/research-other.html   (996 words)

HAF: Preface Here is another example of my preference for abstraction: Some textbooks build Hausdorffness into their definition of "uniform space" or "topological vector space" or "locally convex space" because most spaces used in applications are in fact Hausdorff. Moreover, it may confuse beginners by entangling concepts that are not inherently related: The basic ideas of Hausdorff spaces are independent from the other basic ideas of uniform spaces, topological spaces, and locally convex spaces; neither set of ideas actually requires the other. The weak topology of an infinite-dimensional Banach space is an important nonmetrizable Hausdorff topology that is best explained as the supremum of a collection of pseudometrizable, non- Hausdorff topologies. www.math.vanderbilt.edu /~schectex/ccc/excerpts/preface.html   (4201 words)

Try your search on: Qwika (all wikis)

About us   |   Why use us?   |   Reviews   |   Press   |   Contact us   Copyright © 2005-2007 www.factbites.com Usage implies agreement with terms.