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Topic: Compactness (topology)

 Relative Compactness from the Bitopological Point of View   (Site not responding. Last check: 2007-10-08)
The notion of relative compactness based on the relation of two topologies on the same set and used by Z. Balogh instead of the above-mentioned idea clearly reveals, even at first glance, the bitopological essence of this notion.
Thus we are able to choose different kinds of bitopological local compactness leading to the relative compactness.
Besides, the strengthening of relative compactness makes it possible to connect the resulting strong compactness with a special type local compactness by the equivalence relation.
www.pmf.ukim.edu.mk /mathematics/dvalishvili.htm   (282 words)

 54: General topology
Topology is the study of sets on which one has a notion of "closeness" -- enough to decide which functions defined on it are continuous.
Thus a general theme in topology is to test the extent to which the axioms force the kind of structure one expects to use and then, as appropriate, introduce other axioms so as to better match the intended application.
Since the axioms of topology are stated in terms of subsets of X, it should be no surprise that one branch of topology is closely related to set theory, particularly "descriptive set theory".
www.math.niu.edu /~rusin/known-math/index/54-XX.html   (2431 words)

The first, continuous topology, centers on the effects of compactness and metrization, is represented here by sections on convergence, compactness, metrization and complete metric spaces, uniform spaces, and function spaces.
The second, geometric topology, focuses on the connectivity properties of topological spaces and provides the core results from general topology that serve as background for subsequent courses in geometry and algebraic topology.
In classical topology, this relation is simple and clear: "An open set is a neighborhood of a point if and only if this point belongs to this open set." In early period of fuzzy topology, "membership relation" was similarly defined.
www.wordtrade.com /science/mathematics/topology.htm   (2132 words)

 General Topology - NoiseFactory Science Archives (http://noisefactory.co.uk)
Compact spaces are very important in general topology for a variety of reasons.
If we assign P the discrete topology, in which every subset is open, these will include all the inverse images of open sets in the various factor spaces.
The standard topologies on N, Z, Q, and R are all (defined to be) their order topologies.
noisefactory.co.uk /maths/topology.html   (4788 words)

 Springer Online Reference Works
the topology of uniform convergence, and in the weak topology, which can be described in terms of pointwise convergence.
This topology is the strongest in a large class of natural topologies on a space of mappings.
First, this topology best reflects the compactness, and compactness is one of the most useful properties of a set of functions.
eom.springer.de /s/s086200.htm   (966 words)

 Math 362 (Functional Analysis)
In many cases, the vector space can be equipped with a natural topology that expresses relations (e.g., convergence theorems) between the elements of that space.
In the most elementary cases, the topology is metrizable and is given by a norm, but some important function spaces are unnormable or even unmetrizable.
But weak topologies generally are not metrizable, so they are conceptually a bit more advanced; for this reason we postpone their study until much later in the course.
www.math.vanderbilt.edu /~schectex/courses/past/banach   (1128 words)

 Algebraic Topology (M24)   (Site not responding. Last check: 2007-10-08)
The idea of topology is that the general shape of a mathematical object is the most important thing.
Algebraic topology is the central core of topology: it explains how to describe shapes in a precise and computable way.
The ideas of algebraic topology are now important in almost all mathematical subjects, even those where no topology is visible at first, such as number theory, algebraic geometry, and representation theory.
www.maths.cam.ac.uk /CASM/courses/02-03/descriptions/node36.html   (329 words)

 PlanetMath: sequentially compact
See Also: compact, limit point compact, Bolzano-Weierstrass theorem
Cross-references: complete, totally bounded, compact, limit point compact, the following are equivalent, metric space, subsequence, convergent, sequence, topological space
This is version 3 of sequentially compact, born on 2002-07-06, modified 2004-02-09.
planetmath.org /encyclopedia/SequentialCompactness.html   (106 words)

 Topology MAT 530
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.
This description of the compactness is the most important ingredient in the proof of the Tychonoff theorem.
www.math.sunysb.edu /~azinger/mat530/fall04/index.htm   (2907 words)

 Topology Course Lecture Notes
Moreover, sequential compactness neither implies nor is implied by compactness.
This topology is 'just right' in the sense that it is barely fine enough to guarantee the continuity of the coordinate projection functions while being just course enough allow the important result of Theorem.
A basic formal distinction between algebra and topology is that although the inverse of a one-one, onto group homomorphism [etc!] is automatically a homomorphism again, the inverse of a one-one, onto continuous map can fail to be continuous.
at.yorku.ca /i/a/a/b/23.dir/index.htm   (8277 words)

 Graduate Math Courses
Topic varies from year to year and will be chosen from: Differential Topology, Euclidean and Non-Euclidean Geometries, Knot Theory, Algebraic Topology, and Projective Geometry.
Prerequisite: Topology (Mathematics 147), or Algebra (Mathematics 171), or permission of instructor.
Students with no background in general topology are urged to take MATH 247 concurrently.
www.cgu.edu /print/628.asp   (2740 words)

 Topologies on spaces of continuous functions by Martin Escardo and Reinhold Heckmann   (Site not responding. Last check: 2007-10-08)
The only prerequisite to this development is a basic knowledge of general topology (continuous functions, product topology and compactness).
Continuity of the function-evaluation map is shown to coincide with a certain approximation property of a topology on the frame of open sets of the exponent space, and the existence of a smallest approximating topology is equivalent to exponentiability of the space.
We show that the intersection of the approximating topologies of any preframe is the Scott topology.
at.yorku.ca /b/a/a/l/38.htm   (257 words)

 Tulane Math Graduate qualifying exam syllabi
General definitions of topological spaces: open and closed sets, bases and subbases for topology, subspaces and subspace topology, continuous functions, homeomorphism.
closed subset of compact is compact,compact Hausdorff is normal, continuous image of compact is compact (should be able to prove and use all of these), Compactness for metric spaces: compactness, sequential compactness, complete and totally bounded sets (SSA), Heine-Borel Theorem (SSA), second countability and Lindelof's Theorem (SSA).
Quotient spaces and quotient topology: definitions and use in examples such as showing a map from a quotient space is continuous, or there is a homeomorphism from the quotient
www.math.tulane.edu /graduate/qualifying/topology.html   (673 words)

 Amazon.com: Topology; A First Course: Books: James R. Munkres   (Site not responding. Last check: 2007-10-08)
I must say I was quite confused when I began the actual topology portion of the book (chapter 2), but this was due to the difficulty of topology, in general, and not the book itself.
Unfortunately, this lack of a completely cohesive approach is unavoidable, since a course in point-set topology ought to provide a stepping stone one can use for further study in topology and not a mountain one can climb and conquer and thus know the subject completely.
If you are searching for an introduction to point-set topology that will give you a solid grounding in the basics of point-set topology, but at the same time will give it to you in an easily approached manner, than this book is for you.
www.amazon.com /Topology-First-Course-James-Munkres/dp/0139254951   (1598 words)

 MA651 Topology 1
The major goal of this course is to cover basic topological ideas such as topological spaces and their products, continuous maps, compactness, connectedness, and metrization.
An additional two lectures illustrating how topology contributes to functional analysis and dynamical systems will be given provided that the mandatory material is covered.
Compact subsets in R. Compactness as a topological invariant.
personal.stevens.edu /~nstrigul/MA651/index_T.html#program   (308 words)

 A Taste of Topology   (Site not responding. Last check: 2007-10-08)
Runde (Univ. of Alberta) writes under the assumption that an educated palate is as necessary for a proper appreciation of topology as it is for olives.
Five chapters and three appendixes cover set theory, metric spaces, the fundamentals of set-theoretic topology (compactness, connectedness, separation properties), the theorems of Stone-Cech and Stone-Weierstrass, homotopy (the fundamental group and covering spaces), the classical Mittag-Leffler theorem, the failure of the Heine-Borel theorem in infinite-dimensional spaces, and the Ascoli-Arzela theorem.
The presentation is everywhere tailored to the upper-division undergraduate mathematics major, though first-year graduate students too will find much profitable reading here--several topics are presented in highly nontraditional ways, and each chapter closes with remarks providing a broader context in which to appreciate the material in the chapter.
www.thattechnicalbookstore.com /b038725790X.htm   (188 words)

 MATH525 - Topics in Topology II: Compactness and Algebraic Structures
MATH525 - Topics in Topology II: Compactness and Algebraic Structures
Topics in Topology II: Compactness and Algebraic Structures
Successful completion of Wesleyan's level I courses in Algebra, Analysis, and Topology; or the equivalent; or permission of the instructor.
www.wesleyan.edu /wesmaps/course0304/math525f.htm   (164 words)

The following examples relate to the use of the concept of function in the context of mappings between vector spaces in Vectorial Analysis and to the introductory concepts of Topology, most notably the notion of compactness.
Topology seems to be one of the areas in which the difficulties of the transition from the concrete to the abstract are mostly apparent.
As a flavour of the basic difficulties the students have with their introduction to Topology, I cite two instances relating to the notion of set of sets.
www.uea.ac.uk /~m011/thesis/chapter3/3iia.htm   (1049 words)

 NISC South Africa
The relations between m-convergences of fuzzy nets and fuzzy filters are investigated as are the relations between Hausdorff fuzzy topology and m-convergences of fuzzy nets and fuzzy filters.
Characterizations of Hausdorff fuzzy topology in terms of neighborhoods and quasi-coincident neighborhoods and of minimal Hausdorff fuzzy topology in terms of m-convergence of fuzzy filters, are established.
We study the relations between m-convergence of fuzzy nets and m-compactness, and between minimal Hausdorff fuzzy topology and m-compactness in a fuzzy topological space.
www.nisc.co.za /oneAbstract?absId=1114   (173 words)

 MIT OpenCourseWare | Mathematics | 18.901 Introduction to Topology, Fall 2004 | Home
A standard example in topology called "the topologist's sine curve." (Image courtesy of Prof.
This course introduces topology, covering topics fundamental to modern analysis and geometry.
It also deals with subjects like topological spaces and continuous functions, connectedness, compactness, separation axioms, and selected further topics such as function spaces, metrization theorems, embedding theorems and the fundamental group.
ocw.mit.edu /OcwWeb/Mathematics/18-901Fall-2004/CourseHome   (103 words)

 Alex Suciu: Topology I
The main properties that are studied are connectedness and compactness.
It starts with covering space theory, and the correspondence between coverings of a space and the subgroups of the fundamental group of that space.
Here are some past qualifying exams in Topology, based in large part on the material covered in this course.
www.math.neu.edu /~suciu/mth3105/top1.f98.html   (334 words)

 Graduate Handbook   (Site not responding. Last check: 2007-10-08)
Fundamentals of point set topology with an introduction to the fundamental group and related topics: topological spaces, product topology, quotient topology, compactness and connectedness, function spaces, homotopy of paths and the fundamental group, covering spaces, Seifert-van Kampen theorem.
Propositional and predicate calculus; the Gödel completeness and compactness theorems; primitive recursive and recursive functions; the Gödel incompleteness theorem; Tarski's theorem; Church's theorem; rescursive undecidability, special topics such as nonstandard analysis.
Topics from completeness and compactness theorems; Löwenheim-Skolem theorems; omitting types and interpolation theorems; homogeneous and saturated models; elimination of quantifiers; Boolean algebras; complete, model complete and decidable theories; ultraproducts; nonstandard analysis.
www.math.purdue.edu /academic/grad/handbook/handbook?id=9   (1957 words)

 Math 436: Topology   (Site not responding. Last check: 2007-10-08)
As major areas of mathematics go, topology is relatively new, since it only emerged as a separate subject in the early part of the twentieth century.
The basic study of topological spaces is known as point-set topology.
Towards the end of the course, though, I would like to spend a little time on algebraic topology, which combines ideas from topology and from abstract algebra.
math.hws.edu /eck/courses/math436_s96.html   (450 words)

 Topology Seminar, Spring 2004
This semester the topic of the Topology seminar will be J-holomorphic curves.
Outline (tentative): The goal of this seminar is to learn some basics of J-holomorphic (aka pseudo-holomorphic) curves in symplectic manifolds.
Abstract: The Faddeev-Hopf functional has been studied numerically by a small number of mathematical physicists, to determine if it could be the basis of a more efficient emperical model of nucleons.
people.brandeis.edu /~levine/topschedsp04.html   (576 words)

 Math. 655   (Site not responding. Last check: 2007-10-08)
655 is an introduction to the basic concepts of modern topology: metric spaces, topological spaces, connectedness, compactness, completeness, quotient spaces, manifolds, and classification of surfaces.
While the course will emphasize the geometric aspects of topology, some applications to analysis will also be discussed, such as the Banach fixed point theorem and the existence of solutions to first order differential equations.
Pictures of various objects we will encounter in the study of topology.
www.math.ohio-state.edu /~fiedorow/math655   (147 words)

 MTH 453, Introduction to Topology   (Site not responding. Last check: 2007-10-08)
This course covers basic point set topology, in particular, connectedness, compactness, and metric spaces.
We aim to cover a bit of algebraic topology, e.g., fundamental groups, as time permits.
The exam is open-text and open-notes, but students are not permitted to work together or to discuss any aspect of the exam with any other person.
www.math.cornell.edu /~brendle/topology/453.html   (496 words)

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