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# Topic: Topological space

###### In the News (Mon 24 Jun 19)

 Regular space X is a regular space iff, given any closed set F and any point x that does not belong to F, there are a neighbourhood U of x and a neighbourhood V of F that are disjoint. Most topological spaces studied in mathematical analysis are regular; in fact, they are usually completely regular, which is a stronger condition. Suppose that A is a set in a topological space X and f is a continuous function from A to a regular space Y. www.ebroadcast.com.au /lookup/encyclopedia/t3/T3_space.html   (917 words)

 Tychonoff space Almost every topological space studied in mathematical analysis is Tychonoff, or at least completely regular. Generalising both the metric spaces and the topological groups, every uniform space is completely regular. Tychonoff spaces are precisely those topological spaces which can be embedded in a compact Hausdorff space. www.ebroadcast.com.au /lookup/encyclopedia/ty/Tychonoff_space.html   (401 words)

 [No title] Given a vector space of functions of a parameter or functions on a manifold, an operator may have a kernel or matrix whose rows and columns are indexed by the parameter or by points on the manifold. loop space Given a topological space X, its loop space is the topological space of all continuous functions from a circle to X. Loop spaces are important examples of new topological spaces formed from old ones, as well as examples of infinite-dimensional spaces in mathematics. PL flow A "piecewise linear" motion on a space or a manifold, akin to a flow given by a vector field, in which every particle in a given simplex of some triangulation moves with constant velocity and in the same direction, so that the particle trajectories are polygons. www.ornl.gov /sci/ortep/topology/defs.txt   (5717 words)

 Topological vector space - Wikipedia, the free encyclopedia As the name suggests the space blends a topological structure (a uniform structure to be precise) with the algebraic concept of a vector space. All normed vector spaces (and therefore all Banach spaces and Hilbert spaces) are examples of topological vector spaces. A vector space is an abelian group with respect to the operation of addition, and in a topological vector space the inverse operation is always continuous (since it is the same as multiplication by −1). en.wikipedia.org /wiki/Topological_vector_space   (1112 words)

 Connected space   (Site not responding. Last check: 2007-11-03) A space is connected iff it cannot be divided into two nonempty closed sets (since the complement of an open set is closed). The space X is said to be path-connected if for any two points x and y in X there exists a continuous function f from the unit interval [0 1] to X with f (0) = x and f (1) = y. The components form partition of the space (that is they disjoint and their union is the whole Every component is a closed subset of the original space. www.freeglossary.com /Connected_space   (686 words)

 Springer Online Reference Works The completeness of the metric space of continuous real-valued functions on a compactum is used, via the contraction-mapping principle, in the proof of the fundamental theorem on the existence of a solution to a differential equation under certain assumptions. The fact that duality between properties of a topological space and topological properties of the space of functions on them with the topology of pointwise convergence is inherited is of special significance. The quest for categories of topological spaces with nice categorical properties, such as Cartesian closedness, has also led to other ways of capturing the idea of  "nearness"  (in analysis, geometry and elsewhere) and has contributed to the emergence of categorical topology and the theory of topological structures, [a5]. eom.springer.de /s/s086200.htm   (966 words)

 Topological Vector Space A topological vector space is a vector space with a topology, such that addition and scaling are continuous. A normed vector space is a topological vector space, deriving its topology from the metric. In a metric space, the translate of an open ball is an open ball, since the distance between two points does not change; but in a topological group, we have to use the properties of continuity to prove translation preserves open sets. www.mathreference.com /top-ban,tvs.html   (1402 words)

 PlanetMath: homogeneous topological space This can be considered a pathological example, as most homogeneous topological spaces encountered in practice are also bihomogeneous. This is true even if we do not require our manifolds to be paracompact, as any two points share a Euclidean neighbourhood, and a suitable homeomorphism for this neighbourhood can be extended to the whole manifold. This is version 2 of homogeneous topological space, born on 2006-10-07, modified 2006-10-14. planetmath.org /encyclopedia/HomogeneousTopologicalSpace.html   (176 words)

 Dynamic Topological Logic   (Site not responding. Last check: 2007-11-03) Topological dynamics studies the asymptotic properties of continuous maps on topological spaces. Let a dynamic topological system be a topological space X together with a continuous function f. Dynamic topological logics are defined for a trimodal language with an S4-ish topological modality □ (interior), and two temporal modalities, ○ (a circle for "next") and * (an asterisk for "henceforth"), both interpreted using the continuous function f. individual.utoronto.ca /philipkremer/DynamicTopologicalLogic.html   (649 words)

 Algebraic Topology: Topology A topological space is a set X together with a collection of subsets OS the members of which are called open, with the property that (i) the union of an arbitrary collection of open sets is open, and (ii) the intersection of a finite collection of open sets is open. Given a topological space (X,OX) and a function f from X to a set B, we call the topology on B determined by f the quotient topology, and f the corresponding quotient map. A topological space is called metric when there is a distance function determining the topology (i.e., open balls for the metric are open sets, and conversely, if a point x lies in an open set U then for some positive e the ball with radius e around x is contained in U. www.win.tue.nl /~aeb/at/algtop-2.html   (1509 words)

 Topological Space   (Site not responding. Last check: 2007-11-03) Such a space, that is one with closure and a metric, may be of any dimension. These spaces are usually defined by their components and we usually call them vector spaces. By completeness (see the Math Appendix) we mean that, in the linear vector space, any function in the space may be expressed as a linear combination of the functions making up the space. www.chem.brown.edu /chem277/space.html   (566 words)

 Topology MAT 530 The next were the formal definitions of metric and topological spaces, bases and subbases in topological spaces (i.e., a description of different ways to define topology on a set). Note that for general topological spaces the first statement is not equivalent to the compactness, and the second statement does not make sense. The Urysohn lemma states that for a normal topological space X and two disjoint closed subsets A and B of it, there exists a continuous function from X to [0,1] that is 0 on A and 1 on B. www.math.sunysb.edu /~azinger/mat530/fall04/index.htm   (2907 words)

 PlanetMath: topological space See Also: neighborhood, metric space, examples of compact spaces, examples of locally compact and not locally compact spaces, site This is version 7 of topological space, born on 2001-10-19, modified 2005-08-13. i can't seem to find a definition for the topology induced by a metric space either, which is something you probably want to add. planetmath.org /encyclopedia/TopologicalSpace.html   (302 words)

 Algebraic Topology: Homotopy Given two spaces X,Y, and a map f from X to Y, let [f] denote the homotopy class of f, that is, the set of all maps from X to Y homotopic to f. A topological space X is called simply connected when it is path-connected and its fundamental group P(X) is trivial. Let X be a topological space that is the union of two path-connected subspaces A and B, where the intersection of A and B is nonempty and path-connected. www.win.tue.nl /~aeb/at/algtop-3.html   (2011 words)

 MAT246: Outline of Point Set Topology A topological space is a set together with a collection of subsets, called the open sets, such that the union of any number of open sets is open, the intersection of a finite number of open sets is open, and the empty set and the entire space are both open. Definition: In a topological space, a point x is a limit point of a subset S if every open set containing x intersects S in a point other than x (x may be in S but need not be). Definition: a topological property is a property that a topological space may have that is preserved under homeomorphism. www.math.toronto.edu /jkorman/Math246Y/top.htm   (766 words)

 3.2 Topological Dimension The dimension on any other space will be defined as one greater that the dimension of the object that could be used to completely separate any part of the first space from the rest. The dimension of a space should be the maximum of its local dimensions where the local dimension is defined as one more than the dimension of the lowest dimensional object with the capacity to separate any neighborhood of the space into two parts. A topological property of an entity is one that remains invariant under continuous, one-to-one transformations or homeomorphisms. hypertextbook.com /chaos/32.shtml   (858 words)

 Topological Properties These differences are not important for basic topological properties, so statements and proofs involving H are often identical to those for R. First an open ball of quaternions needs to be defined to set the stage for an open set. [Wald p424] The quaternion topological space (H, T) is Hausdorff because for each pair of distinct points a, b E H, a not equal to b, one can find open sets Oa, Ob E T such that a E Oa, b I Ob and the intersection of Oa and Ob is the null set. A continuous function from a compact topological space into H is bound and its absolute value attains a maximum and minimum values. www.theworld.com /~sweetser/quaternions/intro/topology/topology.html   (1623 words)

 Good Math, Bad Math : Topological Spaces A function from topological space X to topological space U is continuous if/f for every open sets C ∈ T the inverse image of f on C is an open set. Topological spaces are basically sets with extra structure, and morphisms in that category are simply maps of sets preserving that extra structure, and since injectivity and surjectivity are set theoretic concepts, nothing changes in that respect from Top to Sets. I think your explanation of the relationship between the formal topological notion of continuity and the intuitive notion of a continuous function leaves something to be desired. scienceblogs.com /goodmath/2006/08/topological_spaces.php   (1437 words)

 Notes: Differential Topology, First Steps (Andrew H. Wallace)   (Site not responding. Last check: 2007-11-03) Defintion 1.1: A topological space is a set E along with an assignment to each P /belong to E of a collection of subsets of E, to be called neighborhoods of p, and satisfying the four properties listed earlier. Defintion 1.11: A space E is connected if it cannot be expressed as the union of two nonempty disjoin sets open in E. A set in a topological space is connected if, as a subspace, it is a conneceted space. Defintion 1.13: A covering of a topological space E is a collection of sets in E whose union is E. It is called an open covering if all the sets of the collection are open. www.cs.miami.edu /students/strac/topology/notes01.html   (974 words)

 Springer Online Reference Works The earliest attempts to define the topological product of an infinite set of topological spaces was for the case of metrizable factors. He also proved that the topological product of compact Hausdorff spaces is always a compact Hausdorff space (Tikhonov's theorem). The theorem that follows is of importance in general topology and its applications (particularly in the construction of models of set theory): The Suslin number of the topological product of an arbitrary set of separable topological spaces is countable. eom.springer.de /t/t093100.htm   (757 words)

 Topology Course Lecture Notes 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. A property of topological spaces which when possessed by a space is also possessed by every space homeomorphic to it is called a topological invariant. We learnt that, for metric spaces, sequential convergence was adequate to describe the topology of such spaces (in the sense that the basic primitives of `open set', `neighbourhood', `closure' etc. could be fully characterised in terms of sequential convergence). at.yorku.ca /i/a/a/b/23.dir/ch1.htm   (2430 words)

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