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Path Space (other)
In mathematics, the term path space refers to any topological space of paths from one specified set into another. In particular, it may refer to: * The classical Wiener space of continuous paths * The Skorokhod space of càdlàg paths * For the usage in algebraic topology, see path space (algebraic topology). For Moore's path space, see path space fibration#Moore's path space. See also: loop space In topology, a branch of mathematics, the loop space Ω''X'' of a pointed topological space ''X'' is the space of (based) loops in ''X'', i.e. continuous pointed maps from the pointed circle ''S''1 to ''X'', equipped with the compact-open topolog ..., the space of loops in a topological space {{Mathematical disambiguation ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematics
Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many areas of mathematics, which include number theory (the study of numbers), algebra (the study of formulas and related structures), geometry (the study of shapes and spaces that contain them), Mathematical analysis, analysis (the study of continuous changes), and set theory (presently used as a foundation for all mathematics). Mathematics involves the description and manipulation of mathematical object, abstract objects that consist of either abstraction (mathematics), abstractions from nature orin modern mathematicspurely abstract entities that are stipulated to have certain properties, called axioms. Mathematics uses pure reason to proof (mathematics), prove properties of objects, a ''proof'' consisting of a succession of applications of in ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Topological Space
In mathematics, a topological space is, roughly speaking, a Geometry, geometrical space in which Closeness (mathematics), closeness is defined but cannot necessarily be measured by a numeric Distance (mathematics), distance. More specifically, a topological space is a Set (mathematics), set whose elements are called Point (geometry), points, along with an additional structure called a topology, which can be defined as a set of Neighbourhood (mathematics), neighbourhoods for each point that satisfy some Axiom#Non-logical axioms, axioms formalizing the concept of closeness. There are several equivalent definitions of a topology, the most commonly used of which is the definition through open sets, which is easier than the others to manipulate. A topological space is the most general type of a space (mathematics), mathematical space that allows for the definition of Limit (mathematics), limits, Continuous function (topology), continuity, and Connected space, connectedness. Common types ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Path (topology)
In mathematics, a path in a topological space X is a continuous function from a closed interval into X. Paths play an important role in the fields of topology and mathematical analysis. For example, a topological space for which there exists a path connecting any two points is said to be path-connected. Any space may be broken up into path-connected components. The set of path-connected components of a space X is often denoted \pi_0(X). One can also define paths and loops in pointed spaces, which are important in homotopy theory. If X is a topological space with basepoint x_0, then a path in X is one whose initial point is x_0. Likewise, a loop in X is one that is based at x_0. Definition A ''curve'' in a topological space X is a continuous function f : J \to X from a non-empty and non-degenerate interval J \subseteq \R. A in X is a curve f : , b\to X whose domain , b/math> is a compact non-degenerate interval (meaning a is homeomorphic to , 1 which is why a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Classical Wiener Space
In mathematics, classical Wiener space is the collection of all continuous functions on a given domain (usually a subinterval of the real line), taking values in a metric space (usually ''n''-dimensional Euclidean space). Classical Wiener space is useful in the study of stochastic processes whose sample paths are continuous functions. It is named after the American mathematician Norbert Wiener. Definition Consider E\subseteq \mathbb^n and a metric space (M,d). The classical Wiener space C(E,M) is the space of all continuous functions f:E\to M. That is, for every fixed t\in E, :d(f(s), f(t)) \to 0 as , s - t , \to 0. In almost all applications, one takes E= ,T/math> or E=\R_+= ,T">, +\infty) and M=\mathbb^n for some n\in\mathbb. For brevity, write C for C([0,T; this is a vector space. Write C_0 for the linear subspace consisting only of those function (mathematics), functions that take the value zero at the infimum of the set E. Many authors refer to C_0 as "classical Wiene ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Continuous Function
In mathematics, a continuous function is a function such that a small variation of the argument induces a small variation of the value of the function. This implies there are no abrupt changes in value, known as '' discontinuities''. More precisely, a function is continuous if arbitrarily small changes in its value can be assured by restricting to sufficiently small changes of its argument. A discontinuous function is a function that is . Until the 19th century, mathematicians largely relied on intuitive notions of continuity and considered only continuous functions. The epsilon–delta definition of a limit was introduced to formalize the definition of continuity. Continuity is one of the core concepts of calculus and mathematical analysis, where arguments and values of functions are real and complex numbers. The concept has been generalized to functions between metric spaces and between topological spaces. The latter are the most general continuous functions, and their d ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Càdlàg
In mathematics, a càdlàg (), RCLL ("right continuous with left limits"), or corlol ("continuous on (the) right, limit on (the) left") function is a function defined on the real numbers (or a subset of them) that is everywhere right-continuous and has left limits everywhere. Càdlàg functions are important in the study of stochastic processes that admit (or even require) jumps, unlike Brownian motion, which has continuous sample paths. The collection of càdlàg functions on a given domain is known as Skorokhod space. Two related terms are càglàd, standing for "", the left-right reversal of càdlàg, and càllàl for "" (continuous on one side, limit on the other side), for a function which at each point of the domain is either càdlàg or càglàd. Definition Let (M, d) be a metric space, and let E \subseteq \mathbb. A function f:E \to M is called a càdlàg function if, for every t \in E, * the left limit f(t-) := \lim_f(s) exists; and * the right limit f(t+) := \lim_ ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Path Space (algebraic Topology)
In algebraic topology, a branch of mathematics, the based path space PX of a pointed space (X, *) is the space that consists of all maps f from the interval I = , 1 The comma is a punctuation mark that appears in several variants in different languages. Some typefaces render it as a small line, slightly curved or straight, but inclined from the vertical; others give it the appearance of a miniature fille .../math> to ''X'' such that f(0) = *, called based paths.Martin FranklandMath 527 - Homotopy Theory - Fiber sequences/ref> In other words, it is the mapping space from (I, 0) to (X, *). A space X^I of all maps from I to ''X'', with no distinguished point for the start of the paths, is called the free path space of ''X.'' The maps from I to ''X'' are called free paths. The path space PX is then the pullback of X^I \to X, \, \chi \mapsto \chi(0) along * \hookrightarrow X. The natural map PX \to X, \, \chi \to \chi(1) is a fibration called the path space fibration. See als ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Path Space Fibration
In algebraic topology, the path space fibration over a pointed space (X, *) is a fibration of the form :\Omega X \hookrightarrow PX \overset\to X where *PX is the ''based'' path space of the pointed space (X, *); that is, PX = \ equipped with the compact-open topology. *\Omega X is the fiber of \chi \mapsto \chi(1) over the base point of (X, *); thus it is the loop space of (X, *). The ''free'' path space of ''X'', that is, \operatorname(I, X) = X^I, consists of all maps from ''I'' to ''X'' that do not necessarily begin at a base point, and the fibration X^I \to X given by, say, \chi \mapsto \chi(1), is called the free path space fibration. The path space fibration can be understood to be dual to the mapping cone. The fiber of the based fibration is called the mapping fiber or, equivalently, the homotopy fiber. Mapping path space If f\colon X\to Y is any map, then the mapping path space P_f of f is the pullback of the fibration Y^I \to Y, \, \chi \mapsto \chi(1) along f. (A map ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Loop Space
In topology, a branch of mathematics, the loop space Ω''X'' of a pointed topological space ''X'' is the space of (based) loops in ''X'', i.e. continuous pointed maps from the pointed circle ''S''1 to ''X'', equipped with the compact-open topology. Two loops can be multiplied by concatenation. With this operation, the loop space is an ''A''∞-space. That is, the multiplication is homotopy-coherently associative. The set of path components of Ω''X'', i.e. the set of based-homotopy equivalence classes of based loops in ''X'', is a group, the fundamental group ''π''1(''X''). The iterated loop spaces of ''X'' are formed by applying Ω a number of times. There is an analogous construction for topological spaces without basepoint. The free loop space of a topological space ''X'' is the space of maps from the circle ''S''1 to ''X'' with the compact-open topology. The free loop space of ''X'' is often denoted by \mathcalX. As a functor, the free loop space construction is rig ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
