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Comb Space
In mathematics, particularly topology, a comb space is a particular subspace of \R^2 that resembles a comb. The comb space has properties that serve as a number of counterexamples. The topologist's sine curve has similar properties to the comb space. The deleted comb space is a variation on the comb space. Formal definition Consider \R^2 with its standard topology and let ''K'' be the set \. The set ''C'' defined by: :(\ \times ,1) \cup (K \times ,1 \cup ( ,1\times \) considered as a subspace of \R^2 equipped with the subspace topology is known as the comb space. The deleted comb space, D, is defined by: :\ \cup (K \times ,1 \cup ( ,1\times \) . This is the comb space with the line segment \ \times contractible,_but_not_locally_contractible,_Locally_connected_space.html" "title="Contractible_space.html" "title="locally_connected_space.html" "title=",1) deleted. Topological properties The comb space and the deleted comb space have some interesting topological prope ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Topology
In mathematics, topology (from the Greek language, Greek words , and ) is concerned with the properties of a mathematical object, geometric object that are preserved under Continuous function, continuous Deformation theory, deformations, such as Stretch factor, stretching, Twist (mathematics), twisting, crumpling, and bending; that is, without closing holes, opening holes, tearing, gluing, or passing through itself. A topological space is a set (mathematics), set endowed with a structure, called a ''Topology (structure), topology'', which allows defining continuous deformation of subspaces, and, more generally, all kinds of continuity (mathematics), continuity. Euclidean spaces, and, more generally, metric spaces are examples of a topological space, as any distance or metric defines a topology. The deformations that are considered in topology are homeomorphisms and homotopy, homotopies. A property that is invariant under such deformations is a topological property. Basic exampl ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Continuous Function
In mathematics, a continuous function is a function such that a continuous variation (that is a change without jump) of the argument induces a continuous variation of the value of the function. This means that 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 . Up 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 mo ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Prentice Hall
Prentice Hall was an American major educational publisher owned by Savvas Learning Company. Prentice Hall publishes print and digital content for the 6–12 and higher-education market, and distributes its technical titles through the Safari Books Online e-reference service. History On October 13, 1913, law professor Charles Gerstenberg and his student Richard Ettinger founded Prentice Hall. Gerstenberg and Ettinger took their mothers' maiden names, Prentice and Hall, to name their new company. Prentice Hall became known as a publisher of trade books by authors such as Norman Vincent Peale; elementary, secondary, and college textbooks; loose-leaf information services; and professional books. Prentice Hall acquired the training provider Deltak in 1979. Prentice Hall was acquired by Gulf+Western in 1984, and became part of that company's publishing division Simon & Schuster. S&S sold several Prentice Hall subsidiaries: Deltak and Resource Systems were sold to National Education ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Topologist's Sine Curve
In the branch of mathematics known as topology, the topologist's sine curve or Warsaw sine curve is a topological space with several interesting properties that make it an important textbook example. It can be defined as the graph of the function sin(1/''x'') on the half-open interval (0, 1], together with the origin, under the topology subspace topology, induced from the Euclidean plane: : T = \left\ \cup \. Properties The topologist's sine curve ''T'' is connected but neither locally connected nor path connected. This is because it includes the point (0,0) but there is no way to link the function to the origin so as to make a path. The space ''T'' is the continuous image of a locally compact space (namely, let ''V'' be the space ∪ (0, 1], and use the map ''f'' from ''V'' to ''T'' defined by = (0,0) and = for ''x'' > 0), but ''T'' is not locally compact itself. The topological dimension of ''T'' is 1. Variants Two variants of the topologist's sine cur ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Order Topology
In mathematics, an order topology is a certain topology that can be defined on any totally ordered set. It is a natural generalization of the topology of the real numbers to arbitrary totally ordered sets. If ''X'' is a totally ordered set, the order topology on ''X'' is generated by the subbase of "open rays" :\ :\ for all ''a, b'' in ''X''. Provided ''X'' has at least two elements, this is equivalent to saying that the open intervals :(a,b) = \ together with the above rays form a base for the order topology. The open sets in ''X'' are the sets that are a union of (possibly infinitely many) such open intervals and rays. A topological space ''X'' is called orderable or linearly orderable if there exists a total order on its elements such that the order topology induced by that order and the given topology on ''X'' coincide. The order topology makes ''X'' into a completely normal Hausdorff space. The standard topologies on R, Q, Z, and N are the order topologies. Indu ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
List Of Topologies
The following is a list of named topologies or topological spaces, many of which are counterexamples in topology and related branches of mathematics. This is not a list of properties that a topology or topological space might possess; for that, see List of general topology topics and Topological property. Widely known topologies * The Baire space − \N^ with the product topology, where \N denotes the natural numbers endowed with the discrete topology. It is the space of all sequences of natural numbers. * Cantor set − A subset of the closed interval , 1/math> with remarkable properties. ** Cantor dust * Discrete topology − All subsets are open. * Euclidean topology − The natural topology on Euclidean space \Reals^n induced by the Euclidean metric, which is itself induced by the Euclidean norm. ** Real line − \Reals ** Space-filling curve ** Unit interval − , 1/math> * Extended real number line * Hilbert cube − , 1/1\times , 1/2\times , 1/3\times \cdots with t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Infinite Broom
In topology, a branch of mathematics, the infinite broom is a subset of the Euclidean plane that is used as an example distinguishing various notions of connectedness. The closed infinite broom is the closure of the infinite broom, and is also referred to as the broom space.Chapter 6 exercise 3.5 of Definition The infinite broom is the subset of the Euclidean plane that consists of all closed line segments joining the origin to the point as ''n'' varies over all positive integers, together with the interval (½, 1] on the ''x''-axis. The closed infinite broom is then the infinite broom together with the interval (0, ½] on the ''x''-axis. In other words, it consists of all closed line segments joining the origin to the point or to the point . Properties Both the infinite broom and its closure are Connected space, connected, as every open set in the plane which contains the segment on the ''x''-axis must intersect slanted segments. Neither are locally connected. Despite t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hedgehog Space
In mathematics, a hedgehog space is a topological space consisting of a set of spines joined at a point. For any cardinal number \kappa, the \kappa-hedgehog space is formed by taking the disjoint union of \kappa real unit intervals identified at the origin (though its topology is not the quotient topology, but that defined by the metric below). Each unit interval is referred to as one of the hedgehog's ''spines.'' A \kappa-hedgehog space is sometimes called a hedgehog space of spininess \kappa. The hedgehog space is a metric space, when endowed with the hedgehog metric d(x,y)=\left, x - y \ if x and y lie in the same spine, and by d(x,y)=\left, x\ + \left, y\ if x and y lie in different spines. Although their disjoint union makes the origins of the intervals distinct, the metric makes them equivalent by assigning them 0 distance. Hedgehog spaces are examples of real trees. Paris metric The metric on the plane in which the distance between any two points is their Euclidean dista ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Connected Space
In topology and related branches of mathematics, a connected space is a topological space that cannot be represented as the union of two or more disjoint non-empty open subsets. Connectedness is one of the principal topological properties that are used to distinguish topological spaces. A subset of a topological space X is a if it is a connected space when viewed as a subspace of X. Some related but stronger conditions are path connected, simply connected, and n-connected. Another related notion is ''locally connected'', which neither implies nor follows from connectedness. Formal definition A topological space X is said to be if it is the union of two disjoint non-empty open sets. Otherwise, X is said to be connected. A subset of a topological space is said to be connected if it is connected under its subspace topology. Some authors exclude the empty set (with its unique topology) as a connected space, but this article does not follow that practice. For a topologi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Retraction (topology)
In topology, a branch of mathematics, a retraction is a continuous mapping from a topological space into a subspace that preserves the position of all points in that subspace. The subspace is then called a retract of the original space. A deformation retraction is a mapping that captures the idea of ''continuously shrinking'' a space into a subspace. An absolute neighborhood retract (ANR) is a particularly well-behaved type of topological space. For example, every topological manifold is an ANR. Every ANR has the homotopy type of a very simple topological space, a CW complex. Definitions Retract Let ''X'' be a topological space and ''A'' a subspace of ''X''. Then a continuous map :r\colon X \to A is a retraction if the restriction of ''r'' to ''A'' is the identity map on ''A''; that is, r(a) = a for all ''a'' in ''A''. Equivalently, denoting by :\iota\colon A \hookrightarrow X the inclusion, a retraction is a continuous map ''r'' such that :r \circ \iota = \operatorname_A, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Homotopy
In topology, a branch of mathematics, two continuous functions from one topological space to another are called homotopic (from grc, ὁμός "same, similar" and "place") if one can be "continuously deformed" into the other, such a deformation being called a homotopy (, ; , ) between the two functions. A notable use of homotopy is the definition of homotopy groups and cohomotopy groups, important invariants in algebraic topology. In practice, there are technical difficulties in using homotopies with certain spaces. Algebraic topologists work with compactly generated spaces, CW complexes, or spectra. Formal definition Formally, a homotopy between two continuous functions ''f'' and ''g'' from a topological space ''X'' to a topological space ''Y'' is defined to be a continuous function H: X \times ,1\to Y from the product of the space ''X'' with the unit interval , 1to ''Y'' such that H(x,0) = f(x) and H(x,1) = g(x) for all x \in X. If we think of the second ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Separated Sets
In topology and related branches of mathematics, separated sets are pairs of subsets of a given topological space that are related to each other in a certain way: roughly speaking, neither overlapping nor touching. The notion of when two sets are separated or not is important both to the notion of connected spaces (and their connected components) as well as to the separation axioms for topological spaces. Separated sets should not be confused with separated spaces (defined below), which are somewhat related but different. Separable spaces are again a completely different topological concept. Definitions There are various ways in which two subsets of a topological space ''X'' can be considered to be separated. * ''A'' and ''B'' are disjoint if their intersection is the empty set. This property has nothing to do with topology as such, but only set theory. It is included here because it is the weakest in the sequence of different notions. ** ''A'' and ''B'' are separated in ' ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
