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Broom Space
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] |
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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 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Dover Publications
Dover Publications, also known as Dover Books, is an American book publisher founded in 1941 by Hayward and Blanche Cirker. It primarily reissues books that are out of print from their original publishers. These are often, but not always, books in the public domain. The original published editions may be scarce or historically significant. Dover republishes these books, making them available at a significantly reduced cost. Classic reprints Dover reprints classic works of literature, classical sheet music, and public-domain images from the 18th and 19th centuries. Dover also publishes an extensive collection of mathematical, scientific, and engineering texts. It often targets its reprints at a niche market, such as woodworking. Starting in 2015, the company branched out into graphic novel reprints, overseen by Dover acquisitions editor and former comics writer and editor Drew Ford. Most Dover reprints are photo facsimiles of the originals, retaining the original pagination an ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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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 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Integer Broom Topology
In general topology, a branch of mathematics, the integer broom topology is an example of a topology on the so-called integer broom space ''X''. Definition of the integer broom space The integer broom space ''X'' is a subset of the plane R2. Assume that the plane is parametrised by polar coordinates. The integer broom contains the origin and the points such that ''n'' is a non-negative integer and , where Z+ is the set of positive integers. The image on the right gives an illustration for and . Geometrically, the space consists of a collection of convergent sequences. For a fixed ''n'', we have a sequence of points − lying on circle with centre (0, 0) and radius ''n'' − that converges to the point (''n'', 0). Definition of the integer broom topology We define the topology on ''X'' by means of a product topology. The integer broom space is given by the polar coordinates :(n, \theta) \in \ \times \ \, . Let us write for simplicity. The integer broom topology on ''X'' ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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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 Locally connected space">locally path connected, or locally connected. 2. The deleted comb space, D, is connected: ::Let E be the comb space without \ \times (0,1. E is also path connected and the closure of E is the comb space. As E \subset D \subset the closure of E, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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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_ ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Path Connected
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 topolog ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Locally Connected
In topology and other branches of mathematics, a topological space ''X'' is locally connected if every point admits a neighbourhood basis consisting entirely of open, connected sets. Background Throughout the history of topology, connectedness and compactness have been two of the most widely studied topological properties. Indeed, the study of these properties even among subsets of Euclidean space, and the recognition of their independence from the particular form of the Euclidean metric, played a large role in clarifying the notion of a topological property and thus a topological space. However, whereas the structure of ''compact'' subsets of Euclidean space was understood quite early on via the Heine–Borel theorem, ''connected'' subsets of \R^n (for ''n'' > 1) proved to be much more complicated. Indeed, while any compact Hausdorff space is locally compact, a connected space—and even a connected subset of the Euclidean plane—need not be locally connected (see below ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Open Set
In mathematics, open sets are a generalization of open intervals in the real line. In a metric space (a set along with a distance defined between any two points), open sets are the sets that, with every point , contain all points that are sufficiently near to (that is, all points whose distance to is less than some value depending on ). More generally, one defines open sets as the members of a given collection of subsets of a given set, a collection that has the property of containing every union of its members, every finite intersection of its members, the empty set, and the whole set itself. A set in which such a collection is given is called a topological space, and the collection is called a topology. These conditions are very loose, and allow enormous flexibility in the choice of open sets. For example, ''every'' subset can be open (the discrete topology), or no set can be open except the space itself and the empty set (the indiscrete topology). In practice ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Counterexamples In Topology
''Counterexamples in Topology'' (1970, 2nd ed. 1978) is a book on mathematics by topologists Lynn Steen and J. Arthur Seebach, Jr. In the process of working on problems like the metrization problem, topologists (including Steen and Seebach) have defined a wide variety of topological properties. It is often useful in the study and understanding of abstracts such as topological spaces to determine that one property does not follow from another. One of the easiest ways of doing this is to find a counterexample which exhibits one property but not the other. In ''Counterexamples in Topology'', Steen and Seebach, together with five students in an undergraduate research project at St. Olaf College, Minnesota in the summer of 1967, canvassed the field of topology for such counterexamples and compiled them in an attempt to simplify the literature. For instance, an example of a first-countable space which is not second-countable is counterexample #3, the discrete topology on an un ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Topology
In mathematics, topology (from the Greek words , and ) is concerned with the properties of a geometric object that are preserved under continuous deformations, such as stretching, twisting, crumpling, and bending; that is, without closing holes, opening holes, tearing, gluing, or passing through itself. A topological space is a set endowed with a structure, called a ''topology'', which allows defining continuous deformation of subspaces, and, more generally, all kinds of 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 homotopies. A property that is invariant under such deformations is a topological property. Basic examples of topological properties are: the dimension, which allows distinguishing between a line and a surface; compactness, which allows distinguishing between a line and a circle; connectedne ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |