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Nemytskii Plane
In mathematics, the Moore plane, also sometimes called Niemytzki plane (or Nemytskii plane, Nemytskii's tangent disk topology), is a topological space. It is a completely regular Hausdorff space (also called Tychonoff space) that is not normal. It is named after Robert Lee Moore and Viktor Vladimirovich Nemytskii. Definition If \Gamma is the (closed) upper half-plane \Gamma = \, then a topology may be defined on \Gamma by taking a local basis \mathcal(p,q) as follows: *Elements of the local basis at points (x,y) with y>0 are the open discs in the plane which are small enough to lie within \Gamma. *Elements of the local basis at points p = (x,0) are sets \\cup A where ''A'' is an open disc in the upper half-plane which is tangent to the ''x'' axis at ''p''. That is, the local basis is given by :\mathcal(p,q) = \begin \, & \mbox q > 0; \\ \, & \mbox q = 0. \end Thus the subspace topology inherited by \Gamma\backslash \ is the same as the subspace topology inherited from the s ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics with the major subdisciplines of number theory, algebra, geometry, and analysis, respectively. There is no general consensus among mathematicians about a common definition for their academic discipline. Most mathematical activity involves the discovery of properties of abstract objects and the use of pure reason to prove them. These objects consist of either abstractions from nature orin modern mathematicsentities that are stipulated to have certain properties, called axioms. A ''proof'' consists of a succession of applications of deductive rules to already established results. These results include previously proved theorems, axioms, andin case of abstraction from naturesome basic properties that are considered true starting points of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
First Countable
In topology, a branch of mathematics, a first-countable space is a topological space satisfying the "first axiom of countability". Specifically, a space X is said to be first-countable if each point has a countable neighbourhood basis (local base). That is, for each point x in X there exists a sequence N_1, N_2, \ldots of neighbourhoods of x such that for any neighbourhood N of x there exists an integer i with N_i contained in N. Since every neighborhood of any point contains an open neighborhood of that point, the neighbourhood basis can be chosen without loss of generality to consist of open neighborhoods. Examples and counterexamples The majority of 'everyday' spaces in mathematics are first-countable. In particular, every metric space is first-countable. To see this, note that the set of open balls centered at x with radius 1/n for integers form a countable local base at x. An example of a space which is not first-countable is the cofinite topology on an uncountable set ( ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Springer-Verlag
Springer Science+Business Media, commonly known as Springer, is a German multinational publishing company of books, e-books and peer-reviewed journals in science, humanities, technical and medical (STM) publishing. Originally founded in 1842 in Berlin, it expanded internationally in the 1960s, and through mergers in the 1990s and a sale to venture capitalists it fused with Wolters Kluwer and eventually became part of Springer Nature in 2015. Springer has major offices in Berlin, Heidelberg, Dordrecht, and New York City. History Julius Springer founded Springer-Verlag in Berlin in 1842 and his son Ferdinand Springer grew it from a small firm of 4 employees into Germany's then second largest academic publisher with 65 staff in 1872.Chronology ". Springer Science+Business Media. In 1964, Springer expanded its business internationally, o ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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 uncoun ... [...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] |
Moore Space (other) , a regular, developable topological space.
{{Mathdab ...
In mathematics, Moore space may refer to: * Moore space (algebraic topology) * Moore space (topology) In mathematics, more specifically point-set topology, a Moore space is a developable regular Hausdorff space. That is, a topological space ''X'' is a Moore space if the following conditions hold: * Any two distinct points can be separated by neig ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Tietze Extension Theorem
In topology, the Tietze extension theorem (also known as the Tietze–Urysohn–Brouwer extension theorem) states that continuous functions on a closed subset of a normal topological space can be extended to the entire space, preserving boundedness if necessary. Formal statement If X is a normal space and f : A \to \R is a continuous map from a closed subset A of X into the real numbers \R carrying the standard topology, then there exists a of f to X; that is, there exists a map F : X \to \R continuous on all of X with F(a) = f(a) for all a \in A. Moreover, F may be chosen such that \sup \ ~=~ \sup \, that is, if f is bounded then F may be chosen to be bounded (with the same bound as f). History L. E. J. Brouwer and Henri Lebesgue proved a special case of the theorem, when X is a finite-dimensional real vector space. Heinrich Tietze extended it to all metric spaces, and Pavel Urysohn proved the theorem as stated here, for normal topological spaces. Equivalent statements This the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Sorgenfrey Plane
In topology, the Sorgenfrey plane is a frequently-cited counterexample to many otherwise plausible-sounding conjectures. It consists of the product of two copies of the Sorgenfrey line, which is the real line \mathbb under the half-open interval topology. The Sorgenfrey line and plane are named for the American mathematician Robert Sorgenfrey. A basis for the Sorgenfrey plane, denoted \mathbb from now on, is therefore the set of rectangles that include the west edge, southwest corner, and south edge, and omit the southeast corner, east edge, northeast corner, north edge, and northwest corner. Open sets in \mathbb are unions of such rectangles. \mathbb is an example of a space that is a product of Lindelöf spaces that is not itself a Lindelöf space. The so-called anti-diagonal \Delta = \ is an uncountable discrete subset of this space, and this is a non- separable subset of the separable space \mathbb. It shows that separability does not inherit to closed subspaces. Note ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Metacompact
In the mathematical field of general topology, a topological space is said to be metacompact if every open cover has a point-finite open refinement. That is, given any open cover of the topological space, there is a refinement that is again an open cover with the property that every point is contained only in finitely many sets of the refining cover. A space is countably metacompact if every countable open cover has a point-finite open refinement. Properties The following can be said about metacompactness in relation to other properties of topological spaces: * Every paracompact space is metacompact. This implies that every compact space is metacompact, and every metric space is metacompact. The converse does not hold: a counter-example is the Dieudonné plank. * Every metacompact space is orthocompact. * Every metacompact normal space is a shrinking space * The product of a compact space and a metacompact space is metacompact. This follows from the tube lemma. * An easy exampl ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Countably Metacompact
In the mathematical field of general topology, a topological space is said to be metacompact if every open cover has a point-finite open refinement. That is, given any open cover of the topological space, there is a refinement that is again an open cover with the property that every point is contained only in finitely many sets of the refining cover. A space is countably metacompact if every countable open cover has a point-finite open refinement. Properties The following can be said about metacompactness in relation to other properties of topological spaces: * Every paracompact space is metacompact. This implies that every compact space is metacompact, and every metric space is metacompact. The converse does not hold: a counter-example is the Dieudonné plank. * Every metacompact space is orthocompact. * Every metacompact normal space is a shrinking space * The product of a compact space and a metacompact space is metacompact. This follows from the tube lemma. * An easy exampl ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Locally Compact
In topology and related branches of mathematics, a topological space is called locally compact if, roughly speaking, each small portion of the space looks like a small portion of a compact space. More precisely, it is a topological space in which every point has a compact neighborhood. In mathematical analysis locally compact spaces that are Hausdorff are of particular interest; they are abbreviated as LCH spaces. Formal definition Let ''X'' be a topological space. Most commonly ''X'' is called locally compact if every point ''x'' of ''X'' has a compact neighbourhood, i.e., there exists an open set ''U'' and a compact set ''K'', such that x\in U\subseteq K. There are other common definitions: They are all equivalent if ''X'' is a Hausdorff space (or preregular). But they are not equivalent in general: :1. every point of ''X'' has a compact neighbourhood. :2. every point of ''X'' has a closed compact neighbourhood. :2′. every point of ''X'' has a relatively compact neighbourhoo ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Lindelöf Space
In mathematics, a Lindelöf space is a topological space in which every open cover has a countable subcover. The Lindelöf property is a weakening of the more commonly used notion of '' compactness'', which requires the existence of a ''finite'' subcover. A hereditarily Lindelöf space is a topological space such that every subspace of it is Lindelöf. Such a space is sometimes called strongly Lindelöf, but confusingly that terminology is sometimes used with an altogether different meaning. The term ''hereditarily Lindelöf'' is more common and unambiguous. Lindelöf spaces are named after the Finnish mathematician Ernst Leonard Lindelöf. Properties of Lindelöf spaces * Every compact space, and more generally every σ-compact space, is Lindelöf. In particular, every countable space is Lindelöf. * A Lindelöf space is compact if and only if it is countably compact. * Every second-countable space is Lindelöf, but not conversely. For example, there are many compact spaces ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
