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Alexandroff Plank
Alexandroff plank in topology, an area of mathematics, is a topological space that serves as an instructive example. Definition The construction of the Alexandroff plank starts by defining the topological space (X,\tau) to be the Cartesian product of ,\omega_1/math> and 1,1 where \omega_1 is the first uncountable ordinal, and both carry the interval topology. The topology \tau is extended to a topology \sigma by adding the sets of the form U(\alpha,n) = \ \cup (\alpha,\omega_1] \times (0,1/n) where p = (\omega_1,0) \in X. The Alexandroff plank is the topological space (X,\sigma). It is called plank for being constructed from a subspace of the product of two spaces. Properties The space (X,\sigma) has the following properties: # It is Urysohn and completely Hausdorff spaces, Urysohn, since (X,\tau) is regular. The space (X,\sigma) is not regular, since C = \ is a closed set not containing (\omega_1,0), while every neighbourhood of C intersects every neighbourhood of ( ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Alexandroff Plank
Alexandroff plank in topology, an area of mathematics, is a topological space that serves as an instructive example. Definition The construction of the Alexandroff plank starts by defining the topological space (X,\tau) to be the Cartesian product of ,\omega_1/math> and 1,1 where \omega_1 is the first uncountable ordinal, and both carry the interval topology. The topology \tau is extended to a topology \sigma by adding the sets of the form U(\alpha,n) = \ \cup (\alpha,\omega_1] \times (0,1/n) where p = (\omega_1,0) \in X. The Alexandroff plank is the topological space (X,\sigma). It is called plank for being constructed from a subspace of the product of two spaces. Properties The space (X,\sigma) has the following properties: # It is Urysohn and completely Hausdorff spaces, Urysohn, since (X,\tau) is regular. The space (X,\sigma) is not regular, since C = \ is a closed set not containing (\omega_1,0), while every neighbourhood of C intersects every neighbourhood of ( ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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; co ... [...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 t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Topological Space
In mathematics, a topological space is, roughly speaking, a geometrical space in which closeness is defined but cannot necessarily be measured by a numeric distance. More specifically, a topological space is a set whose elements are called points, along with an additional structure called a topology, which can be defined as a set of neighbourhoods for each point that satisfy some 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 mathematical space that allows for the definition of limits, continuity, and connectedness. Common types of topological spaces include Euclidean spaces, metric spaces and manifolds. Although very general, the concept of topological spaces is fundamental, and used in virtually every branch of modern mathematics. The study of topologic ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Product Topology
In topology and related areas of mathematics, a product space is the Cartesian product of a family of topological spaces equipped with a natural topology called the product topology. This topology differs from another, perhaps more natural-seeming, topology called the box topology, which can also be given to a product space and which agrees with the product topology when the product is over only finitely many spaces. However, the product topology is "correct" in that it makes the product space a categorical product of its factors, whereas the box topology is too fine; in that sense the product topology is the natural topology on the Cartesian product. Definition Throughout, I will be some non-empty index set and for every index i \in I, let X_i be a topological space. Denote the Cartesian product of the sets X_i by X := \prod X_ := \prod_ X_i and for every index i \in I, denote the i-th by \begin p_i :\;&& \prod_ X_j &&\;\to\; & X_i \\ .3ex && \l ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
First Uncountable Ordinal
In mathematics, the first uncountable ordinal, traditionally denoted by \omega_1 or sometimes by \Omega, is the smallest ordinal number that, considered as a set, is uncountable. It is the supremum (least upper bound) of all countable ordinals. When considered as a set, the elements of \omega_1 are the countable ordinals (including finite ordinals), of which there are uncountably many. Like any ordinal number (in von Neumann's approach), \omega_1 is a well-ordered set, with set membership serving as the order relation. \omega_1 is a limit ordinal, i.e. there is no ordinal \alpha such that \omega_1 = \alpha+1. The cardinality of the set \omega_1 is the first uncountable cardinal number, \aleph_1 (aleph-one). The ordinal \omega_1 is thus the initial ordinal of \aleph_1. Under the continuum hypothesis, the cardinality of \omega_1 is \beth_1, the same as that of \mathbb—the set of real numbers. In most constructions, \omega_1 and \aleph_1 are considered equal as sets. To ge ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Overlapping Interval Topology
In mathematics, the overlapping interval topology is a topology which is used to illustrate various topological principles. Definition Given the closed interval 1,1/math> of the real number line, the open sets of the topology are generated from the half-open intervals (a,1] with a 0. The topology therefore consists of intervals of the form 1,b)_with_b_>_0._The_topology_therefore_consists_of_intervals_of_the_form_[-1,b),_(a,b),_and_(a,1/math>_with_a_1,b) with b > 0. The topology therefore consists of intervals of the form [-1,b), (a,b), and (a,1/math> with a distinct points in 1,1/math> are topologically distinguishable under the overlapping interval topology as one can always find an open set containing one but not the other point. However, every non-empty open set contains the point 0 which can therefore not be separated from any other point in 1,1/math>, making 1,1/math> with the overlapping interval topology an example of a T0 space that is not a T1 space. The overlap ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Urysohn And Completely Hausdorff Spaces
In topology, a discipline within mathematics, an Urysohn space, or T2½ space, is a topological space in which any two distinct points can be separated by closed neighborhoods. A completely Hausdorff space, or functionally Hausdorff space, is a topological space in which any two distinct points can be separated by a continuous function. These conditions are separation axioms that are somewhat stronger than the more familiar Hausdorff axiom T2. Definitions Suppose that ''X'' is a topological space. Let ''x'' and ''y'' be points in ''X''. *We say that ''x'' and ''y'' can be '' separated by closed neighborhoods'' if there exists a closed neighborhood ''U'' of ''x'' and a closed neighborhood ''V'' of ''y'' such that ''U'' and ''V'' are disjoint (''U'' ∩ ''V'' = ∅). (Note that a "closed neighborhood of ''x''" is a closed set that contains an open set containing ''x''.) *We say that ''x'' and ''y'' can be ''separated by a function'' if there exists a continuous function ''f ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Regular Space
In topology and related fields of mathematics, a topological space ''X'' is called a regular space if every closed subset ''C'' of ''X'' and a point ''p'' not contained in ''C'' admit non-overlapping open neighborhoods. Thus ''p'' and ''C'' can be separated by neighborhoods. This condition is known as Axiom T3. The term "T3 space" usually means "a regular Hausdorff space". These conditions are examples of separation axioms. Definitions A topological space ''X'' is a regular space if, given any closed set ''F'' and any point ''x'' that does not belong to ''F'', there exists a neighbourhood ''U'' of ''x'' and a neighbourhood ''V'' of ''F'' that are disjoint. Concisely put, it must be possible to separate ''x'' and ''F'' with disjoint neighborhoods. A or is a topological space that is both regular and a Hausdorff space. (A Hausdorff space or T2 space is a topological space in which any two distinct points are separated by neighbourhoods.) It turns out that a space is T3 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Semiregular Space
A semiregular space is a topological space whose regular open sets (sets that equal the interiors of their closures) form a base for the topology. Examples and sufficient conditions Every regular space is semiregular, and every topological space may be embedded into a semiregular space.. The space X = \Reals^2 \cup \ with the double origin topology In mathematics, more specifically general topology, the double origin topology is an example of a topology given to the plane R2 with an extra point, say 0*, added. In this case, the double origin topology gives a topology on the set , where ∐ d ... and the Arens squareSteen & Seebach, example #80 are examples of spaces that are Hausdorff semiregular, but not regular. See also * Notes References * Lynn Arthur Steen and J. Arthur Seebach, Jr., ''Counterexamples in Topology''. Springer-Verlag, New York, 1978. Reprinted by Dover Publications, New York, 1995. (Dover edition). * Properties of topological spaces Separation ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Base (topology)
In mathematics, a base (or basis) for the topology of a topological space is a family \mathcal of open subsets of such that every open set of the topology is equal to the union of some sub-family of \mathcal. For example, the set of all open intervals in the real number line \R is a basis for the Euclidean topology on \R because every open interval is an open set, and also every open subset of \R can be written as a union of some family of open intervals. Bases are ubiquitous throughout topology. The sets in a base for a topology, which are called , are often easier to describe and use than arbitrary open sets. Many important topological definitions such as continuity and convergence can be checked using only basic open sets instead of arbitrary open sets. Some topologies have a base of open sets with specific useful properties that may make checking such topological definitions easier. Not all families of subsets of a set X form a base for a topology on X. Under some c ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Countably Compact Space
In mathematics a topological space is called countably compact if every countable open cover has a finite subcover. Equivalent definitions A topological space ''X'' is called countably compact if it satisfies any of the following equivalent conditions: :(1) Every countable open cover of ''X'' has a finite subcover. :(2) Every infinite ''set'' ''A'' in ''X'' has an ω-accumulation point in ''X''. :(3) Every ''sequence'' in ''X'' has an accumulation point in ''X''. :(4) Every countable family of closed subsets of ''X'' with an empty intersection has a finite subfamily with an empty intersection. (1) \Rightarrow (2): Suppose (1) holds and ''A'' is an infinite subset of ''X'' without \omega-accumulation point. By taking a subset of ''A'' if necessary, we can assume that ''A'' is countable. Every x\in X has an open neighbourhood O_x such that O_x\cap A is finite (possibly empty), since ''x'' is ''not'' an ω-accumulation point. For every finite subset ''F'' of ''A'' define O_F = \cu ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
