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Subbase
In topology, a subbase (or subbasis, prebase, prebasis) for a topological space X with topology T is a subcollection B of T that generates T, in the sense that T is the smallest topology containing B. A slightly different definition is used by some authors, and there are other useful equivalent formulations of the definition; these are discussed below. Definition Let X be a topological space with topology T. A subbase of T is usually defined as a subcollection B of T satisfying one of the two following equivalent conditions: #The subcollection B ''generates'' the topology T. This means that T is the smallest topology containing B: any topology T^\prime on X containing B must also contain T. #The collection of open sets consisting of all finite intersections of elements of B, together with the set X, forms a basis for T. This means that every proper open set in T can be written as a union of finite intersections of elements of B. Explicitly, given a point x in an open set U \ ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Ultrafilter Principle
In the mathematical field of set theory, an ultrafilter is a ''maximal proper filter'': it is a filter U on a given non-empty set X which is a certain type of non-empty family of subsets of X, that is not equal to the power set \wp(X) of X (such filters are called ) and that is also "maximal" in that there does not exist any other proper filter on X that contains it as a proper subset. Said differently, a proper filter U is called an ultrafilter if there exists proper filter that contains it as a subset, that proper filter (necessarily) being U itself. More formally, an ultrafilter U on X is a proper filter that is also a maximal filter on X with respect to set inclusion, meaning that there does not exist any proper filter on X that contains U as a proper subset. Ultrafilters on sets are an important special instance of ultrafilters on partially ordered sets, where the partially ordered set consists of the power set \wp(X) and the partial order is subset inclusion \,\su ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Net (mathematics)
In mathematics, more specifically in general topology and related branches, a net or Moore–Smith sequence is a generalization of the notion of a sequence. In essence, a sequence is a function whose domain is the natural numbers. The codomain of this function is usually some topological space. The motivation for generalizing the notion of a sequence is that, in the context of topology, sequences do not fully encode all information about functions between topological spaces. In particular, the following two conditions are, in general, not equivalent for a map f between topological spaces X and Y: #The map f is continuous in the topological sense; #Given any point x in X, and any sequence in X converging to x, the composition of f with this sequence converges to f(x) (continuous in the sequential sense). While it is necessarily true that condition 1 implies condition 2 (The truth of the condition 1 ensures the truth of the conditions 2.), the reverse implication is not ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Basis (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 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Compact-open Topology
In mathematics, the compact-open topology is a topology defined on the set of continuous maps between two topological spaces. The compact-open topology is one of the commonly used topologies on function spaces, and is applied in homotopy theory and functional analysis. It was introduced by Ralph Fox in 1945. If the codomain of the functions under consideration has a uniform structure or a metric structure then the compact-open topology is the "topology of uniform convergence on compact sets." That is to say, a sequence of functions limit (mathematics), converges in the compact-open topology precisely when it converges uniformly on every compact subset of the Domain of a function, domain. Definition Let and be two topological spaces, and let denote the set of all continuous maps between and . Given a compact set, compact subset of and an open set, open subset of , let denote the set of all functions such that In other words, V(K, U) = C(K, U) \times_ C(X, Y). Then ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Initial Topology
In general topology and related areas of mathematics, the initial topology (or induced topology or weak topology or limit topology or projective topology) on a set X, with respect to a family of functions on X, is the coarsest topology on ''X'' that makes those functions continuous. The subspace topology and product topology constructions are both special cases of initial topologies. Indeed, the initial topology construction can be viewed as a generalization of these. The dual notion is the final topology, which for a given family of functions mapping to a set X is the finest topology on X that makes those functions continuous. Definition Given a set X and an indexed family \left(Y_i\right)_ of topological spaces with functions f_i : X \to Y_i, the initial topology \tau on X is the coarsest topology on X such that each f_i : (X, \tau) \to Y_i is continuous. Definition in terms of open sets If \left(\tau_i\right)_ is a family of topologies X indexed by I \neq \varnothing, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Tychonoff's Theorem
In mathematics, Tychonoff's theorem states that the product of any collection of compact topological spaces is compact with respect to the product topology. The theorem is named after Andrey Nikolayevich Tikhonov (whose surname sometimes is transcribed ''Tychonoff''), who proved it first in 1930 for powers of the closed unit interval and in 1935 stated the full theorem along with the remark that its proof was the same as for the special case. The earliest known published proof is contained in a 1935 article of Tychonoff, A., "Uber einen Funktionenraum", Mathematical Annals, 111, pp. 762–766 (1935). (This reference is mentioned in "Topology" by Hocking and Young, Dover Publications, Ind.) Tychonoff's theorem is often considered as perhaps the single most important result in general topology (along with Urysohn's lemma). The theorem is also valid for topological spaces based on fuzzy sets.Joseph Goguen, "The Fuzzy Tychonoff Theorem", Journal of Mathematical Analys ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
T1 Separation Axiom
In topology and related branches of mathematics, a T1 space is a topological space in which, for every pair of distinct points, each has a neighborhood not containing the other point. An R0 space is one in which this holds for every pair of topologically distinguishable points. The properties T1 and R0 are examples of separation axioms. Definitions Let ''X'' be a topological space and let ''x'' and ''y'' be points in ''X''. We say that ''x'' and ''y'' are if each lies in a neighbourhood that does not contain the other point. * ''X'' is called a T1 space if any two distinct points in ''X'' are separated. * ''X'' is called an R0 space if any two topologically distinguishable points in ''X'' are separated. A T1 space is also called an accessible space or a space with Fréchet topology and an R0 space is also called a symmetric space. (The term also has an entirely different meaning in functional analysis. For this reason, the term ''T1 space'' is preferred. There is also a n ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
T1 Space
In topology and related branches of mathematics, a T1 space is a topological space in which, for every pair of distinct points, each has a neighborhood not containing the other point. An R0 space is one in which this holds for every pair of topologically distinguishable points. The properties T1 and R0 are examples of separation axioms. Definitions Let ''X'' be a topological space and let ''x'' and ''y'' be points in ''X''. We say that ''x'' and ''y'' are if each lies in a neighbourhood that does not contain the other point. * ''X'' is called a T1 space if any two distinct points in ''X'' are separated. * ''X'' is called an R0 space if any two topologically distinguishable points in ''X'' are separated. A T1 space is also called an accessible space or a space with Fréchet topology and an R0 space is also called a symmetric space. (The term also has an entirely different meaning in functional analysis. For this reason, the term ''T1 space'' is preferred. There is also a not ... [...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 && \left(x_j ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Power Set
In mathematics, the power set (or powerset) of a set is the set of all subsets of , including the empty set and itself. In axiomatic set theory (as developed, for example, in the ZFC axioms), the existence of the power set of any set is postulated by the axiom of power set. The powerset of is variously denoted as , , , \mathbb(S), or . The notation , meaning the set of all functions from S to a given set of two elements (e.g., ), is used because the powerset of can be identified with, equivalent to, or bijective to the set of all the functions from to the given two elements set. Any subset of is called a '' family of sets'' over . Example If is the set , then all the subsets of are * (also denoted \varnothing or \empty, the empty set or the null set) * * * * * * * and hence the power set of is . Properties If is a finite set with the cardinality (i.e., the number of all elements in the set is ), then the number of all the subsets of is . This fact ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hausdorff Space
In topology and related branches of mathematics, a Hausdorff space ( , ), separated space or T2 space is a topological space where, for any two distinct points, there exist neighbourhoods of each which are disjoint from each other. Of the many separation axioms that can be imposed on a topological space, the "Hausdorff condition" (T2) is the most frequently used and discussed. It implies the uniqueness of limits of sequences, nets, and filters. Hausdorff spaces are named after Felix Hausdorff, one of the founders of topology. Hausdorff's original definition of a topological space (in 1914) included the Hausdorff condition as an axiom. Definitions Points x and y in a topological space X can be '' separated by neighbourhoods'' if there exists a neighbourhood U of x and a neighbourhood V of y such that U and V are disjoint (U\cap V=\varnothing). X is a Hausdorff space if any two distinct points in X are separated by neighbourhoods. This condition is the third separati ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Topology (structure)
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] |
