|
Collectionwise Normal
In mathematics, a topological space X is called collectionwise normal if for every discrete family ''F''''i'' (''i'' ∈ ''I'') of closed subsets of X there exists a pairwise disjoint family of open sets ''U''''i'' (''i'' ∈ ''I''), such that ''F''''i'' ⊆ ''U''''i''. Here a family \mathcal of subsets of X is called ''discrete'' when every point of X has a neighbourhood that intersects at most one of the sets from \mathcal. An equivalent definition of collectionwise normal demands that the above ''U''''i'' (''i'' ∈ ''I'') themselves form a discrete family, which is stronger than pairwise disjoint. Some authors assume that X is also a T1 space as part of the definition. The property is intermediate in strength between paracompactness and normality, and occurs in metrization theorems. Properties *A collectionwise normal space is collectionwise Hausdorff. *A collectionwise normal space is normal. *A Hausdorff paracompact space is collectionwise normal.Note: The ... [...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] |
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] |
Monotonically Normal Space
In mathematics, specifically in the field of topology, a monotonically normal space is a particular kind of normal space, defined in terms of a monotone normality operator. It satisfies some interesting properties; for example metric spaces and linearly ordered spaces are monotonically normal, and every monotonically normal space is hereditarily normal. Definition A topological space X is called monotonically normal if it satisfies any of the following equivalent definitions: Definition 1 The space X is T1 and there is a function G that assigns to each ordered pair (A,B) of disjoint closed sets in X an open set G(A,B) such that: :(i) A\subseteq G(A,B)\subseteq \overline\subseteq X\setminus B; :(ii) G(A,B)\subseteq G(A',B') whenever A\subseteq A' and B'\subseteq B. Condition (i) says X is a normal space, as witnessed by the function G. Condition (ii) says that G(A,B) varies in a monotone fashion, hence the terminology ''monotonically normal''. The operator G is called a mon ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Proc
Proc may refer to: * Proč, a village in eastern Slovakia * ''Proč?'', a 1987 Czech film * procfs or proc filesystem, a special file system (typically mounted to ) in Unix-like operating systems for accessing process information * Protein C (PROC) * Proc, a term in video game terminology * Procedures or process, in the programming language ALGOL 68 * People's Republic of China, the formal name of China China, officially the People's Republic of China (PRC), is a country in East Asia. It is the world's List of countries and dependencies by population, most populous country, with a Population of China, population exceeding 1.4 billion, slig ... * the official acronym for the Canadian House of Commons Standing Committee on Procedure and House Affairs {{disambiguation ... [...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. In ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Closure (topology)
In topology, the closure of a subset of points in a topological space consists of all points in together with all limit points of . The closure of may equivalently be defined as the union of and its boundary, and also as the intersection of all closed sets containing . Intuitively, the closure can be thought of as all the points that are either in or "near" . A point which is in the closure of is a point of closure of . The notion of closure is in many ways dual to the notion of interior. Definitions Point of closure For S as a subset of a Euclidean space, x is a point of closure of S if every open ball centered at x contains a point of S (this point can be x itself). This definition generalizes to any subset S of a metric space X. Fully expressed, for X as a metric space with metric d, x is a point of closure of S if for every r > 0 there exists some s \in S such that the distance d(x, s) < r ( is allowed). Another way to express this i ... [...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] |
Hereditarily Normal Space
In topology and related branches of mathematics, a normal space is a topological space ''X'' that satisfies Axiom T4: every two disjoint closed sets of ''X'' have disjoint open neighborhoods. A normal Hausdorff space is also called a T4 space. These conditions are examples of separation axioms and their further strengthenings define completely normal Hausdorff spaces, or T5 spaces, and perfectly normal Hausdorff spaces, or T6 spaces. Definitions A topological space ''X'' is a normal space if, given any disjoint closed sets ''E'' and ''F'', there are neighbourhoods ''U'' of ''E'' and ''V'' of ''F'' that are also disjoint. More intuitively, this condition says that ''E'' and ''F'' can be separated by neighbourhoods. A T4 space is a T1 space ''X'' that is normal; this is equivalent to ''X'' being normal and Hausdorff. A completely normal space, or , is a topological space ''X'' such that every subspace of ''X'' with subspace topology is a normal space. It turns out that ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Metrizable
In topology and related areas of mathematics, a metrizable space is a topological space that is homeomorphic to a metric space. That is, a topological space (X, \mathcal) is said to be metrizable if there is a metric d : X \times X \to , \infty) such that the topology induced by d is \mathcal. Metrization theorems are theorems that give sufficient conditions for a topological space to be metrizable. Properties Metrizable spaces inherit all topological properties from metric spaces. For example, they are Hausdorff paracompact spaces (and hence normal and Tychonoff) and first-countable. However, some properties of the metric, such as completeness, cannot be said to be inherited. This is also true of other structures linked to the metric. A metrizable uniform space, for example, may have a different set of contraction maps than a metric space to which it is homeomorphic. Metrization theorems One of the first widely recognized metrization theorems was . This states that every ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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 neighbourhoods, and any closed set and any point in its complement can be separated by neighbourhoods. (''X'' is a regular Hausdorff space.) * There is a countable collection of open covers of ''X'', such that for any closed set ''C'' and any point ''p'' in its complement there exists a cover in the collection such that every neighbourhood of ''p'' in the cover is disjoint from ''C''. (''X'' is a developable space.) Moore spaces are generally interesting in mathematics because they may be applied to prove interesting metrization theorems. The concept of a Moore space was formulated by R. L. Moore in the earlier part of the 20th century. Examples and properties #Every metrizable space, ''X'', is a Moore space. If is the open cover of ''X'' ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Subspace Topology
In topology and related areas of mathematics, a subspace of a topological space ''X'' is a subset ''S'' of ''X'' which is equipped with a topology induced from that of ''X'' called the subspace topology (or the relative topology, or the induced topology, or the trace topology). Definition Given a topological space (X, \tau) and a subset S of X, the subspace topology on S is defined by :\tau_S = \lbrace S \cap U \mid U \in \tau \rbrace. That is, a subset of S is open in the subspace topology if and only if it is the intersection of S with an open set in (X, \tau). If S is equipped with the subspace topology then it is a topological space in its own right, and is called a subspace of (X, \tau). Subsets of topological spaces are usually assumed to be equipped with the subspace topology unless otherwise stated. Alternatively we can define the subspace topology for a subset S of X as the coarsest topology for which the inclusion map :\iota: S \hookrightarrow X is continuous. More ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
