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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] |
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] |
Axiom Of Choice
In mathematics, the axiom of choice, or AC, is an axiom of set theory equivalent to the statement that ''a Cartesian product of a collection of non-empty sets is non-empty''. Informally put, the axiom of choice says that given any collection of sets, each containing at least one element, it is possible to construct a new set by arbitrarily choosing one element from each set, even if the collection is infinite. Formally, it states that for every indexed family (S_i)_ of nonempty sets, there exists an indexed set (x_i)_ such that x_i \in S_i for every i \in I. The axiom of choice was formulated in 1904 by Ernst Zermelo in order to formalize his proof of the well-ordering theorem. In many cases, a set arising from choosing elements arbitrarily can be made without invoking the axiom of choice; this is, in particular, the case if the number of sets from which to choose the elements is finite, or if a canonical rule on how to choose the elements is available – some distinguis ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hereditarily 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] |
Completely Normal Hausdorff 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 tha ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hereditary Property
In mathematics, a hereditary property is a property of an object that is inherited by all of its subobjects, where the meaning of ''subobject'' depends on the context. These properties are particularly considered in topology and graph theory, but also in set theory. In topology In topology, a topological property is said to be ''hereditary'' if whenever a topological space has that property, then so does every subspace of it. If the latter is true only for closed subspaces, then the property is called ''weakly hereditary'' or ''closed-hereditary''. For example, second countability and metrisability are hereditary properties. Sequentiality and Hausdorff compactness are weakly hereditary, but not hereditary. Connectivity is not weakly hereditary. If ''P'' is a property of a topological space ''X'' and every subspace also has property ''P'', then ''X'' is said to be "hereditarily ''P''". In combinatorics and graph theory The notion of hereditary properties occurs throughout co ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Generalised Metric
In mathematics, the concept of a generalised metric is a generalisation of that of a metric, in which the distance is not a real number but taken from an arbitrary ordered field. In general, when we define metric space the distance function is taken to be a real-valued function. The real numbers form an ordered field which is Archimedean and order complete. These metric spaces have some nice properties like: in a metric space compactness, sequential compactness and countable compactness are equivalent etc. These properties may not, however, hold so easily if the distance function is taken in an arbitrary ordered field, instead of in \scriptstyle \R. Preliminary definition Let (F, +, \cdot, <) be an arbitrary ordered field, and a nonempty set; a function is called a metric on if the following conditions hold: # if and only if ; # |
Double Arrow Space
In topology, the split interval, or double arrow space, is a topological space that results from splitting each point in a closed interval into two adjacent points and giving the resulting ordered set the order topology. It satisfies various interesting properties and serves as a useful counterexample in general topology. Definition The split interval can be defined as the lexicographic product , 1\times\ equipped with the order topology. Equivalently, the space can be constructed by taking the closed interval ,1/math> with its usual order, splitting each point a into two adjacent points a^- linearly ordered topological space ...
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Sorgenfrey Line
In mathematics, the lower limit topology or right half-open interval topology is a topology defined on the set \mathbb of real numbers; it is different from the standard topology on \mathbb (generated by the open intervals) and has a number of interesting properties. It is the topology generated by the basis of all half-open intervals ''a'',''b''),_where_''a''_and_''b''_are_real_numbers. The_resulting_topological_space.html" ;"title="/nowiki>''a'',''b''), where ''a'' and ''b'' are real numbers. The resulting topological space">/nowiki>''a'',''b''), where ''a'' and ''b'' are real numbers. The resulting topological space is called the Sorgenfrey line after Robert Sorgenfrey or the arrow and is sometimes written \mathbb_l. Like the Cantor set and the long line (topology), long line, the Sorgenfrey line often serves as a useful counterexample to many otherwise plausible-sounding conjectures in general topology. The product of \mathbb_l with itself is also a useful counterexample, ... [...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] |
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 Space
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 H ... [...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] |
