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 is said to be metrizable if there is a metric such that the topology induced by is Metrization theorems are theorems that give sufficient conditions for a topological space to be metrizable.
This is a glossary of some terms used in the branch of mathematics known as topology. Although there is no absolute distinction between different areas of topology,the focus here is on general topology. The following definitions are also fundamental to algebraic topology,differential topology and geometric topology.
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.
In mathematics,a paracompact space is a topological space in which every open cover has an open refinement that is locally finite. These spaces were introduced by Dieudonné(1944). Every compact space is paracompact. Every paracompact Hausdorff space is normal,and a Hausdorff space is paracompact if and only if it admits partitions of unity subordinate to any open cover. Sometimes paracompact spaces are defined so as to always be Hausdorff.
In mathematics,the lower limit topology or right half-open interval topology is a topology defined on ,the set of real numbers;it is different from the standard topology on 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.
In topology,the long line is a topological space somewhat similar to the real line,but in a certain way "longer". It behaves locally just like the real line,but has different large-scale properties. Therefore,it serves as an important counterexample in topology. Intuitively,the usual real-number line consists of a countable number of line segments laid end-to-end,whereas the long line is constructed from an uncountable number of such segments.
In topology,a topological space with the trivial topology is one where the only open sets are the empty set and the entire space. Such spaces are commonly called indiscrete,anti-discrete,concrete or codiscrete. Intuitively,this has the consequence that all points of the space are "lumped together" and cannot be distinguished by topological means. Every indiscrete space is a pseudometric space in which the distance between any two points is zero.
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 under the half-open interval topology. The Sorgenfrey line and plane are named for the American mathematician Robert Sorgenfrey.
In topology,a second-countable space,also called a completely separable space,is a topological space whose topology has a countable base. More explicitly,a topological space is second-countable if there exists some countable collection of open subsets of such that any open subset of can be written as a union of elements of some subfamily of . A second-countable space is said to satisfy the second axiom of countability. Like other countability axioms,the property of being second-countable restricts the number of open sets that a space can have.
In topology and related areas of mathematics,a topological property or topological invariant is a property of a topological space that is invariant under homeomorphisms. Alternatively,a topological property is a proper class of topological spaces which is closed under homeomorphisms. That is,a property of spaces is a topological property if whenever a space X possesses that property every space homeomorphic to X possesses that property. Informally,a topological property is a property of the space that can be expressed using open sets.
In topology,a branch of mathematics,a topological manifold is a topological space that locally resembles real n-dimensional Euclidean space. Topological manifolds are an important class of topological spaces,with applications throughout mathematics. All manifolds are topological manifolds by definition. Other types of manifolds are formed by adding structure to a topological manifold. Every manifold has an "underlying" topological manifold,obtained by simply "forgetting" the added structure. However,not every topological manifold can be endowed with a particular additional structure. For example,the E8 manifold is a topological manifold which cannot be endowed with a differentiable structure.
Mary Ellen Rudin was an American mathematician known for her work in set-theoretic topology. In 2013,Elsevier established the Mary Ellen Rudin Young Researcher Award,which is awarded annually to a young researcher,mainly in fields adjacent to general topology.
In mathematics,in the field of topology,a topological space is said to be pseudonormal if given two disjoint closed sets in it,one of which is countable,there are disjoint open sets containing them. Note the following:
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:
In mathematics a topological space is called countably compact if every countable open cover has a finite subcover.
In mathematics,a topological space is called collectionwise normal if for every discrete family Fi (i ∈I) of closed subsets of there exists a pairwise disjoint family of open sets Ui (i ∈I),such that Fi ⊆Ui. Here a family of subsets of is called discrete when every point of has a neighbourhood that intersects at most one of the sets from . An equivalent definition of collectionwise normal demands that the above Ui (i ∈I) themselves form a discrete family,which is stronger than pairwise disjoint.
The Morita conjectures in general topology are certain problems about normal spaces,now solved in the affirmative. The conjectures,formulated by Kiiti Morita in 1976,asked
- If is normal for every normal space Y,is X a discrete space?
- If is normal for every normal P-space Y,is X metrizable?
- If is normal for every normal countably paracompact space Y,is X metrizable and sigma-locally compact?
In mathematics,particularly topology,a Gδ space is a topological space in which closed sets are in a way ‘separated’from their complements using only countably many open sets. A Gδ space may thus be regarded as a space satisfying a different kind of separation axiom. In fact normal Gδ spaces are referred to as perfectly normal spaces,and satisfy the strongest of separation axioms.
A braille e-book is a refreshable braille display using electroactive polymers or heated wax rather than mechanical pins to raise braille dots on a display. Though not inherently expensive,due to the small scale of production they have not been shown to be economical.
