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# 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. [1]

58 relations: Algebraic geometry, Algebraic variety, Arthur Harold Stone, Closed set, Compact space, Continuous function, Cover (topology), Disjoint sets, Elsevier, Function (mathematics), Gδ set, Hausdorff space, History of the separation axioms, Image (mathematics), Kolmogorov space, Lindelöf space, Locally normal space, Mathematical analysis, Mathematics, Metric space, Metrization theorem, Moore plane, Neighbourhood (mathematics), Order topology, Paracompact space, Partition of unity, Pointwise convergence, Prentice Hall, Product topology, Pseudometric space, Pseudonormal space, Real line, Real number, Regular space, Robert Sorgenfrey, Second-countable space, Separated sets, Separation axiom, Sierpiński space, Sorgenfrey plane, Spectrum of a ring, Stone–Čech compactification, Subspace topology, T1 space, Tietze extension theorem, Topological manifold, Topological space, Topological vector space, Topology, Total order, ... Expand index (8 more) »

## Algebraic geometry

Algebraic geometry is a branch of mathematics, classically studying zeros of multivariate polynomials.

## Algebraic variety

Algebraic varieties are the central objects of study in algebraic geometry.

## Arthur Harold Stone

Arthur Harold Stone (30 September 1916 &ndash; 6 August 2000) was a British mathematician born in London, who worked mostly in topology.

## Closed set

In geometry, topology, and related branches of mathematics, a closed set is a set whose complement is an open set.

## Compact space

In mathematics, and more specifically in general topology, compactness is a property that generalizes the notion of a subset of Euclidean space being closed (that is, containing all its limit points) and bounded (that is, having all its points lie within some fixed distance of each other).

## Continuous function

In mathematics, a continuous function is a function for which sufficiently small changes in the input result in arbitrarily small changes in the output.

## Cover (topology)

In mathematics, a cover of a set X is a collection of sets whose union contains X as a subset.

## Disjoint sets

In mathematics, two sets are said to be disjoint sets if they have no element in common.

## Elsevier

Elsevier is an information and analytics company and one of the world's major providers of scientific, technical, and medical information.

## Function (mathematics)

In mathematics, a function was originally the idealization of how a varying quantity depends on another quantity.

## Gδ set

In the mathematical field of topology, a Gδ set is a subset of a topological space that is a countable intersection of open sets.

## Hausdorff space

In topology and related branches of mathematics, a Hausdorff space, separated space or T2 space is a topological space in which distinct points have disjoint neighbourhoods.

## History of the separation axioms

The history of the separation axioms in general topology has been convoluted, with many meanings competing for the same terms and many terms competing for the same concept.

## Image (mathematics)

In mathematics, an image is the subset of a function's codomain which is the output of the function from a subset of its domain.

## Kolmogorov space

In topology and related branches of mathematics, a topological space X is a T0 space or Kolmogorov space (named after Andrey Kolmogorov) if for every pair of distinct points of X, at least one of them has a neighborhood not containing the other.

## Lindelöf space

In mathematics, a Lindelöf space is a topological space in which every open cover has a countable subcover.

## Locally normal space

In mathematics, particularly topology, a topological space X is locally normal if intuitively it looks locally like a normal space.

## Mathematical analysis

Mathematical analysis is the branch of mathematics dealing with limits and related theories, such as differentiation, integration, measure, infinite series, and analytic functions.

## Mathematics

Mathematics (from Greek μάθημα máthēma, "knowledge, study, learning") is the study of such topics as quantity, structure, space, and change.

## Metric space

In mathematics, a metric space is a set for which distances between all members of the set are defined.

## Metrization theorem

In topology and related areas of mathematics, a metrizable space is a topological space that is homeomorphic to a metric space.

## Moore plane

In mathematics, the Moore plane, also sometimes called Niemytzki plane (or Nemytskii plane, Nemytskii's tangent disk topology), is a topological space.

## Neighbourhood (mathematics)

In topology and related areas of mathematics, a neighbourhood (or neighborhood) is one of the basic concepts in a topological space.

## Order topology

In mathematics, an order topology is a certain topology that can be defined on any totally ordered set.

## Paracompact space

In mathematics, a paracompact space is a topological space in which every open cover has an open refinement that is locally finite.

## Partition of unity

In mathematics, a partition of unity of a topological space X is a set R of continuous functions from X to the unit interval such that for every point, x\in X,.

## Pointwise convergence

In mathematics, pointwise convergence is one of various senses in which a sequence of functions can converge to a particular function.

## Prentice Hall

Prentice Hall is a major educational publisher owned by Pearson plc.

## 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.

## Pseudometric space

In mathematics, a pseudometric space is a generalized metric space in which the distance between two distinct points can be zero.

## Pseudonormal space

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.

## Real line

In mathematics, the real line, or real number line is the line whose points are the real numbers.

## Real number

In mathematics, a real number is a value of a continuous quantity that can represent a distance along a line.

## 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.

## Robert Sorgenfrey

Robert Henry Sorgenfrey (1915 – January 6, 1995) was an American mathematician and Professor Emeritus of Mathematics at the University of California, Los Angeles.

## Second-countable space

In topology, a second-countable space, also called a completely separable space, is a topological space whose topology has a countable base.

## 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.

## Separation axiom

In topology and related fields of mathematics, there are several restrictions that one often makes on the kinds of topological spaces that one wishes to consider.

## Sierpiński space

In mathematics, the Sierpiński space (or the connected two-point set) is a finite topological space with two points, only one of which is closed.

## Sorgenfrey plane

In topology, the Sorgenfrey plane is a frequently-cited counterexample to many otherwise plausible-sounding conjectures.

## Spectrum of a ring

In abstract algebra and algebraic geometry, the spectrum of a commutative ring R, denoted by \operatorname(R), is the set of all prime ideals of R. It is commonly augmented with the Zariski topology and with a structure sheaf, turning it into a locally ringed space.

## Stone–Čech compactification

In the mathematical discipline of general topology, Stone–Čech compactification (or Čech–Stone compactification) is a technique for constructing a universal map from a topological space X to a compact Hausdorff space &beta;X.

## 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).

## 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.

## Tietze extension theorem

In topology, the Tietze extension theorem (also known as the Tietze–Urysohn–Brouwer extension theorem) states that continuous functions on a closed subset of a normal topological space can be extended to the entire space, preserving boundedness if necessary.

## Topological manifold

In topology, a branch of mathematics, a topological manifold is a topological space (which may also be a separated space) which locally resembles real n-dimensional space in a sense defined below.

## Topological space

In topology and related branches of mathematics, a topological space may be defined as a set of points, along with a set of neighbourhoods for each point, satisfying a set of axioms relating points and neighbourhoods.

## Topological vector space

In mathematics, a topological vector space (also called a linear topological space) is one of the basic structures investigated in functional analysis.

## Topology

In mathematics, topology (from the Greek τόπος, place, and λόγος, study) is concerned with the properties of space that are preserved under continuous deformations, such as stretching, crumpling and bending, but not tearing or gluing.

## Total order

In mathematics, a linear order, total order, simple order, or (non-strict) ordering is a binary relation on some set X, which is antisymmetric, transitive, and a connex relation.

## Tychonoff plank

In topology, the Tychonoff plank is a topological space defined using ordinal spaces that is a counterexample to several plausible-sounding conjectures.

## Tychonoff space

In topology and related branches of mathematics, Tychonoff spaces and completely regular spaces are kinds of topological spaces.

## 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.

## Uncountable set

In mathematics, an uncountable set (or uncountably infinite set) is an infinite set that contains too many elements to be countable.

## Unit interval

In mathematics, the unit interval is the closed interval, that is, the set of all real numbers that are greater than or equal to 0 and less than or equal to 1.

## Urysohn's lemma

In topology, Urysohn's lemma is a lemma that states that a topological space is normal if and only if any two disjoint closed subsets can be separated by a continuous function.

## Zariski topology

In algebraic geometry and commutative algebra, the Zariski topology is a topology on algebraic varieties, introduced primarily by Oscar Zariski and later generalized for making the set of prime ideals of a commutative ring a topological space, called the spectrum of the ring.

## Zero of a function

In mathematics, a zero, also sometimes called a root, of a real-, complex- or generally vector-valued function f is a member x of the domain of f such that f(x) vanishes at x; that is, x is a solution of the equation f(x).

## References

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