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Separation axiom

Index 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. [1]

40 relations: Abbreviation, Andrey Nikolayevich Tikhonov, Axiom, Axiomatic system, Base (topology), Closure (topology), Continuous function, Cover (topology), Disjoint sets, Distinct (mathematics), Equality (mathematics), Fréchet space, Functional analysis, General topology, German language, Glossary of topology, Hausdorff space, History of the separation axioms, Image (mathematics), Kolmogorov space, Mathematics, Neighbourhood (mathematics), Normal space, Open set, Paracompact space, Real line, Regular space, Semiregular space, Separated sets, Sober space, Star refinement, Subset, T1 space, Topological indistinguishability, Topological space, Topology, Tychonoff space, Urysohn and completely Hausdorff spaces, Urysohn's lemma, Weak Hausdorff space.

Abbreviation

An abbreviation (from Latin brevis, meaning short) is a shortened form of a word or phrase.

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Andrey Nikolayevich Tikhonov

Andrey Nikolayevich Tikhonov (Андре́й Никола́евич Ти́хонов; October 30, 1906 – October 7, 1993) was a Soviet and Russian mathematician and geophysicist known for important contributions to topology, functional analysis, mathematical physics, and ill-posed problems.

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Axiom

An axiom or postulate is a statement that is taken to be true, to serve as a premise or starting point for further reasoning and arguments.

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Axiomatic system

In mathematics, an axiomatic system is any set of axioms from which some or all axioms can be used in conjunction to logically derive theorems.

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Base (topology)

In mathematics, a base (or basis) B for a topological space X with topology T is a collection of open sets in T such that every open set in T can be written as a union of elements of B.We are using a convention that the union of empty collection of sets is the empty set.

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Closure (topology)

In mathematics, the closure of a subset S of points in a topological space consists of all points in S together with all limit points of S. The closure of S may equivalently be defined as the union of S and its boundary, and also as the intersection of all closed sets containing S. Intuitively, the closure can be thought of as all the points that are either in S or "near" S. A point which is in the closure of S is a point of closure of S. The notion of closure is in many ways dual to the notion of interior.

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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.

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Cover (topology)

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

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Disjoint sets

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

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Distinct (mathematics)

In mathematics, two things are called distinct if they are not equal.

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Equality (mathematics)

In mathematics, equality is a relationship between two quantities or, more generally two mathematical expressions, asserting that the quantities have the same value, or that the expressions represent the same mathematical object.

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Fréchet space

In functional analysis and related areas of mathematics, Fréchet spaces, named after Maurice Fréchet, are special topological vector spaces.

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Functional analysis

Functional analysis is a branch of mathematical analysis, the core of which is formed by the study of vector spaces endowed with some kind of limit-related structure (e.g. inner product, norm, topology, etc.) and the linear functions defined on these spaces and respecting these structures in a suitable sense.

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General topology

In mathematics, general topology is the branch of topology that deals with the basic set-theoretic definitions and constructions used in topology.

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German language

German (Deutsch) is a West Germanic language that is mainly spoken in Central Europe.

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Glossary of topology

This is a glossary of some terms used in the branch of mathematics known as topology.

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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.

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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.

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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.

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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.

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Mathematics

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

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Neighbourhood (mathematics)

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

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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.

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Open set

In topology, an open set is an abstract concept generalizing the idea of an open interval in the real line.

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Paracompact space

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

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Real line

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

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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.

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Semiregular space

A semiregular space is a topological space whose regular open sets (sets that equal the interiors of their closures) form a base.

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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.

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Sober space

In mathematics, a sober space is a topological space X such that every irreducible closed subset of X is the closure of exactly one point of X: that is, this closed subset has a unique generic point.

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Star refinement

In mathematics, specifically in the study of topology and open covers of a topological space X, a star refinement is a particular kind of refinement of an open cover of X. The general definition makes sense for arbitrary coverings and does not require a topology.

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Subset

In mathematics, a set A is a subset of a set B, or equivalently B is a superset of A, if A is "contained" inside B, that is, all elements of A are also elements of B. A and B may coincide.

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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.

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Topological indistinguishability

In topology, two points of a topological space X are topologically indistinguishable if they have exactly the same neighborhoods.

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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.

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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.

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Tychonoff space

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

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Urysohn and completely Hausdorff spaces

In topology, a discipline within mathematics, an Urysohn space, or T2½ space, is a topological space in which any two distinct points can be separated by closed neighborhoods.

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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.

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Weak Hausdorff space

In mathematics, a weak Hausdorff space or weakly Hausdorff space is a topological space where the image of every continuous map from a compact Hausdorff space into the space is closed.

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Redirects here:

Separation axioms, Seperation axiom, Seperation axioms, T-axiom, T-axioms, Trennungsaxiom, Trennungsaxiome, Tychonoff condition, Tychonoff conditions, Tychonoff separation axiom, Tychonoff separation axioms.

References

[1] https://en.wikipedia.org/wiki/Separation_axiom

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