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Hausdorff space

Index 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. [1]

74 relations: Abstract algebra, Algebraic geometry, Algebraic variety, Axiom, Banach–Mazur compactum, Banach–Stone theorem, C*-algebra, Cauchy space, Closed set, Cofiniteness, Compact space, Complete lattice, Complete metric space, Continuous function, Cover (topology), Dense set, Disjoint sets, Equaliser (mathematics), Existential quantification, Felix Hausdorff, Filter (mathematics), Fixed-point space, Graph of a function, Heyting algebra, History of the separation axioms, If and only if, Infinite set, Intuitionistic logic, Kernel (set theory), Kolmogorov space, Lev Pontryagin, Limit of a sequence, Locally compact space, Mathematical analysis, Mathematics, Metric space, Model theory, Neighbourhood (mathematics), Net (mathematics), Nicolas Bourbaki, Noncommutative geometry, Normal space, Open and closed maps, Open set, Paracompact space, Partition of unity, Product topology, Pseudometric space, Quasitopological space, Quotient space (topology), ..., Real number, Regular space, Scott domain, Separated sets, Separation axiom, Sierpiński space, Singleton (mathematics), Spectrum of a ring, Subspace topology, Surjective function, T1 space, Tietze extension theorem, Topological group, Topological indistinguishability, Topological manifold, Topological space, Topology, Topology and Its Applications, Tychonoff space, Uniform space, University of Bonn, Urysohn's lemma, Weak Hausdorff space, Zariski topology. Expand index (24 more) »

Abstract algebra

In algebra, which is a broad division of mathematics, abstract algebra (occasionally called modern algebra) is the study of algebraic structures.

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Algebraic geometry

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

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Algebraic variety

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

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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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Banach–Mazur compactum

In the mathematical study of functional analysis, the Banach–Mazur distance is a way to define a distance on the set Q(n) of n-dimensional normed spaces.

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Banach–Stone theorem

In mathematics, the Banach–Stone theorem is a classical result in the theory of continuous functions on topological spaces, named after the mathematicians Stefan Banach and Marshall Stone.

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C*-algebra

C∗-algebras (pronounced "C-star") are an area of research in functional analysis, a branch of mathematics.

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Cauchy space

In general topology and analysis, a Cauchy space is a generalization of metric spaces and uniform spaces for which the notion of Cauchy convergence still makes sense.

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Closed set

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

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Cofiniteness

In mathematics, a cofinite subset of a set X is a subset A whose complement in X is a finite set.

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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).

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Complete lattice

In mathematics, a complete lattice is a partially ordered set in which all subsets have both a supremum (join) and an infimum (meet).

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Complete metric space

In mathematical analysis, a metric space M is called complete (or a Cauchy space) if every Cauchy sequence of points in M has a limit that is also in M or, alternatively, if every Cauchy sequence in M converges in M. Intuitively, a space is complete if there are no "points missing" from it (inside or at the boundary).

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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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Dense set

In topology and related areas of mathematics, a subset A of a topological space X is called dense (in X) if every point x in X either belongs to A or is a limit point of A, that is the closure of A is constituting the whole set X. Informally, for every point in X, the point is either in A or arbitrarily "close" to a member of A — for instance, every real number either is a rational number or has a rational number arbitrarily close to it (see Diophantine approximation).

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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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Equaliser (mathematics)

In mathematics, an equalizer is a set of arguments where two or more functions have equal values.

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Existential quantification

In predicate logic, an existential quantification is a type of quantifier, a logical constant which is interpreted as "there exists", "there is at least one", or "for some".

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Felix Hausdorff

Felix Hausdorff (November 8, 1868 – January 26, 1942) was a German mathematician who is considered to be one of the founders of modern topology and who contributed significantly to set theory, descriptive set theory, measure theory, function theory, and functional analysis.

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Filter (mathematics)

In mathematics, a filter is a special subset of a partially ordered set.

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Fixed-point space

In mathematics, a Hausdorff space X is called a fixed-point space if every continuous function f:X\rightarrow X has a fixed point.

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Graph of a function

In mathematics, the graph of a function f is, formally, the set of all ordered pairs, and, in practice, the graphical representation of this set.

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Heyting algebra

In mathematics, a Heyting algebra is a bounded lattice (with join and meet operations written ∨ and ∧ and with least element 0 and greatest element 1) equipped with a binary operation a → b of implication such that c ∧ a ≤ b is equivalent to c ≤ a → b. From a logical standpoint, A → B is by this definition the weakest proposition for which modus ponens, the inference rule A → B, A ⊢ B, is sound.

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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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If and only if

In logic and related fields such as mathematics and philosophy, if and only if (shortened iff) is a biconditional logical connective between statements.

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Infinite set

In set theory, an infinite set is a set that is not a finite set.

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Intuitionistic logic

Intuitionistic logic, sometimes more generally called constructive logic, refers to systems of symbolic logic that differ from the systems used for classical logic by more closely mirroring the notion of constructive proof.

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Kernel (set theory)

In set theory, the kernel of a function f may be taken to be either.

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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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Lev Pontryagin

Lev Semyonovich Pontryagin (Лев Семёнович Понтрягин, also written Pontriagin or Pontrjagin) (3 September 1908 – 3 May 1988) was a Soviet mathematician.

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Limit of a sequence

As the positive integer n becomes larger and larger, the value n\cdot \sin\bigg(\frac1\bigg) becomes arbitrarily close to 1.

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Locally compact space

In topology and related branches of mathematics, a topological space is called locally compact if, roughly speaking, each small portion of the space looks like a small portion of a compact space.

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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.

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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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Metric space

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

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Model theory

In mathematics, model theory is the study of classes of mathematical structures (e.g. groups, fields, graphs, universes of set theory) from the perspective of mathematical logic.

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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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Net (mathematics)

In mathematics, more specifically in general topology and related branches, a net or Moore–Smith sequence is a generalization of the notion of a sequence.

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Nicolas Bourbaki

Nicolas Bourbaki is the collective pseudonym under which a group of (mainly French) 20th-century mathematicians, with the aim of reformulating mathematics on an extremely abstract and formal but self-contained basis, wrote a series of books beginning in 1935.

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Noncommutative geometry

Noncommutative geometry (NCG) is a branch of mathematics concerned with a geometric approach to noncommutative algebras, and with the construction of spaces that are locally presented by noncommutative algebras of functions (possibly in some generalized sense).

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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 and closed maps

In topology, an open map is a function between two topological spaces which maps open sets to open sets.

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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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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,.

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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.

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Pseudometric space

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

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Quasitopological space

In mathematics, a quasi-topology on a set X is a function that associates to every compact Hausdorff space C a collection of mappings from C to X satisfying certain natural conditions.

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Quotient space (topology)

In topology and related areas of mathematics, a quotient space (also called an identification space) is, intuitively speaking, the result of identifying or "gluing together" certain points of a given topological space.

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Real number

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

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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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Scott domain

In the mathematical fields of order and domain theory, a Scott domain is an algebraic, bounded-complete cpo.

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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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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.

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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.

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Singleton (mathematics)

In mathematics, a singleton, also known as a unit set, is a set with exactly one element.

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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.

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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).

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Surjective function

In mathematics, a function f from a set X to a set Y is surjective (or onto), or a surjection, if for every element y in the codomain Y of f there is at least one element x in the domain X of f such that f(x).

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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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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.

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Topological group

In mathematics, a topological group is a group G together with a topology on G such that the group's binary operation and the group's inverse function are continuous functions with respect to the topology.

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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 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.

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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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Topology and Its Applications

Topology and Its Applications is a peer-reviewed mathematics journal publishing research on topology.

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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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Uniform space

In the mathematical field of topology, a uniform space is a set with a uniform structure.

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University of Bonn

The University of Bonn (Rheinische Friedrich-Wilhelms-Universität Bonn) is a public research university located in Bonn, Germany.

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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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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.

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Hausdorff Space, Hausdorff axiom, Hausdorff axioms, Hausdorff property, Hausdorff separation axiom, Hausdorff spaces, Hausdorff topological space, Hausdorff topology, Hausdorffness, Preregular space, Preregular topology, R1 space, R1 topology, Separated space, Separated topology, Separated uniformity, T2 space, T2 topology, T2-separation axiom, T2-space, T₂ space.

References

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

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