Hausdorff space, the Glossary

Table of Contents

1. 83 relations: Abstract algebra, Addison-Wesley, Algebraic geometry, Algebraic variety, Axiom, Banach–Mazur compactum, Banach–Stone theorem, C*-algebra, Cauchy space, Closed set, Cocountable topology, Cofiniteness, Compact space, Complete lattice, Complete metric space, Continuous function, Convergence space, Cover (topology), Dense set, Disjoint sets, Dover Publications, Equaliser (mathematics), Existential quantification, Felix Hausdorff, Filters in topology, Graph of a function, Heyting algebra, History of the separation axioms, If and only if, Infinite set, Intuitionistic logic, Kernel (set theory), Kolmogorov space, Lifting property, Limit of a sequence, Locally compact space, Mathematical analysis, Mathematics, Metric space, Model theory, Neighbourhood (mathematics), Net (mathematics), Noncommutative geometry, Normal space, Open and closed maps, Open set, Paracompact space, Partition of unity, Prentice Hall, Product topology, ... Expand index (33 more) »

2. Separation axioms

Abstract algebra

In mathematics, more specifically algebra, abstract algebra or modern algebra is the study of algebraic structures, which are sets with specific operations acting on their elements.

See Hausdorff space and Abstract algebra

Addison-Wesley

Addison–Wesley is an American publisher of textbooks and computer literature.

See Hausdorff space and Addison-Wesley

Algebraic geometry

Algebraic geometry is a branch of mathematics which uses abstract algebraic techniques, mainly from commutative algebra, to solve geometrical problems.

See Hausdorff space and Algebraic geometry

Algebraic variety

Algebraic varieties are the central objects of study in algebraic geometry, a sub-field of mathematics.

See Hausdorff space and Algebraic variety

Axiom

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

See Hausdorff space and Axiom

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.

See Hausdorff space and Banach–Mazur compactum

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.

See Hausdorff space and Banach–Stone theorem

C*-algebra

In mathematics, specifically in functional analysis, a C∗-algebra (pronounced "C-star") is a Banach algebra together with an involution satisfying the properties of the adjoint.

See Hausdorff space and C*-algebra

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.

See Hausdorff space and Cauchy space

Closed set

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

See Hausdorff space and Closed set

Cocountable topology

The cocountable topology or countable complement topology on any set X consists of the empty set and all cocountable subsets of X, that is all sets whose complement in X is countable.

See Hausdorff space and Cocountable topology

Cofiniteness

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

See Hausdorff space and Cofiniteness

Compact space

In mathematics, specifically general topology, compactness is a property that seeks to generalize the notion of a closed and bounded subset of Euclidean space. Hausdorff space and compact space are properties of topological spaces.

See Hausdorff space and Compact space

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).

See Hausdorff space and Complete lattice

Complete metric space

In mathematical analysis, a metric space is called complete (or a Cauchy space) if every Cauchy sequence of points in has a limit that is also in.

See Hausdorff space and Complete metric space

Continuous function

In mathematics, a continuous function is a function such that a small variation of the argument induces a small variation of the value of the function.

See Hausdorff space and Continuous function

Convergence space

In mathematics, a convergence space, also called a generalized convergence, is a set together with a relation called a that satisfies certain properties relating elements of X with the family of filters on X. Convergence spaces generalize the notions of convergence that are found in point-set topology, including metric convergence and uniform convergence.

See Hausdorff space and Convergence space

Cover (topology)

In mathematics, and more particularly in set theory, a cover (or covering) of a set X is a family of subsets of X whose union is all of X. More formally, if C.

See Hausdorff space and Cover (topology)

Dense set

In topology and related areas of mathematics, a subset A of a topological space X is said to be dense in X if every point of X either belongs to A or else is arbitrarily "close" to a member of A — for instance, the rational numbers are a dense subset of the real numbers because every real number either is a rational number or has a rational number arbitrarily close to it (see Diophantine approximation).

See Hausdorff space and Dense set

Disjoint sets

In set theory in mathematics and formal logic, two sets are said to be disjoint sets if they have no element in common.

See Hausdorff space and Disjoint sets

Dover Publications

Dover Publications, also known as Dover Books, is an American book publisher founded in 1941 by Hayward and Blanche Cirker.

See Hausdorff space and Dover Publications

Equaliser (mathematics)

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

See Hausdorff space and Equaliser (mathematics)

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".

See Hausdorff space and Existential quantification

Felix Hausdorff

Felix Hausdorff (November 8, 1868 – January 26, 1942) was a German mathematician, pseudonym Paul Mongré (à mon (Fr.).

See Hausdorff space and Felix Hausdorff

Filters in topology

Filters in topology, a subfield of mathematics, can be used to study topological spaces and define all basic topological notions such as convergence, continuity, compactness, and more.

See Hausdorff space and Filters in topology

Graph of a function

In mathematics, the graph of a function f is the set of ordered pairs (x, y), where f(x).

See Hausdorff space and Graph of a function

Heyting algebra

In mathematics, a Heyting algebra (also known as pseudo-Boolean 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).

See Hausdorff space and Heyting algebra

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. Hausdorff space and history of the separation axioms are separation axioms.

See Hausdorff space and History of the separation axioms

If and only if

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

See Hausdorff space and If and only if

Infinite set

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

See Hausdorff space and Infinite set

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.

See Hausdorff space and Intuitionistic logic

Kernel (set theory)

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

See Hausdorff space and Kernel (set theory)

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. Hausdorff space and Kolmogorov space are properties of topological spaces and separation axioms.

See Hausdorff space and Kolmogorov space

Lifting property

In mathematics, in particular in category theory, the lifting property is a property of a pair of morphisms in a category.

See Hausdorff space and Lifting property

Limit of a sequence

As the positive integer n becomes larger and larger, the value n\times \sin\left(\tfrac1\right) becomes arbitrarily close to 1.

See Hausdorff space and Limit of a sequence

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. Hausdorff space and locally compact space are properties of topological spaces.

See Hausdorff space and Locally compact space

Mathematical analysis

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

See Hausdorff space and Mathematical analysis

Mathematics

Mathematics is a field of study that discovers and organizes abstract objects, methods, theories and theorems that are developed and proved for the needs of empirical sciences and mathematics itself.

See Hausdorff space and Mathematics

Metric space

In mathematics, a metric space is a set together with a notion of distance between its elements, usually called points.

See Hausdorff space and Metric space

Model theory

In mathematical logic, model theory is the study of the relationship between formal theories (a collection of sentences in a formal language expressing statements about a mathematical structure), and their models (those structures in which the statements of the theory hold).

See Hausdorff space and Model theory

Neighbourhood (mathematics)

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

See Hausdorff space and Neighbourhood (mathematics)

Net (mathematics)

In mathematics, more specifically in general topology and related branches, a net or Moore–Smith sequence is a function whose domain is a directed set.

See Hausdorff space and Net (mathematics)

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.

See Hausdorff space and Noncommutative geometry

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. Hausdorff space and normal space are properties of topological spaces and separation axioms.

See Hausdorff space and Normal space

Open and closed maps

In mathematics, more specifically in topology, an open map is a function between two topological spaces that maps open sets to open sets.

See Hausdorff space and Open and closed maps

Open set

In mathematics, an open set is a generalization of an open interval in the real line.

See Hausdorff space and Open 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. Hausdorff space and paracompact space are properties of topological spaces and separation axioms.

See Hausdorff space and Paracompact space

Partition of unity

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

See Hausdorff space and Partition of unity

Prentice Hall

Prentice Hall was a major American educational publisher.

See Hausdorff space and Prentice Hall

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.

See Hausdorff space and Product topology

Pseudometric space

In mathematics, a pseudometric space is a generalization of a metric space in which the distance between two distinct points can be zero. Hausdorff space and pseudometric space are properties of topological spaces.

See Hausdorff space and Pseudometric space

Pun

A pun, also known as a paranomasia in the context of linguistics, is a form of word play that exploits multiple meanings of a term, or of similar-sounding words, for an intended humorous or rhetorical effect.

See Hausdorff space and Pun

Quarterly Journal of Mathematics

The Quarterly Journal of Mathematics is a quarterly peer-reviewed mathematics journal established in 1930 from the merger of The Quarterly Journal of Pure and Applied Mathematics and the Messenger of Mathematics.

See Hausdorff space and Quarterly Journal of Mathematics

Quotient space (topology)

In topology and related areas of mathematics, the quotient space of a topological space under a given equivalence relation is a new topological space constructed by endowing the quotient set of the original topological space with the quotient topology, that is, with the finest topology that makes continuous the canonical projection map (the function that maps points to their equivalence classes).

See Hausdorff space and Quotient space (topology)

Real number

In mathematics, a real number is a number that can be used to measure a continuous one-dimensional quantity such as a distance, duration or temperature.

See Hausdorff space and Real number

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 have non-overlapping open neighborhoods. Hausdorff space and regular space are properties of topological spaces and separation axioms.

See Hausdorff space and Regular space

Scott domain

In the mathematical fields of order and domain theory, a Scott domain is an algebraic, bounded-complete and directed-complete partial order (dcpo).

See Hausdorff space and Scott domain

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. Hausdorff space and separated sets are separation axioms.

See Hausdorff space and Separated sets

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. Hausdorff space and Separation axiom are separation axioms.

See Hausdorff space and Separation axiom

Sequential space

In topology and related fields of mathematics, a sequential space is a topological space whose topology can be completely characterized by its convergent/divergent sequences. Hausdorff space and sequential space are properties of topological spaces.

See Hausdorff space and Sequential space

Sierpiński space

In mathematics, the Sierpiński space is a finite topological space with two points, only one of which is closed.

See Hausdorff space and Sierpiński space

Singleton (mathematics)

In mathematics, a singleton (also known as a unit set or one-point set) is a set with exactly one element.

See Hausdorff space and Singleton (mathematics)

Sober space

In mathematics, a sober space is a topological space X such that every (nonempty) irreducible closed subset of X is the closure of exactly one point of X: that is, every irreducible closed subset has a unique generic point. Hausdorff space and sober space are properties of topological spaces and separation axioms.

See Hausdorff space and Sober space

Spectrum of a ring

In commutative algebra, the prime spectrum (or simply the spectrum) of a commutative ring R is the set of all prime ideals of R, and is usually denoted by \operatorname; in algebraic geometry it is simultaneously a topological space equipped with the sheaf of rings \mathcal.

See Hausdorff space and Spectrum of a ring

Springer Science+Business Media, commonly known as Springer, is a German multinational publishing company of books, e-books and peer-reviewed journals in science, humanities, technical and medical (STM) publishing.

See Hausdorff space and Springer Science+Business Media

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).

See Hausdorff space and Subspace topology

Surjective function

In mathematics, a surjective function (also known as surjection, or onto function) is a function such that, for every element of the function's codomain, there exists one element in the function's domain such that.

See Hausdorff space and Surjective function

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 point. Hausdorff space and t1 space are properties of topological spaces and separation axioms.

See Hausdorff space and T1 space

The American Mathematical Monthly

The American Mathematical Monthly is a mathematical journal founded by Benjamin Finkel in 1894.

See Hausdorff space and The American Mathematical Monthly

Tietze extension theorem

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

See Hausdorff space and Tietze extension theorem

Topological group

In mathematics, topological groups are the combination of groups and topological spaces, i.e. they are groups and topological spaces at the same time, such that the continuity condition for the group operations connects these two structures together and consequently they are not independent from each other.

See Hausdorff space and Topological group

Topological indistinguishability

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

See Hausdorff space and Topological indistinguishability

Topological manifold

In topology, a branch of mathematics, a topological manifold is a topological space that locally resembles real n-dimensional Euclidean space. Hausdorff space and topological manifold are properties of topological spaces.

See Hausdorff space and Topological manifold

Topological space

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.

See Hausdorff space and Topological space

Topology

Topology (from the Greek words, and) is the branch of mathematics 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.

See Hausdorff space and Topology

Topology and Its Applications

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

See Hausdorff space and Topology and Its Applications

Tychonoff space

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

See Hausdorff space and Tychonoff space

Uncountable set

In mathematics, an uncountable set, informally, is an infinite set that contains too many elements to be countable.

See Hausdorff space and Uncountable set

Uniform space

In the mathematical field of topology, a uniform space is a set with additional structure that is used to define uniform properties, such as completeness, uniform continuity and uniform convergence.

See Hausdorff space and Uniform space

University of Bonn

The University of Bonn, officially the Rhenish Friedrich Wilhelm University of Bonn (Rheinische Friedrich-Wilhelms-Universität Bonn), is a public research university located in Bonn, North Rhine-Westphalia, Germany.

See Hausdorff space and University of Bonn

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. Hausdorff space and Urysohn's lemma are separation axioms.

See Hausdorff space and Urysohn's lemma

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. Hausdorff space and weak Hausdorff space are properties of topological spaces and separation axioms.

See Hausdorff space and Weak Hausdorff space

Zariski topology

In algebraic geometry and commutative algebra, the Zariski topology is a topology defined on geometric objects called varieties.

See Hausdorff space and Zariski topology

See also

Separation axioms

• Dowker space
• Hausdorff space
• History of the separation axioms
• Kolmogorov space
• Locally Hausdorff space
• Normal space
• Paracompact space
• Regular space
• Semiregular space
• Separated sets
• Separation axiom
• Sober space
• T1 space
• Topological indistinguishability
• Tychonoff space
• Urysohn and completely Hausdorff spaces
• Urysohn's lemma
• Weak Hausdorff space

References

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

Also known as Hausdorff axiom, Hausdorff axioms, Hausdorff property, Hausdorff separation axiom, Hausdorff spaces, Hausdorff topological space, Hausdorff topology, Hausdorffness, Non-Hausdorff, Preregular space, Preregular topology, R1 space, R1 topology, Separated space, Separated topology, Separated uniformity, T2 axiom, T2 space, T2 topology, T2-separation axiom, T2-space, T₂ space.

, Pseudometric space, Pun, Quarterly Journal of Mathematics, Quotient space (topology), Real number, Regular space, Scott domain, Separated sets, Separation axiom, Sequential space, Sierpiński space, Singleton (mathematics), Sober space, Spectrum of a ring, Springer Science+Business Media, Subspace topology, Surjective function, T1 space, The American Mathematical Monthly, Tietze extension theorem, Topological group, Topological indistinguishability, Topological manifold, Topological space, Topology, Topology and Its Applications, Tychonoff space, Uncountable set, Uniform space, University of Bonn, Urysohn's lemma, Weak Hausdorff space, Zariski topology.