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65 relations: Addison-Wesley, Algorithm, Axiom of countability, Banach space, Banach–Mazur theorem, Bounded variation, Cardinality, Cardinality of the continuum, Closure (topology), Compact space, Connected space, Constructivism (mathematics), Continuous function, Countable set, Counterexamples in Topology, Dense set, Discrete space, Dover Publications, Euclidean space, Finite set, First uncountable ordinal, First-countable space, Fundamenta Mathematicae, Hahn–Banach theorem, Hausdorff space, Hilbert cube, Hilbert space, Homeomorphism, If and only if, Isometry, Kolmogorov space, Lindelöf space, Linear subspace, Lower limit topology, Lp space, Mathematics, Measure (mathematics), Metric (mathematics), Metric space, Metrization theorem, Moore plane, Normal space, Numerical analysis, Open set, Order topology, Orthonormal basis, Product topology, Quotient space (topology), Rational number, Real line, ..., Second-countable space, Sequence, Sigma-algebra, Sorgenfrey plane, Springer Science+Business Media, Stefan Banach, Stone–Weierstrass theorem, Subspace topology, Symmetric difference, Topological space, Trivial topology, Uniform convergence, Uniform norm, Unit interval, Urysohn universal space. Expand index (15 more) »
Addison-Wesley
Addison-Wesley is a publisher of textbooks and computer literature.
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Algorithm
In mathematics and computer science, an algorithm is an unambiguous specification of how to solve a class of problems.
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Axiom of countability
In mathematics, an axiom of countability is a property of certain mathematical objects (usually in a category) that asserts the existence of a countable set with certain properties.
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Banach space
In mathematics, more specifically in functional analysis, a Banach space (pronounced) is a complete normed vector space.
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Banach–Mazur theorem
In mathematics, the Banach–Mazur theorem is a theorem of functional analysis.
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Bounded variation
In mathematical analysis, a function of bounded variation, also known as function, is a real-valued function whose total variation is bounded (finite): the graph of a function having this property is well behaved in a precise sense.
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Cardinality
In mathematics, the cardinality of a set is a measure of the "number of elements of the set".
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Cardinality of the continuum
In set theory, the cardinality of the continuum is the cardinality or “size” of the set of real numbers \mathbb R, sometimes called the continuum.
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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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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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Connected space
In topology and related branches of mathematics, a connected space is a topological space that cannot be represented as the union of two or more disjoint nonempty open subsets.
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Constructivism (mathematics)
In the philosophy of mathematics, constructivism asserts that it is necessary to find (or "construct") a mathematical object to prove that it exists.
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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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Countable set
In mathematics, a countable set is a set with the same cardinality (number of elements) as some subset of the set of natural numbers.
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Counterexamples in Topology
Counterexamples in Topology (1970, 2nd ed. 1978) is a book on mathematics by topologists Lynn Steen and J. Arthur Seebach, Jr. In the process of working on problems like the metrization problem, topologists (including Steen and Seebach) have defined a wide variety of topological properties.
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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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Discrete space
In topology, a discrete space is a particularly simple example of a topological space or similar structure, one in which the points form a discontinuous sequence, meaning they are isolated from each other in a certain sense.
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Dover Publications
Dover Publications, also known as Dover Books, is an American book publisher founded in 1941 by Hayward Cirker and his wife, Blanche.
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Euclidean space
In geometry, Euclidean space encompasses the two-dimensional Euclidean plane, the three-dimensional space of Euclidean geometry, and certain other spaces.
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Finite set
In mathematics, a finite set is a set that has a finite number of elements.
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First uncountable ordinal
In mathematics, the first uncountable ordinal, traditionally denoted by ω1 or sometimes by Ω, is the smallest ordinal number that, considered as a set, is uncountable.
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First-countable space
In topology, a branch of mathematics, a first-countable space is a topological space satisfying the "first axiom of countability".
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Fundamenta Mathematicae
Fundamenta Mathematicae is a peer-reviewed scientific journal of mathematics with a special focus on the foundations of mathematics, concentrating on set theory, mathematical logic, topology and its interactions with algebra, and dynamical systems.
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Hahn–Banach theorem
In mathematics, the Hahn–Banach theorem is a central tool in functional analysis.
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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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Hilbert cube
In mathematics, the Hilbert cube, named after David Hilbert, is a topological space that provides an instructive example of some ideas in topology.
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Hilbert space
The mathematical concept of a Hilbert space, named after David Hilbert, generalizes the notion of Euclidean space.
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Homeomorphism
In the mathematical field of topology, a homeomorphism or topological isomorphism or bi continuous function is a continuous function between topological spaces that has a continuous inverse function.
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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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Isometry
In mathematics, an isometry (or congruence, or congruent transformation) is a distance-preserving transformation between metric spaces, usually assumed to be bijective.
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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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Lindelöf space
In mathematics, a Lindelöf space is a topological space in which every open cover has a countable subcover.
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Linear subspace
In linear algebra and related fields of mathematics, a linear subspace, also known as a vector subspace, or, in the older literature, a linear manifold, is a vector space that is a subset of some other (higher-dimension) vector space.
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Lower limit topology
In mathematics, the lower limit topology or right half-open interval topology is a topology defined on the set \mathbb of real numbers; it is different from the standard topology on \mathbb (generated by the open intervals) and has a number of interesting properties.
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Lp space
In mathematics, the Lp spaces are function spaces defined using a natural generalization of the ''p''-norm for finite-dimensional vector spaces.
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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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Measure (mathematics)
In mathematical analysis, a measure on a set is a systematic way to assign a number to each suitable subset of that set, intuitively interpreted as its size.
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Metric (mathematics)
In mathematics, a metric or distance function is a function that defines a distance between each pair of elements of a set.
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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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Metrization theorem
In topology and related areas of mathematics, a metrizable space is a topological space that is homeomorphic to a metric space.
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Moore plane
In mathematics, the Moore plane, also sometimes called Niemytzki plane (or Nemytskii plane, Nemytskii's tangent disk topology), is 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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Numerical analysis
Numerical analysis is the study of algorithms that use numerical approximation (as opposed to general symbolic manipulations) for the problems of mathematical analysis (as distinguished from discrete mathematics).
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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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Order topology
In mathematics, an order topology is a certain topology that can be defined on any totally ordered set.
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Orthonormal basis
In mathematics, particularly linear algebra, an orthonormal basis for an inner product space V with finite dimension is a basis for V whose vectors are orthonormal, that is, they are all unit vectors and orthogonal to each other.
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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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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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Rational number
In mathematics, a rational number is any number that can be expressed as the quotient or fraction of two integers, a numerator and a non-zero denominator.
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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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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.
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Sequence
In mathematics, a sequence is an enumerated collection of objects in which repetitions are allowed.
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Sigma-algebra
In mathematical analysis and in probability theory, a σ-algebra (also σ-field) on a set X is a collection Σ of subsets of X that includes the empty subset, is closed under complement, and is closed under countable unions and countable intersections.
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Sorgenfrey plane
In topology, the Sorgenfrey plane is a frequently-cited counterexample to many otherwise plausible-sounding conjectures.
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Springer Science+Business Media
Springer Science+Business Media or Springer, part of Springer Nature since 2015, is a global publishing company that publishes books, e-books and peer-reviewed journals in science, humanities, technical and medical (STM) publishing.
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Stefan Banach
Stefan Banach (30 March 1892 – 31 August 1945) was a Polish mathematician who is generally considered one of the world's most important and influential 20th-century mathematicians.
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Stone–Weierstrass theorem
In mathematical analysis, the Weierstrass approximation theorem states that every continuous function defined on a closed interval can be uniformly approximated as closely as desired by a polynomial function.
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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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Symmetric difference
In mathematics, the symmetric difference, also known as the disjunctive union, of two sets is the set of elements which are in either of the sets and not in their intersection.
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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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Trivial topology
In topology, a topological space with the trivial topology is one where the only open sets are the empty set and the entire space.
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Uniform convergence
In the mathematical field of analysis, uniform convergence is a type of convergence of functions stronger than pointwise convergence.
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Uniform norm
In mathematical analysis, the uniform norm (or sup norm) assigns to real- or complex-valued bounded functions f defined on a set S the non-negative number This norm is also called the supremum norm, the Chebyshev norm, or the infinity norm. The name "uniform norm" derives from the fact that a sequence of functions \ converges to f under the metric derived from the uniform norm if and only if f_n converges to f uniformly.
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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.
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Urysohn universal space
The Urysohn universal space is a certain metric space that contains all separable metric spaces in a particularly nice manner.
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Redirects here:
Separable (topology), Separable metric space, Separable topological space, Seperable space.
References
[1] https://en.wikipedia.org/wiki/Separable_space
