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List of general topology topics

Index List of general topology topics

This is a list of general topology topics, by Wikipedia page. [1]

166 relations: Adjunction space, Alexandrov topology, Algebraic topology, Atlas (topology), Axiom of countability, Baire category theorem, Baire space, Ball (mathematics), Banach fixed-point theorem, Banach–Mazur game, Barycentric subdivision, Base (topology), Boundary (topology), Bounded set, Cantor cube, Cantor set, Cantor space, Category of metric spaces, Cauchy sequence, Clopen set, Closed set, Closeness (mathematics), Closure (topology), Cofiniteness, Combinatorial topology, Compact space, Compact-open topology, Compactification (mathematics), Compactly generated space, Comparison of topologies, Complete metric space, Cone (topology), Connected space, Continuous function, Continuum (topology), Cover (topology), Covering space, CW complex, Dense set, Descriptive set theory, Dimension theory, Discrete space, Disjoint union (topology), Equivalence class, Extended real number line, Filter (mathematics), Finite intersection property, First-countable space, Fσ set, General topology, ..., Germ (mathematics), Gδ set, Glossary of topology, Group action, Hausdorff distance, Hausdorff space, Heine–Borel theorem, Hilbert cube, Homeomorphism, Homotopy, Homotopy lifting property, Inductive dimension, Intrinsic metric, Σ-compact space, Kolmogorov space, Kuratowski closure axioms, Lawson topology, Lebesgue covering dimension, Lebesgue's number lemma, Limit point, Lindelöf space, Lipschitz continuity, List of algebraic topology topics, List of geometric topology topics, List of topology topics, Local homeomorphism, Locally compact space, Locally constant function, Long line (topology), Lower limit topology, Manifold, Meagre set, Measure of non-compactness, Metric space, Metrization theorem, Neighbourhood (mathematics), Nerve of a covering, Net (mathematics), Normal space, Nowhere dense set, Open and closed maps, Open set, Order theory, P-adic analysis, P-adic number, Paracompact space, Path (topology), Pointed space, Pointwise convergence, Polish space, Polytope, Product topology, Quotient space (topology), Real tree, Regular space, Relatively compact subspace, Restricted product, Scott continuity, Second-countable space, Semi-locally simply connected, Separable space, Separated sets, Separation axiom, Sequential space, Sheaf (mathematics), Sierpiński space, Simplex, Simplicial approximation theorem, Simplicial complex, Simply connected space, Smash product, Sober space, Solenoid (mathematics), Sorgenfrey plane, Space-filling curve, Specialization (pre)order, Spectral space, Sperner's lemma, Stone duality, Stone's representation theorem for Boolean algebras, Strong topology (polar topology), Subbase, Subspace topology, T1 space, Taxicab geometry, Tietze extension theorem, Topological abelian group, Topological algebra, Topological group, Topological module, Topological property, Topological ring, Topological space, Topological tensor product, Topological vector space, Topologist's sine curve, Triangulation, Trivial topology, Tychonoff space, Tychonoff's theorem, Ultrafilter, Ultrametric space, Unicoherent space, Uniform continuity, Uniform isomorphism, Uniform norm, Uniform property, Uniform space, Uniformly connected space, Unit interval, Upper topology, Urysohn and completely Hausdorff spaces, Urysohn's lemma, Weak topology, Wedge sum, Zariski topology. Expand index (116 more) »

Adjunction space

In mathematics, an adjunction space (or attaching space) is a common construction in topology where one topological space is attached or "glued" onto another.

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Alexandrov topology

In topology, an Alexandrov topology is a topology in which the intersection of any family of open sets is open.

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Algebraic topology

Algebraic topology is a branch of mathematics that uses tools from abstract algebra to study topological spaces.

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Atlas (topology)

In mathematics, particularly topology, one describes a manifold using an atlas.

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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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Baire category theorem

The Baire category theorem (BCT) is an important tool in general topology and functional analysis.

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Baire space

In mathematics, a Baire space is a topological space such that every intersection of a countable collection of open dense sets in the space is also dense.

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Ball (mathematics)

In mathematics, a ball is the space bounded by a sphere.

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Banach fixed-point theorem

In mathematics, the Banach fixed-point theorem (also known as the contraction mapping theorem or contraction mapping principle) is an important tool in the theory of metric spaces; it guarantees the existence and uniqueness of fixed points of certain self-maps of metric spaces, and provides a constructive method to find those fixed points.

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Banach–Mazur game

In general topology, set theory and game theory, a Banach–Mazur game is a topological game played by two players, trying to pin down elements in a set (space).

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Barycentric subdivision

In geometry, the barycentric subdivision is a standard way of dividing an arbitrary convex polygon into triangles, a convex polyhedron into tetrahedra, or, in general, a convex polytope into simplices with the same dimension, by connecting the barycenters of their faces in a specific way.

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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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Boundary (topology)

In topology and mathematics in general, the boundary of a subset S of a topological space X is the set of points which can be approached both from S and from the outside of S. More precisely, it is the set of points in the closure of S not belonging to the interior of S. An element of the boundary of S is called a boundary point of S. The term boundary operation refers to finding or taking the boundary of a set.

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Bounded set

In mathematical analysis and related areas of mathematics, a set is called bounded, if it is, in a certain sense, of finite size.

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Cantor cube

In mathematics, a Cantor cube is a topological group of the form A for some index set A. Its algebraic and topological structures are the group direct product and product topology over the cyclic group of order 2 (which is itself given the discrete topology).

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Cantor set

In mathematics, the Cantor set is a set of points lying on a single line segment that has a number of remarkable and deep properties.

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Cantor space

In mathematics, a Cantor space, named for Georg Cantor, is a topological abstraction of the classical Cantor set: a topological space is a Cantor space if it is homeomorphic to the Cantor set.

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Category of metric spaces

In category-theoretic mathematics, Met is a category that has metric spaces as its objects and metric maps (continuous functions between metric spaces that do not increase any pairwise distance) as its morphisms.

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Cauchy sequence

In mathematics, a Cauchy sequence, named after Augustin-Louis Cauchy, is a sequence whose elements become arbitrarily close to each other as the sequence progresses.

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Clopen set

In topology, a clopen set (a portmanteau of closed-open set) in a topological space is a set which is both open and closed.

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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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Closeness (mathematics)

Closeness is a basic concept in topology and related areas in mathematics.

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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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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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Combinatorial topology

In mathematics, combinatorial topology was an older name for algebraic topology, dating from the time when topological invariants of spaces (for example the Betti numbers) were regarded as derived from combinatorial decompositions of spaces, such as decomposition into simplicial complexes.

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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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Compact-open topology

In mathematics, the compact-open topology is a topology defined on the set of continuous maps between two topological spaces.

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Compactification (mathematics)

In mathematics, in general topology, compactification is the process or result of making a topological space into a compact space.

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Compactly generated space

In topology, a compactly generated space (or k-space) is a topological space whose topology is coherent with the family of all compact subspaces.

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Comparison of topologies

In topology and related areas of mathematics, the set of all possible topologies on a given set forms a partially ordered set.

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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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Cone (topology)

In topology, especially algebraic topology, the cone CX of a topological space X is the quotient space: of the product of X with the unit interval I.

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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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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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Continuum (topology)

In the mathematical field of point-set topology, a continuum (plural: "continua") is a nonempty compact connected metric space, or, less frequently, a compact connected Hausdorff space.

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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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Covering space

In mathematics, more specifically algebraic topology, a covering map (also covering projection) is a continuous function p from a topological space, C, to a topological space, X, such that each point in X has an open neighbourhood evenly covered by p (as shown in the image); the precise definition is given below.

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CW complex

In topology, a CW complex is a type of topological space introduced by J. H. C. Whitehead to meet the needs of homotopy theory.

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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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Descriptive set theory

In mathematical logic, descriptive set theory (DST) is the study of certain classes of "well-behaved" subsets of the real line and other Polish spaces.

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Dimension theory

In mathematics, dimension theory is a branch of general topology dealing with dimensional invariants of topological spaces.

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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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Disjoint union (topology)

In general topology and related areas of mathematics, the disjoint union (also called the direct sum, free union, free sum, topological sum, or coproduct) of a family of topological spaces is a space formed by equipping the disjoint union of the underlying sets with a natural topology called the disjoint union topology.

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Equivalence class

In mathematics, when the elements of some set S have a notion of equivalence (formalized as an equivalence relation) defined on them, then one may naturally split the set S into equivalence classes.

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Extended real number line

In mathematics, the affinely extended real number system is obtained from the real number system by adding two elements: and (read as positive infinity and negative infinity respectively).

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Filter (mathematics)

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

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Finite intersection property

In general topology, a branch of mathematics, a collection A of subsets of a set X is said to have the finite intersection property (FIP) if the intersection over any finite subcollection of A is nonempty.

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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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Fσ set

In mathematics, an Fσ set (said F-sigma set) is a countable union of closed sets.

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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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Germ (mathematics)

In mathematics, the notion of a germ of an object in/on a topological space is an equivalence class of that object and others of the same kind which captures their shared local properties.

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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.

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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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Group action

In mathematics, an action of a group is a formal way of interpreting the manner in which the elements of the group correspond to transformations of some space in a way that preserves the structure of that space.

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Hausdorff distance

In mathematics, the Hausdorff distance, or Hausdorff metric, also called Pompeiu–Hausdorff distance, measures how far two subsets of a metric space are from each other.

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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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Heine–Borel theorem

In real analysis the Heine–Borel theorem, named after Eduard Heine and Émile Borel, states: For a subset S of Euclidean space Rn, the following two statements are equivalent.

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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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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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Homotopy

In topology, two continuous functions from one topological space to another are called homotopic (from Greek ὁμός homós "same, similar" and τόπος tópos "place") if one can be "continuously deformed" into the other, such a deformation being called a homotopy between the two functions.

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Homotopy lifting property

In mathematics, in particular in homotopy theory within algebraic topology, the homotopy lifting property (also known as an instance of the right lifting property or the covering homotopy axiom) is a technical condition on a continuous function from a topological space E to another one, B. It is designed to support the picture of E "above" B by allowing a homotopy taking place in B to be moved "upstairs" to E. For example, a covering map has a property of unique local lifting of paths to a given sheet; the uniqueness is because the fibers of a covering map are discrete spaces.

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Inductive dimension

In the mathematical field of topology, the inductive dimension of a topological space X is either of two values, the small inductive dimension ind(X) or the large inductive dimension Ind(X).

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Intrinsic metric

In the mathematical study of metric spaces, one can consider the arclength of paths in the space.

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Σ-compact space

In mathematics, a topological space is said to be σ-compact if it is the union of countably many compact subspaces.

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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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Kuratowski closure axioms

In topology and related branches of mathematics, the Kuratowski closure axioms are a set of axioms that can be used to define a topological structure on a set.

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Lawson topology

In mathematics and theoretical computer science the Lawson topology, named after J. D. Lawson, is a topology on partially ordered sets used in the study of domain theory.

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Lebesgue covering dimension

In mathematics, the Lebesgue covering dimension or topological dimension of a topological space is one of several different ways of defining the dimension of the space in a topologically invariant way.

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Lebesgue's number lemma

In topology, Lebesgue's number lemma, named after Henri Lebesgue, is a useful tool in the study of compact metric spaces.

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Limit point

In mathematics, a limit point (or cluster point or accumulation point) of a set S in a topological space X is a point x that can be "approximated" by points of S in the sense that every neighbourhood of x with respect to the topology on X also contains a point of S other than x itself.

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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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Lipschitz continuity

In mathematical analysis, Lipschitz continuity, named after Rudolf Lipschitz, is a strong form of uniform continuity for functions.

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List of algebraic topology topics

This is a list of algebraic topology topics, by Wikipedia page.

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List of geometric topology topics

This is a list of geometric topology topics, by Wikipedia page.

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List of topology topics

This is a list of topology topics, by Wikipedia page.

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Local homeomorphism

In mathematics, more specifically topology, a local homeomorphism is a function between topological spaces that, intuitively, preserves local (though not necessarily global) structure.

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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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Locally constant function

In mathematics, a function f from a topological space A to a set B is called locally constant, if for every a in A there exists a neighborhood U of a, such that f is constant on U. Every constant function is locally constant.

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Long line (topology)

In topology, the long line (or Alexandroff line) is a topological space somewhat similar to the real line, but in a certain way "longer".

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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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Manifold

In mathematics, a manifold is a topological space that locally resembles Euclidean space near each point.

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Meagre set

In the mathematical fields of general topology and descriptive set theory, a meagre set (also called a meager set or a set of first category) is a set that, considered as a subset of a (usually larger) topological space, is in a precise sense small or negligible.

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Measure of non-compactness

In functional analysis, two measures of non-compactness are commonly used; these associate numbers to sets in such a way that compact sets all get the measure 0, and other sets get measures that are bigger according to "how far" they are removed from compactness.

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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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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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Nerve of a covering

In topology, the nerve of an open covering is a construction of an abstract simplicial complex from an open covering of a topological space X that captures many of the interesting topological properties in an algorithmic or combinatorial way.

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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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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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Nowhere dense set

In mathematics, a nowhere dense set on a topological space is a set whose closure has empty interior.

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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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Order theory

Order theory is a branch of mathematics which investigates the intuitive notion of order using binary relations.

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P-adic analysis

In mathematics, p-adic analysis is a branch of number theory that deals with the mathematical analysis of functions of ''p''-adic numbers.

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P-adic number

In mathematics, the -adic number system for any prime number extends the ordinary arithmetic of the rational numbers in a different way from the extension of the rational number system to the real and complex number systems.

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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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Path (topology)

In mathematics, a path in a topological space X is a continuous function f from the unit interval I.

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Pointed space

In mathematics, a pointed space is a topological space with a distinguished point, the basepoint.

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Pointwise convergence

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

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Polish space

In the mathematical discipline of general topology, a Polish space is a separable completely metrizable topological space; that is, a space homeomorphic to a complete metric space that has a countable dense subset.

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Polytope

In elementary geometry, a polytope is a geometric object with "flat" sides.

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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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Real tree

In mathematics, real trees (also called \mathbb R-trees) are a class of metric spaces generalising simplicial trees.

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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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Relatively compact subspace

In mathematics, a relatively compact subspace (or relatively compact subset, or precompact subset) Y of a topological space X is a subset whose closure is compact.

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Restricted product

In mathematics, the restricted product is a construction in the theory of topological groups.

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Scott continuity

In mathematics, given two partially ordered sets P and Q, a function f \colon P \rightarrow Q between them is Scott-continuous (named after the mathematician Dana Scott) if it preserves all directed suprema, i.e. if for every directed subset D of P with supremum in P its image has a supremum in Q, and that supremum is the image of the supremum of D: that is, \sqcup f.

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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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Semi-locally simply connected

In mathematics, specifically algebraic topology, semi-locally simply connected is a certain local connectedness condition that arises in the theory of covering spaces.

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Separable space

In mathematics, a topological space is called separable if it contains a countable, dense subset; that is, there exists a sequence \_^ of elements of the space such that every nonempty open subset of the space contains at least one element of the sequence.

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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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Sequential space

In topology and related fields of mathematics, a sequential space is a topological space that satisfies a very weak axiom of countability.

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Sheaf (mathematics)

In mathematics, a sheaf is a tool for systematically tracking locally defined data attached to the open sets of a topological space.

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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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Simplex

In geometry, a simplex (plural: simplexes or simplices) is a generalization of the notion of a triangle or tetrahedron to arbitrary dimensions.

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Simplicial approximation theorem

In mathematics, the simplicial approximation theorem is a foundational result for algebraic topology, guaranteeing that continuous mappings can be (by a slight deformation) approximated by ones that are piecewise of the simplest kind.

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Simplicial complex

In mathematics, a simplicial complex is a set composed of points, line segments, triangles, and their ''n''-dimensional counterparts (see illustration).

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Simply connected space

In topology, a topological space is called simply connected (or 1-connected, or 1-simply connected) if it is path-connected and every path between two points can be continuously transformed (intuitively for embedded spaces, staying within the space) into any other such path while preserving the two endpoints in question.

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Smash product

In mathematics, the smash product of two pointed spaces (i.e. topological spaces with distinguished basepoints) X and Y is the quotient of the product space X × Y under the identifications (x, y0) ∼ (x0, y) for all x ∈ X and y ∈ Y.

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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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Solenoid (mathematics)

In mathematics, a solenoid is a compact connected topological space (i.e. a continuum) that may be obtained as the inverse limit of an inverse system of topological groups and continuous homomorphisms where each Si is a circle and fi is the map that uniformly wraps the circle Si+1 ni times (ni ≥ 2) around the circle Si.

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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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Space-filling curve

In mathematical analysis, a space-filling curve is a curve whose range contains the entire 2-dimensional unit square (or more generally an n-dimensional unit hypercube).

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Specialization (pre)order

In the branch of mathematics known as topology, the specialization (or canonical) preorder is a natural preorder on the set of the points of a topological space.

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Spectral space

In mathematics, a spectral space (sometimes called a coherent space) is a topological space that is homeomorphic to the spectrum of a commutative ring.

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Sperner's lemma

In mathematics, Sperner's lemma is a combinatorial analog of the Brouwer fixed point theorem, which is equivalent to it.

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Stone duality

In mathematics, there is an ample supply of categorical dualities between certain categories of topological spaces and categories of partially ordered sets.

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Stone's representation theorem for Boolean algebras

In mathematics, Stone's representation theorem for Boolean algebras states that every Boolean algebra is isomorphic to a certain field of sets.

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Strong topology (polar topology)

In functional analysis and related areas of mathematics the strong topology is the finest polar topology, the topology with the most open sets, on a dual pair.

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Subbase

In topology, a subbase (or subbasis) for a topological space with topology is a subcollection of that generates, in the sense that is the smallest topology containing.

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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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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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Taxicab geometry

A taxicab geometry is a form of geometry in which the usual distance function or metric of Euclidean geometry is replaced by a new metric in which the distance between two points is the sum of the absolute differences of their Cartesian coordinates.

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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 abelian group

In mathematics, a topological abelian group, or TAG, is a topological group that is also an abelian group.

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Topological algebra

In mathematics, a topological algebra A is an algebra and at the same time a topological space, where the algebraic and the topological structures are coherent in a specified sense.

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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 module

In mathematics, a topological module is a module over a topological ring such that scalar multiplication and addition are continuous.

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Topological property

In topology and related areas of mathematics a topological property or topological invariant is a property of a topological space which is invariant under homeomorphisms.

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Topological ring

In mathematics, a topological ring is a ring R which is also a topological space such that both the addition and the multiplication are continuous as maps where R × R carries the product topology.

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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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Topological tensor product

In mathematics, there are usually many different ways to construct a topological tensor product of two topological vector spaces.

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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.

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Topologist's sine curve

In the branch of mathematics known as topology, the topologist's sine curve or Warsaw sine curve is a topological space with several interesting properties that make it an important textbook example.

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Triangulation

In trigonometry and geometry, triangulation is the process of determining the location of a point by forming triangles to it from known points.

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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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Tychonoff space

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

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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.

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Ultrafilter

In the mathematical field of set theory, an ultrafilter on a given partially ordered set (poset) P is a maximal filter on P, that is, a filter on P that cannot be enlarged.

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Ultrametric space

In mathematics, an ultrametric space is a metric space in which the triangle inequality is strengthened to d(x,z)\leq\max\left\.

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Unicoherent space

In mathematics, a unicoherent space is a topological space X that is connected and in which the following property holds: For any closed, connected A, B \subset X with X.

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Uniform continuity

In mathematics, a function f is uniformly continuous if, roughly speaking, it is possible to guarantee that f(x) and f(y) be as close to each other as we please by requiring only that x and y are sufficiently close to each other; unlike ordinary continuity, the maximum distance between f(x) and f(y) cannot depend on x and y themselves.

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Uniform isomorphism

In the mathematical field of topology a uniform isomorphism or uniform homeomorphism is a special isomorphism between uniform spaces which respects uniform properties.

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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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Uniform property

In the mathematical field of topology a uniform property or uniform invariant is a property of a uniform space which is invariant under uniform isomorphisms.

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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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Uniformly connected space

In topology and related areas of mathematics a uniformly connected space or Cantor connected space is a uniform space U such that every uniformly continuous function from U to a discrete uniform space is constant.

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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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Upper topology

In mathematics, the upper topology on a partially ordered set X is the coarsest topology in which the closure of a singleton \ is the order section a.

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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 topology

In mathematics, weak topology is an alternative term for certain initial topologies, often on topological vector spaces or spaces of linear operators, for instance on a Hilbert space.

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Wedge sum

In topology, the wedge sum is a "one-point union" of a family of topological spaces.

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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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Outline of general topology.

References

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

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