Basic concepts

• Topological space
  Topological space
  Topological spaces are mathematical structures that allow the formal definition of concepts such as convergence, connectedness, and continuity. They appear in virtually every branch of modern mathematics and are a central unifying notion...
  
  
• Topological property
  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. That is, a property of spaces is a topological property if whenever a space X possesses that property every space homeomorphic...
  
  
• Open set
  Open set
  The concept of an open set is fundamental to many areas of mathematics, especially point-set topology and metric topology. Intuitively speaking, a set U is open if any point x in U can be "moved" a small amount in any direction and still be in the set U...
  
  , closed set
  Closed set
  In geometry, topology, and related branches of mathematics, a closed set is a set whose complement is an open set. In a topological space, a closed set can be defined as a set which contains all its limit points...
  
  
  • Clopen set
    Clopen set
    In topology, a clopen set in a topological space is a set which is both open and closed. That this is possible for a set is not as counter-intuitive as it might seem if the terms open and closed were thought of as antonyms; in fact they are not...
    
    
  • Closure (topology)
    Closure (topology)
    In mathematics, the closure of a subset S in a topological space consists of all points in S plus the limit points of S. Intuitively, these are all the points that are "near" S. A point which is in the closure of S is a point of closure of S...
    
    
  • Boundary (topology)
    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...
    
    
  • Dense (topology)
  • G-delta set
    G-delta 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. The notation originated in Germany with G for Gebiet meaning open set in this case and δ for Durchschnitt .The term inner limiting set is also used...
    
    , F-sigma set
    F-sigma set
    In mathematics, an Fσ set is a countable union of closed sets. The notation originated in France with F for fermé and σ for somme ....
    
    
  • closeness (mathematics)
    Closeness (mathematics)
    In topology and related areas in mathematics closeness is one of the basic concepts in a topological space. Intuitively we say two sets are close if they are arbitrarily near to each other...
    
    
  • neighbourhood (mathematics)
    Neighbourhood (mathematics)
    In topology and related areas of mathematics, a neighbourhood is one of the basic concepts in a topological space. Intuitively speaking, a neighbourhood of a point is a set containing the point where you can move that point some amount without leaving the set.This concept is closely related to the...
    
    
• Continuity (topology)
  • Homeomorphism
    Homeomorphism
    In the mathematical field of topology, a homeomorphism or topological isomorphism or bicontinuous function is a continuous function between topological spaces that has a continuous inverse function. Homeomorphisms are the isomorphisms in the category of topological spaces—that is, they are...
    
    
  • Local homeomorphism
    Local homeomorphism
    In mathematics, more specifically topology, a local homeomorphism is intuitively a function, f, between topological spaces that preserves local structure. Equivalently, one can cover the domain of this function by open sets, such that f restricted to each such open set is a homeomorphism onto its...
    
    
  • Open and closed maps
    Open and closed maps
    In topology, an open map is a function between two topological spaces which maps open sets to open sets. That is, a function f : X → Y is open if for any open set U in X, the image f is open in Y. Likewise, a closed map is a function which maps closed sets to closed sets...
    
    
  • Germ (mathematics)
    Germ (mathematics)
    In mathematics, the notion of a germ of an object in/on a topological space captures the local properties of the object. In particular, the objects in question are mostly functions and subsets...
    
    
• Base (topology)
  Base (topology)
  In mathematics, a base 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 say that the base generates the topology T...
  
  , subbase
  Subbase
  In topology, a subbase for a topological space X with topology T is a subcollection B of T which generates T, in the sense that T is the smallest topology containing B...
  
  
• Open cover
• Covering space
• Atlas (topology)
  Atlas (topology)
  In mathematics, particularly topology, one describesa manifold using an atlas. An atlas consists of individualcharts that, roughly speaking, describe individual regionsof the manifold. If the manifold is the surface of the Earth,...
  
  

Topological properties

• Baire category theorem
  Baire category theorem
  The Baire category theorem is an important tool in general topology and functional analysis. The theorem has two forms, each of which gives sufficient conditions for a topological space to be a Baire space....
  
  
  • Nowhere dense
  • Baire space
    Baire space
    In mathematics, a Baire space is a topological space which, intuitively speaking, is very large and has "enough" points for certain limit processes. It is named in honor of René-Louis Baire who introduced the concept.- Motivation :...
    
    
  • Banach-Mazur game
  • Meagre set
    Meagre set
    In the mathematical fields of general topology and descriptive set theory, a meagre set is a set that, considered as a subset of a topological space, is in a precise sense small or negligible...
    
    
  • Comeagre set

Compactness and countability

• Compact space
  Compact space
  In mathematics, specifically general topology and metric topology, a compact space is an abstract mathematical space whose topology has the compactness property, which has many important implications not valid in general spaces...
  
  
  • Relatively compact subspace
    Relatively compact subspace
    In mathematics, a relatively compact subspace Y of a topological space X is a subset whose closure is compact....
    
    
  • Heine-Borel theorem
  • Tychonoff's theorem
    Tychonoff's theorem
    In mathematics, Tychonoff's theorem states that the product of any collection of compact topological spaces is compact. The theorem is named after Andrey Nikolayevich Tychonoff, who proved it first in 1930 for powers of the closed unit interval and in 1935 stated the full theorem along with the...
    
    
  • Finite intersection property
    Finite intersection property
    In general topology, a branch of mathematics, the finite intersection property is a property of a collection of subsets of a set X. A collection has this property if the intersection over any finite subcollection of the collection is nonempty....
    
    
  • Compactification
    Compactification
    Compactification may refer to:* Compactification , making a topological space compact* Compactification , the "curling up" of extra dimensions in string theory* Compaction...
    
    
  • Measure of non-compactness
    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.The underlying idea is the...
    
    
• Paracompact space
  Paracompact space
  In mathematics, a paracompact space is a topological space in which every open cover admits a locally finite open refinement. Paracompact spaces are sometimes also required to be Hausdorff. Paracompact spaces were introduced by ....
  
  
• Locally compact space
  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.-Formal definition:...
  
  
• Compactly generated space
  Compactly generated space
  In topology, a compactly generated space is a topological space whose topology is coherent with the family of all compact subspaces. Specifically, a topological space X is compactly generated if it satisfies the following condition:Equivalently, one can replace closed with open in this definition...
  
  
• Axiom of countability
  Axiom of countability
  In mathematics, an axiom of countability is a property of certain mathematical objects that requires the existence of a countable set with certain properties, while without it such sets might not exist....
  
  
• Sequential space
  Sequential space
  In topology and related fields of mathematics, a sequential space is a topological space that satisfies a very weak axiom of countability. Sequential spaces are the most general class of spaces for which sequences suffice to determine the topology....
  
  
• First-countable space
  First-countable space
  In topology, a branch of mathematics, a first-countable space is a topological space satisfying the "first axiom of countability". Specifically, a space X is said to be first-countable if each point has a countable neighbourhood basis...
  
  
• Second-countable space
  Second-countable space
  In topology, a second-countable space, also called a completely separable space, is a topological space satisfying the second axiom of countability. A space is said to be second-countable if its topology has a countable base...
  
  
• Separable space
• Lindelöf space
  Lindelöf space
  In mathematics, a Lindelöf space is a topological space in which every open cover has a countable subcover. The Lindelöf property is a weakening of the more commonly used notion of compactness, which requires the existence of a finite subcover....
  
  
• Sigma-compact space

Separation axioms

• T0 space
• T1 space
  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 an open neighborhood not containing the other. An R0 space is one in which this holds for every pair of topologically distinguishable points...
  
  
• Hausdorff space
  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. Of the many separation axioms that can be imposed on a topological space, the "Hausdorff condition" is the most frequently...
  
  
  • Completely Hausdorff space
    Completely Hausdorff space
    In topology, an Urysohn space, or T2½ space, is a topological space in which any two distinct points can be separated by closed neighborhoods. A completely Hausdorff space, or functionally Hausdorff space, is a topological space in which any two distinct points can be separated by a continuous...
    
    
• Regular space
  Regular space
  In topology and related fields of mathematics, a topological space X is called a regular space if every non-empty closed subset C of X and a point p not contained in C admit non-overlapping open neighborhoods. Thus p and C...
  
  
• Tychonoff space
  Tychonoff space
  In topology and related branches of mathematics, Tychonoff spaces and completely regular spaces are kinds of topological spaces.These conditions are examples of separation axioms....
  
  
• Normal space
  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. A normal Hausdorff space is also called a T4 space...
  
  
• Urysohn's lemma
  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 function....
  
  
• Tietze extension theorem
• Paracompact
• Separated sets
  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....
  
  

Topological constructions

• direct sum and the dual construction product
• subspace and the dual construction quotient
• Topological tensor product
  Topological tensor product
  In mathematics, there are usually many different ways to construct a topological tensor product of two topological vector spaces. For Hilbert spaces or nuclear spaces there is a simple well-behaved theory of tensor products , but for general Banach spaces or locally convex topological vector space...
  
  

Examples

• Discrete space
  Discrete space
  In topology, a discrete space is a particularly simple example of a topological space or similar structure, one in which the points are "isolated" from each other in a certain sense.- Definitions :Given a set X:...
  
  
  • Locally constant function
    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....
    
    
• Trivial topology
  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. Such a space is sometimes called an indiscrete space, and its topology sometimes called an indiscrete topology...
  
  
• Cofinite topology
• Finer topology
• Product topology
  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...
  
  
  • Restricted product
• Quotient space
  Quotient space
  In topology and related areas of mathematics, a quotient space is, intuitively speaking, the result of identifying or "gluing together" certain points of a given space. The points to be identified are specified by an equivalence relation...
  
  
• Unit interval
  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...
  
  
• Continuum (topology)
  Continuum (topology)
  In the mathematical field of point-set topology, a continuum is a nonempty compact connected metric space, or less frequently, a compact connected Hausdorff topological space...
  
  
• Extended real number line
  Extended real number line
  In mathematics, the affinely extended real number system is obtained from the real number system R by adding two elements: +∞ and −∞ . The projective extended real number system adds a single object, ∞ and makes no distinction between "positive" or "negative" infinity...
  
  
• Long line (topology)
  Long line (topology)
  In topology, the long line is a topological space somewhat similar to the real line, but in a certain way "longer". It behaves locally just like the real line, but has different large-scale properties. Therefore it serves as one of the basic counterexamples of topology...
  
  
• Sierpinski space
  Sierpinski space
  In mathematics, the Sierpiński space is a finite topological space with two points, only one of which is closed.It is the smallest example of a topological space which is neither trivial nor discrete...
  
  
• Cantor set
  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. It was discovered in 1875 by Henry John Stephen Smith and introduced by German mathematician Georg Cantor in 1883....
  
  , Cantor space
  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. In set theory, the topological space 2ω is called "the" Cantor space...
  
  , Cantor cube
• Space-filling curve
  Space-filling curve
  In mathematical analysis, a space-filling curve is a curve whose range contains the entire 2-dimensional unit square...
  
  
• Topologist's sine curve
  Topologist's sine curve
  In the branch of mathematics known as topology, the topologist's sine curve is a topological space with several interesting properties that make it an important textbook example....
  
  
• Uniform norm
• Weak topology
  Weak topology
  In mathematics, weak topology is an alternative term for initial topology. The term is most commonly used for the initial topology of a topological vector space with respect to its continuous dual...
  
  
• Strong topology
  Strong topology
  In mathematics, a strong topology is a topology which is stronger than some other "default" topology. This term is used to describe different topologies depending on context, and it may refer to:* the final topology on the disjoint union...
  
  
• Hilbert cube
  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...
  
  
• Lower limit topology
  Lower limit topology
  In mathematics, the lower limit topology or right half-open interval topology is a topology defined on the set R of real numbers; it is different from the standard topology on R and has a number of interesting properties...
  
  
• Sorgenfrey plane
• Real tree
  Real tree
  A real tree, or an \mathbb R-tree, is a metric space such thatfor any x, y in M there is a unique arc from x to y and this arc is a geodesic segment. Here by an arc from x to y we mean the image in M of a topological embedding f from an interval [a,b] to M such that f=x and f=y...
  
  
• Compact-open topology
  Compact-open topology
  In mathematics, the compact-open topology is a topology defined on the set of continuous maps between two topological spaces. The compact-open topology is one of the commonly-used topologies on function spaces, and is applied in homotopy theory and functional analysis...
  
  
• Zariski topology
  Zariski topology
  In algebraic geometry, the Zariski topology is a particular topology chosen for algebraic varieties that reflects the algebraic nature of their definition. It is due to Oscar Zariski and took a place of particular importance in the field around 1950...
  
  
• Kuratowski closure axioms
  Kuratowski closure axioms
  In topology and related branches of mathematics, the Kuratowski closure axioms are a set of axioms which can be used to define a topological structure on a set. They are equivalent to the more commonly used open set definition...
  
  
• Unicoherent
  Unicoherent
  A topological space X is said to be unicoherent if it is connected and the following property holds:For any closed, connected A, B \subset X with X=A \cup B, the intersection A \cap B is connected....
  
  
• Solenoid (mathematics)
  Solenoid (mathematics)
  In mathematics, a solenoid is a compact connected topological space that may be obtained as the inverse limit of an inverse system of topological groups and continuous homomorphisms...
  
  

Uniform space

Uniform space

In the mathematical field of topology, a uniform space is a set with a uniform structure. Uniform spaces are topological spaces with additional structure which is used to define uniform properties such as completeness, uniform continuity and uniform convergence.The conceptual difference between...

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• Uniform continuity
  Uniform continuity
  In mathematics, a function f is uniformly continuous if, roughly speaking, it is possible to guarantee that f and f 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 x and y cannot...
  
  
• Lipschitz continuity
  Lipschitz continuity
  In mathematical analysis, Lipschitz continuity, named after Rudolf Lipschitz, is a strong form of uniform continuity for functions. Intuitively, a Lipschitz continuous function is limited in how fast it can change: for every pair of points on the graph of this function, the absolute value of the...
  
  
• Uniform isomorphism
  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.-Definition:...
  
  
• Uniform property
  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....
  
  
• Uniformly connected space
  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....
  
  

Metric space

Metric space

In mathematics, a metric space is a set where a notion of distance between elements of the set is defined.The metric space which most closely corresponds to our intuitive understanding of space is the 3-dimensional Euclidean space...

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• Metric topology
• Manhattan distance
• Ultrametric space
  • P-adic numbers, p-adic analysis
    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....
    
    
• Open ball
• Bounded subset
• Pointwise convergence
  Pointwise convergence
  In mathematics, pointwise convergence is one of various senses in which a sequence of functions can converge to a particular function.-Definition:...
  
  
• Metrization theorems
• Complete space
  Complete space
  In mathematical analysis, a metric space M is called complete 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....
  
  
  • Cauchy sequence
    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...
    
    
  • Banach fixed point theorem
    Banach fixed point theorem
    In mathematics, the Banach fixed-point theorem 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...
    
    
• Polish space
  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. Polish spaces are so named because they were first extensively studied by Polish...
  
  
• Hausdorff distance
  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. It turns the set of non-empty compact subsets of a metric space into a metric space in its own right...
  
  
• Intrinsic metric
  Intrinsic metric
  In the mathematical study of metric spaces, one can consider the arclength of paths in the space. If two points are a given distance from each other, it is natural to expect that one should be able to get from one point to another along a path whose arclength is equal to that distance...
  
  
• Category of metric spaces
  Category of metric spaces
  In category-theoretic mathematics, Met is a category that has metric spaces as its objects and metric maps as its morphisms. This is a category because the composition of two metric maps is again a metric map...
  
  

Topology and order theory

Order theory

Order theory is a branch of mathematics which investigates our intuitive notion of order using binary relations. It provides a formal framework for describing statements such as "this is less than that" or "this precedes that". This article introduces the field and gives some basic definitions...

 

• Stone duality
  Stone duality
  In mathematics, there is an ample supply of categorical dualities between certain categories of topological spaces and categories of partially ordered sets. Today, these dualities are usually collected under the label Stone duality, since they form a natural generalization of Stone's representation...
  
  
  • Stone's representation theorem for Boolean algebras
    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 field of sets. The theorem is fundamental to the deeper understanding of Boolean algebra that emerged in the first half of the 20th century. The theorem was first proved by Stone...
    
    
• Specialization (pre)order
  Specialization (pre)order
  In the branch of mathematics known as topology, the specialization preorder is a natural preorder on the set of the points of a topological space. For most spaces that are considered in practice, namely for all those that satisfy the T0 separation axiom, this preorder is even a partial order...
  
  
• Sober space
  Sober space
  In mathematics, a sober space is a topological spacesuch that every irreducible closed subset of X is the closure of exactly one point of X: that is, has a unique generic point.-Properties and examples :...
  
  
• Spectral space
  Spectral space
  In mathematics, a spectral space is a topological space which is homeomorphic to the spectrum of a commutative ring.-Definition:Let X be a topological space and let K\circ be the set of allquasi-compact and open subsets of X...
  
  
• Alexandrov topology
  Alexandrov topology
  In topology, an Alexandrov space is a topological space in which the intersection of any family of open sets is open. It is an axiom of topology that the intersection of any finite family of open sets is open...
  
  
• Upper topology
• Scott topology
  • Scott continuity
    Scott continuity
    In mathematics, given two partially ordered sets P and Q a function f : P \rightarrow Q between them is Scott-continuous if it preserves all directed suprema, i.e...
    
    
• Lawson topology
  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. The lower topology on a poset P is generated by the subbasis consisting of all complements of principal filters on P...
  
  

Dimension theory

Dimension theory

In mathematics, dimension theory is a branch of general topology dealing with dimensional invariants of topological spaces.-See also:*Lebesgue covering dimension*Inductive dimensions *Dimension...

• Inductive dimension
  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 or the large inductive dimension Ind...
  
  
• Lebesgue covering dimension
  Lebesgue covering dimension
  Lebesgue covering dimension or topological dimension is one of several inequivalent notions of assigning a topological invariant dimension to a given topological space.-Definition:...
  
  
• Lebesgue's number lemma
  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. It states:Such a number δ is called a Lebesgue number of this cover. The notion of a Lebesgue number itself is useful in other applications as well.-References:-External links:*...
  
  

Combinatorial topology

Combinatorial topology

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

• Polytope
  Polytope
  In elementary geometry, a polytope is a geometric object with flat sides, which exists in any general number of dimensions. A polygon is a polytope in two dimensions, a polyhedron in three dimensions, and so on in higher dimensions...
  
  
• Simplex
  Simplex
  In geometry, a simplex is a generalization of the notion of a triangle or tetrahedron to arbitrary dimension. Specifically, an n-simplex is an n-dimensional polytope which is the convex hull of its n + 1 vertices. For example, a 2-simplex is a triangle, a 3-simplex is a tetrahedron,...
  
  
• Simplicial complex
  Simplicial complex
  In mathematics, a simplicial complex is a topological space of a certain kind, constructed by "gluing together" points, line segments, triangles, and their n-dimensional counterparts...
  
  
• CW complex
  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. This class of spaces is broader and has some better categorical properties than simplicial complexes, but still retains a combinatorial naturethat allows for...
  
  
• Manifold
  Manifold
  In mathematics , a manifold is a topological space that on a small enough scale resembles the Euclidean space of a specific dimension, called the dimension of the manifold....
  
  
• Triangulation
  Triangulation
  In trigonometry and geometry, triangulation is the process of determining the location of a point by measuring angles to it from known points at either end of a fixed baseline, rather than measuring distances to the point directly...
  
  
• Barycentric subdivision
  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.The name...
  
  
• Sperner's lemma
  Sperner's lemma
  In mathematics, Sperner's lemma is a combinatorial analog of the Brouwer fixed point theorem, which follows from it. Sperner's lemma states that every Sperner coloring of a triangulation of an n-dimensional simplex contains a cell colored with a complete set of colors...
  
  
• Simplicial approximation theorem
  Simplicial approximation theorem
  In mathematics, the simplicial approximation theorem is a foundational result for algebraic topology, guaranteeing that continuous mappings can be approximated by ones that are piecewise of the simplest kind. It applies to mappings between spaces that are built up from simplices — that is,...
  
  
• Nerve of an open covering
  Nerve of an open covering
  In mathematics, the nerve of an open covering is a construction in topology, of an abstract simplicial complex from an open covering of a topological space X.The notion of nerve was introduced by Pavel Alexandrov....
  
  

Foundations of algebraic topology

Algebraic topology

Algebraic topology is a branch of mathematics which uses tools from abstract algebra to study topological spaces. The basic goal is to find algebraic invariants that classify topological spaces up to homeomorphism, though usually most classify up to homotopy equivalence.Although algebraic topology...

• Simply connected
• Semi-locally simply connected
  Semi-locally simply connected
  In mathematics, specifically algebraic topology, the phrase semi-locally simply connected refers to a certain local connectedness condition that arises in the theory of covering spaces. Roughly speaking, a topological space X is semi-locally simply connected if there is a lower bound on the sizes...
  
  
• Path (topology)
  Path (topology)
  In mathematics, a path in a topological space X is a continuous map f from the unit interval I = [0,1] to XThe initial point of the path is f and the terminal point is f. One often speaks of a "path from x to y" where x and y are the initial and terminal points of the path...
  
  
• Homotopy
  Homotopy
  In topology, two continuous functions from one topological space to another are called homotopic if one can be "continuously deformed" into the other, such a deformation being called a homotopy between the two functions...
  
  
• Homotopy lifting property
  Homotopy lifting property
  In mathematics, in particular in homotopy theory within algebraic topology, the homotopy lifting property is a technical condition on a continuous function from a topological space E to another one, B...
  
  
• Pointed space
  Pointed space
  In mathematics, a pointed space is a topological space X with a distinguished basepoint x0 in X. Maps of pointed spaces are continuous maps preserving basepoints, i.e. a continuous map f : X → Y such that f = y0...
  
  
• Wedge sum
  Wedge sum
  In topology, the wedge sum is a "one-point union" of a family of topological spaces. Specifically, if X and Y are pointed spaces the wedge sum of X and Y is the quotient of the disjoint union of X and Y by the identification x0 ∼ y0:X\vee Y = \;/ \sim,\,where ∼ is the...
  
  
• smash product
  Smash product
  In mathematics, the smash product of two pointed spaces X and Y is the quotient of the product space X × Y under the identifications  ∼  for all x ∈ X and y ∈ Y. The smash product is usually denoted X ∧ Y...
  
  
• Cone (topology)
  Cone (topology)
  In topology, especially algebraic topology, the cone CX of a topological space X is the quotient space:CX = /\,of the product of X with the unit interval I = [0, 1]....
  
  
• Adjunction space
  Adjunction space
  In mathematics, an adjunction space is a common construction in topology where one topological space is attached or "glued" onto another. Specifically, let X and Y be a topological spaces with A a subspace of Y. Let f : A → X be a continuous map...
  
  

Topology and algebra

• Topological algebra
  Topological algebra
  In mathematics, a topological algebra A over a topological field K is a topological vector space together with a continuous multiplication\cdot :A\times A \longrightarrow A\longmapsto a\cdot bthat makes it an algebra over K...
  
  
• Topological group
  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. A topological group is a mathematical object with both an algebraic structure and a...
  
  
• Topological ring
  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 mapswhere R × R carries the product topology.- General comments :...
  
  
• Topological vector space
  Topological vector space
  In mathematics, a topological vector space is one of the basic structures investigated in functional analysis...
  
  
• Topological module
  Topological module
  In mathematics, a topological module is a module over a topological ring such that scalar multiplication and addition are continuous.- Examples :A topological vector space is a topological module over a topological field....
  
  
• Topological abelian group
  Topological abelian group
  In mathematics, a topological abelian group, or TAG, is a topological group that is also an abelian group.That is, a TAG is both a group and a topological space, the group operations are continuous, and the group's binary operation is commutative....
  
  
• Properly discontinuous
  Properly discontinuous
  In topology and related branches of mathematics, an action of a group G on a topological space X is called proper if the map from G×X to X×X taking to is proper, and is called properly discontinuous if in addition G is discrete...
  
  
• Sheaf space

See also

• Topology glossary
  Topology glossary
  This is a glossary of some terms used in the branch of mathematics known as topology. Although there is no absolute distinction between different areas of topology, the focus here is on general topology...
  
  
• List of topology topics
• List of geometric topology topics
• List of algebraic topology topics
• Publications in topology