This is a list of

general topology In mathematics, general topology (or point set topology) is the branch of topology that deals with the basic set-theoretic definitions and constructions used in topology. It is the foundation of most other branches of topology, including differ ...

topics.

Basic concepts

Topological space In mathematics, a topological space is, roughly speaking, a Geometry, geometrical space in which Closeness (mathematics), closeness is defined but cannot necessarily be measured by a numeric Distance (mathematics), distance. More specifically, a to ...

Topological property In topology and related areas of mathematics, a topological property or topological invariant is a property of a topological space that is invariant under homeomorphisms. Alternatively, a topological property is a proper class of topological space ...

Open set In mathematics, an open set is a generalization of an Interval (mathematics)#Definitions_and_terminology, open interval in the real line. In a metric space (a Set (mathematics), set with a metric (mathematics), distance defined between every two ...

closed set In geometry, topology, and related branches of mathematics, a closed set is a Set (mathematics), set whose complement (set theory), complement is an open set. In a topological space, a closed set can be defined as a set which contains all its lim ...

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. That this is possible may seem counterintuitive, as the common meanings of and are antonyms, but their mathematical de ...

** Closure ** Boundary **

Density Density (volumetric mass density or specific mass) is the ratio of a substance's mass to its volume. The symbol most often used for density is ''ρ'' (the lower case Greek letter rho), although the Latin letter ''D'' (or ''d'') can also be u ...

** G-delta set, F-sigma set ** Closeness **

Neighborhood A neighbourhood (Commonwealth English) or neighborhood (American English) is a geographically localized community within a larger town, city, suburb or rural area, sometimes consisting of a single street and the buildings lining it. Neigh ...

Continuity (topology) In mathematics, a continuous function is a function such that a small variation of the argument induces a small variation of the value of the function. This implies there are no abrupt changes in value, known as '' discontinuities''. More preci ...

Homeomorphism In mathematics and more specifically in topology, a homeomorphism ( from Greek roots meaning "similar shape", named by Henri Poincaré), also called topological isomorphism, or bicontinuous function, is a bijective and continuous function ...

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. If f : X \to Y is a local homeomorphism, X is said to be an � ...

** Open and closed maps **

Embedding In mathematics, an embedding (or imbedding) is one instance of some mathematical structure contained within another instance, such as a group (mathematics), group that is a subgroup. When some object X is said to be embedded in another object Y ...

Germ Germ or germs may refer to: Science * Germ (microorganism), an informal word for a pathogen * Germ cell, cell that gives rise to the gametes of an organism that reproduces sexually * Germ layer, a primary layer of cells that forms during embry ...

* Basis ** Subbasis *

Open cover In mathematics, and more particularly in set theory, a cover (or covering) of a set X is a family of subsets of X whose union is all of X. More formally, if C = \lbrace U_\alpha : \alpha \in A \rbrace is an indexed family of subsets U_\alpha\su ...

* Locally finite space *

Covering space In topology, a covering or covering projection is a continuous function, map between topological spaces that, intuitively, Local property, locally acts like a Projection (mathematics), projection of multiple copies of a space onto itself. In par ...

Atlas An atlas is a collection of maps; it is typically a bundle of world map, maps of Earth or of a continent or region of Earth. Advances in astronomy have also resulted in atlases of the celestial sphere or of other planets. Atlases have traditio ...

Limits

Limit point In mathematics, a limit point, accumulation point, or cluster 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 contains a point of S other than x itself. A ...

* Net * Filter *

Ultrafilter In the Mathematics, mathematical field of order theory, an ultrafilter on a given partially ordered set (or "poset") P is a certain subset of P, namely a Maximal element, maximal Filter (mathematics), filter on P; that is, a proper filter on P th ...

Topological properties In topology and related areas of mathematics, a topological property or topological invariant is a property of a topological space that is invariant under homeomorphisms. Alternatively, a topological property is a proper class of topological space ...

Baire category theorem The Baire category theorem (BCT) is an important result 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 (a topological space such that th ...

** Nowhere dense **

Baire space In mathematics, a topological space X is said to be a Baire space if countable unions of closed sets with empty interior also have empty interior. According to the Baire category theorem, compact Hausdorff spaces and complete metric spaces are ...

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). The concept of a Banach–Mazur game is closely related to the concept of Baire spa ...

Meagre set In the mathematical Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itse ...

** Comeagre set

Compactness and countability

Compact space In mathematics, specifically general topology, compactness is a property that seeks to generalize the notion of a closed and bounded subset of Euclidean space. The idea is that a compact space has no "punctures" or "missing endpoints", i.e., i ...

** Relatively compact subspace **

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 \mathbb^n, the following two statements are equivalent: *S is compact, that is, every open cover of S has a finite s ...

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. The theorem is named after Andrey Nikolayevich Tikhonov (whose surname sometimes is tra ...

Finite intersection property In general topology, a branch of mathematics, a non-empty family 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 non-empty. It has the strong finite intersect ...

** Compactification ** Measure of non-compactness *

Paracompact space In mathematics, a paracompact space is a topological space in which every open cover has an open refinement that is locally finite. These spaces were introduced by . Every compact space is paracompact. Every paracompact Hausdorff space is normal ...

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. More precisely, it is a topological space in which ...

Compactly generated space In topology, a topological space X is called a compactly generated space or k-space if its topology is determined by compact spaces in a manner made precise below. There is in fact no commonly agreed upon definition for such spaces, as different a ...

Axiom of countability In mathematics, an axiom of countability is a property of certain mathematical objects that asserts the existence of a countable set with certain properties. Without such an axiom, such a set might not provably exist. Important examples Important c ...

Sequential space In topology and related fields of mathematics, a sequential space is a topological space whose topology can be completely characterized by its convergent/divergent sequences. They can be thought of as spaces that satisfy a very weak axiom of count ...

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 (local base ...

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. More explicitly, a topological space T is second-countable if there exists some countable collection \mat ...

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

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'' sub ...

* Sigma-compact space

Connectedness

Connected space In topology and related branches of mathematics, a connected space is a topological space that cannot be represented as the union (set theory), union of two or more disjoint set, disjoint Empty set, non-empty open (topology), open subsets. Conne ...

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 into any other such path while preserving the two endpoi ...

* Path connected space

Separation axioms 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. Some of these restrictions are given by the separation axioms. These are sometimes ...

* T₀ space * T₁ space *

Hausdorff space In topology and related branches of mathematics, a Hausdorff space ( , ), T2 space or separated space, is a topological space where distinct points have disjoint neighbourhoods. Of the many separation axioms that can be imposed on a topologi ...

Completely Hausdorff space Completely may refer to: * ''Completely'' (Diamond Rio album) * ''Completely'' (Christian Bautista album), 2005 * "Completely", a song by American singer and songwriter Michael Bolton * "Completely", a song by Serial Joe from ''(Last Chance) A ...

Regular space In topology and related fields of mathematics, a topological space ''X'' is called a regular space if every closed subset ''C'' of ''X'' and a point ''p'' not contained in ''C'' have non-overlapping open neighborhoods. Thus ''p'' and ''C'' can ...

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. A Tychonoff space is any completely regular space that is also a ...

Normal space Normal(s) or The Normal(s) may refer to: Film and television * Normal (2003 film), ''Normal'' (2003 film), starring Jessica Lange and Tom Wilkinson * Normal (2007 film), ''Normal'' (2007 film), starring Carrie-Anne Moss, Kevin Zegers, Callum Keit ...

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. Section 15. Urysohn's lemma is commonly used to construct contin ...

Tietze extension theorem In topology, the Tietze extension theorem (also known as the Tietze– Urysohn– Brouwer extension theorem or Urysohn-Brouwer lemma) states that any real-valued, continuous function on a closed subset of a normal topological space In mathe ...

Paracompact In mathematics, a paracompact space is a topological space in which every open cover has an open Cover (topology)#Refinement, refinement that is locally finite collection, locally finite. These spaces were introduced by . Every compact space is par ...

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. The notion of when two sets a ...

Topological constructions

Direct sum The direct sum is an operation between structures in abstract algebra, a branch of mathematics. It is defined differently but analogously for different kinds of structures. As an example, the direct sum of two abelian groups A and B is anothe ...

and the dual construction product * Subspace and the dual construction

quotient In arithmetic, a quotient (from 'how many times', pronounced ) is a quantity produced by the division of two numbers. The quotient has widespread use throughout mathematics. It has two definitions: either the integer part of a division (in th ...

* Topological tensor product

Examples

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 , meaning they are '' isolated'' from each other in a certain sense. The discrete topology is the finest to ...

Locally constant function In mathematics, a locally constant function is a function from a topological space into a set with the property that around every point of its domain, there exists some neighborhood of that point on which it restricts to a constant function. ...

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 spaces are commonly called indiscrete, anti-discrete, concrete or codiscrete. Intuitively, this has the conseque ...

* Cofinite topology *

Cocountable topology The cocountable topology, also known as the countable complement topology, is a topology that can be defined on any infinite set X. In this topology, a set is open if its complement in X is either countable or equal to the entire set. Equivalen ...

Finer topology In topology and related areas of mathematics, the set of all possible topologies on a given set forms a partially ordered set. This order relation can be used for comparison of the topologies. Definition A topology on a set may be defined as the ...

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. This topology differs from another, perhaps more natural-seemin ...

Restricted product In mathematics, the restricted product is a construction in the theory of topological groups. Let I be an index set; S a finite subset of I. If G_i is a locally compact group for each i \in I, and K_i \subset G_i is an open compact subgroup for ea ...

* Quotient space *

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. It is often denoted ' (capital letter ). In addition to its role in real analysi ...

* Continuum *

Extended real number line In mathematics, the extended real number system is obtained from the real number system \R by adding two elements denoted +\infty and -\infty that are respectively greater and lower than every real number. This allows for treating the potential ...

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 sense "longer". It behaves locally just like the real line, but has different large-scale properties (e.g., it is neither ...

* Sierpinski space *

Cantor set In mathematics, the Cantor set is a set of points lying on a single line segment that has a number of unintuitive properties. It was discovered in 1874 by Henry John Stephen Smith and mentioned by German mathematician Georg Cantor in 1883. Throu ...

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 cube *

Space-filling curve In mathematical analysis, a space-filling curve is a curve whose Range of a function, range reaches every point in a higher dimensional region, typically the unit square (or more generally an ''n''-dimensional unit hypercube). Because Giuseppe Pea ...

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. It can be defined as the graph of the functi ...

* Tychonoff plank * Comb space *

Uniform norm In mathematical analysis, the uniform norm (or ) assigns, to real- or complex-valued bounded functions defined on a set , the non-negative number :\, f\, _\infty = \, f\, _ = \sup\left\. This norm is also called the , the , the , or, when t ...

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. The term is most commonly used for the initial topology of a ...

* Strong topology *

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. Furthermore, many interesting topological spaces can be embedded in the Hilbert cube; that is, ca ...

Lower limit topology In mathematics, the lower limit topology or right half-open interval topology is a topology defined on \mathbb, the set of real numbers; it is different from the standard topology on \mathbb (generated by the open intervals) and has a number of in ...

Sorgenfrey plane In topology, the Sorgenfrey plane is a frequently-cited counterexample to many otherwise plausible-sounding conjectures. It consists of the product of two copies of the Sorgenfrey line, which is the real line \mathbb under the half-open inter ...

* Real tree *

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 ...

Zariski topology In algebraic geometry and commutative algebra, the Zariski topology is a topology defined on geometric objects called varieties. It is very different from topologies that are commonly used in real or complex analysis; in particular, it is not ...

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 (mathematics), set. They are equivalent to the more commonly used open set definition. The ...

* Unicoherent *

Solenoid (mathematics) :''This page discusses a class of topological groups. For the wrapped loop of wire, see Solenoid.'' 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 ...

Uniform space In the mathematical field of topology, a uniform space is a topological space, set with additional mathematical structure, structure that is used to define ''uniform property, uniform properties'', such as complete space, completeness, uniform con ...
s

Uniform continuity In mathematics, a real function f of real numbers is said to be uniformly continuous if there is a positive real number \delta such that function values over any function domain interval of the size \delta are as close to each other as we want. In ...

Lipschitz continuity In mathematical analysis, Lipschitz continuity, named after Germany, German mathematician Rudolf Lipschitz, is a strong form of uniform continuity for function (mathematics), functions. Intuitively, a Lipschitz continuous function is limited in h ...

* Uniform isomorphism * Uniform property * Uniformly connected space

Metric space In mathematics, a metric space is a Set (mathematics), set together with a notion of ''distance'' between its Element (mathematics), elements, usually called point (geometry), points. The distance is measured by a function (mathematics), functi ...
s

Metric topology In mathematics, a metric space is a set together with a notion of ''distance'' between its elements, usually called points. The distance is measured by a function called a metric or distance function. Metric spaces are a general setting for ...

Manhattan distance Taxicab geometry or Manhattan geometry is geometry where the familiar Euclidean distance is ignored, and the distance between two point (geometry), points is instead defined to be the sum of the absolute differences of their respective Cartesian ...

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\ for all x, y, and z. Sometimes the associated metric is also called a non-Archimedean metric or super-metric. Formal d ...

P-adic numbers In number theory, given a prime number , the -adic numbers form an extension of the rational numbers which is distinct from the real numbers, though with some similar properties; -adic numbers can be written in a form similar to (possibly infin ...

p-adic analysis In mathematics, ''p''-adic analysis is a branch of number theory that studies functions of ''p''-adic numbers. Along with the more classical fields of real and complex analysis, which deal, respectively, with functions on the real and complex ...

Open ball In mathematics, a ball is the solid figure bounded by a ''sphere''; it is also called a solid sphere. It may be a closed ball (including the boundary points that constitute the sphere) or an open ball (excluding them). These concepts are defin ...

* Bounded subset *

Pointwise convergence In mathematics, pointwise convergence is one of Modes of convergence (annotated index), various senses in which a sequence of function (mathematics), functions can Limit (mathematics), converge to a particular function. It is weaker than uniform co ...

* Metrization theorems *

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

Cauchy sequence In mathematics, a Cauchy sequence is a sequence whose elements become arbitrarily close to each other as the sequence progresses. More precisely, given any small positive distance, all excluding a finite number of elements of the sequence are le ...

Banach fixed-point theorem In mathematics, the Banach fixed-point theorem (also known as the contraction mapping theorem or contractive mapping theorem or Banach–Caccioppoli theorem) is an important tool in the theory of metric spaces; it guarantees the existence and uniqu ...

Polish space In the mathematical discipline of general topology, a Polish space is a separable space, separable Completely metrizable space, completely metrizable topological space; that is, a space homeomorphic to a Complete space, complete metric space that h ...

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 set, non-empty compact space, compact subsets o ...

Intrinsic metric In the mathematical study of metric spaces, one can consider the arclength of paths in the space. If two points are at a given distance from each other, it is natural to expect that one should be able to get from the first point to the second a ...

Category of metric spaces In category theory, 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. This is a category because the composition of ...

Topology and
order theory Order theory is a branch of mathematics that investigates the 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 intr ...

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 ...

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. The theorem is fundamental to the deeper understanding of Boolean algebra that emerged in the first ha ...

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. For most spaces that are considered in practice, namely for all those that satisfy the ...

Sober space In mathematics, a sober space is a topological space ''X'' such that every (nonempty) irreducible space, irreducible closed subset of ''X'' is the closure (topology), closure of exactly one point of ''X'': that is, every nonempty irreducible close ...

Spectral space In mathematics, a spectral space is a topological space that is homeomorphic to the spectrum of a commutative ring. It is sometimes also called a coherent space because of the connection to coherent topoi. Definition Let ''X'' be a topological ...

Alexandrov topology In general topology, an Alexandrov topology is a topology in which the intersection of an ''arbitrary'' family of open sets is open (while the definition of a topology only requires this for a ''finite'' family). Equivalently, an Alexandrov top ...

* Upper topology *

Scott topology In mathematics, given two partially ordered sets ''P'' and ''Q'', a function ''f'': ''P'' → ''Q'' between them is Scott-continuous (named after the mathematician Dana Scott) if it preserves all directed suprema. That is, for every directed sub ...

Scott continuity In mathematics, given two partially ordered sets ''P'' and ''Q'', a function ''f'': ''P'' → ''Q'' between them is Scott-continuous (named after the mathematician Dana Scott) if it preserves all directed suprema. That is, for every directed su ...

* Lawson topology

Descriptive set theory In mathematical logic, descriptive set theory (DST) is the study of certain classes of "well-behaved" set (mathematics), subsets of the real line and other Polish spaces. As well as being one of the primary areas of research in set theory, it has a ...

Polish Space In the mathematical discipline of general topology, a Polish space is a separable space, separable Completely metrizable space, completely metrizable topological space; that is, a space homeomorphic to a Complete space, complete metric space that h ...

Dimension theory In physics and mathematics, the dimension of a mathematical space (or object) is informally defined as the minimum number of coordinates needed to specify any point within it. Thus, a line has a dimension of one (1D) because only one coord ...

* Inductive dimension *

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. Informal discussion For ordinary Euclidean ...

* Lebesgue's number lemma

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 a ...

Polytope In elementary geometry, a polytope is a geometric object with flat sides ('' faces''). Polytopes are the generalization of three-dimensional polyhedra to any number of dimensions. Polytopes may exist in any general number of dimensions as an ...

Simplex In geometry, a simplex (plural: simplexes or simplices) is a generalization of the notion of a triangle or tetrahedron to arbitrary dimensions. The simplex is so-named because it represents the simplest possible polytope in any given dimension. ...

Simplicial complex In mathematics, a simplicial complex is a structured Set (mathematics), set composed of Point (geometry), points, line segments, triangles, and their ''n''-dimensional counterparts, called Simplex, simplices, such that all the faces and intersec ...

CW complex In mathematics, and specifically in topology, a CW complex (also cellular complex or cell complex) is a topological space that is built by gluing together topological balls (so-called ''cells'') of different dimensions in specific ways. It generali ...

Manifold In mathematics, a manifold is a topological space that locally resembles Euclidean space near each point. More precisely, an n-dimensional manifold, or ''n-manifold'' for short, is a topological space with the property that each point has a N ...

Triangulation In trigonometry and geometry, triangulation is the process of determining the location of a point by forming triangles to the point from known points. Applications In surveying Specifically in surveying, triangulation involves only angle m ...

Barycentric subdivision In mathematics, the barycentric subdivision is a standard way to subdivide a given simplex into smaller ones. Its extension to simplicial complexes is a canonical method to refining them. Therefore, the barycentric subdivision is an important tool ...

Sperner's lemma In mathematics, Sperner's lemma is a combinatorial result on colorings of triangulations, analogous to the Brouwer fixed point theorem, which is equivalent to it. It states that every Sperner coloring (described below) of a triangulation of an ...

* Simplicial approximation theorem *

Nerve of an open covering In topology, the nerve complex of a set family is an abstract simplicial complex, abstract complex that records the pattern of intersections between the sets in the family. It was introduced by Pavel Alexandrov and now has many variants and genera ...

Foundations of
algebraic topology Algebraic topology is a branch of mathematics that uses tools from abstract algebra to study topological spaces. The basic goal is to find algebraic invariant (mathematics), invariants that classification theorem, classify topological spaces up t ...

Simply connected In topology, a topological space is called simply connected (or 1-connected, or 1-simply connected) if it is path-connected and every Path (topology), path between two points can be continuously transformed into any other such path while preserving ...

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. Roughly speaking, a topological space ''X'' is semi-locally simply connected i ...

Path (topology) In mathematics, a path in a topological space X is a continuous function from a closed interval into X. Paths play an important role in the fields of topology and mathematical analysis. For example, a topological space for which there exists ...

Homotopy In topology, two continuous functions from one topological space to another are called homotopic (from and ) if one can be "continuously deformed" into the other, such a deformation being called a homotopy ( ; ) between the two functions. ...

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 fr ...

Pointed space In mathematics, a pointed space or based space is a topological space with a distinguished point, the basepoint. The distinguished point is just simply one particular point, picked out from the space, and given a name, such as x_0, that remains u ...

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 (i.e. topological spaces with distinguished basepoints x_0 and y_0) the wedge sum of ''X'' and ''Y'' is the ...

Smash product In topology, a branch of mathematics, the smash product of two pointed spaces (i.e. topological spaces with distinguished basepoints) and is the quotient of the product space under the identifications for all in and in . The smash prod ...

Cone (topology) In topology, especially algebraic topology, the cone of a topological space X is intuitively obtained by stretching ''X'' into a cylinder and then collapsing one of its end faces to a point. The cone of X is denoted by CX or by \operatorname(X). ...

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. Specifically, let X and Y be topological spaces, and let A be a subspace of Y. Let f : A ...

Topology and algebra

* Topological algebra *

Topological group In mathematics, topological groups are the combination of groups and topological spaces, i.e. they are groups and topological spaces at the same time, such that the continuity condition for the group operations connects these two structures ...

Topological ring In mathematics, a topological ring is a ring R that is also a topological space such that both the addition and the multiplication are continuous as maps: R \times R \to R where R \times R carries the product topology. That means R is an additive ...

Topological vector space In mathematics, a topological vector space (also called a linear topological space and commonly abbreviated TVS or t.v.s.) is one of the basic structures investigated in functional analysis. A topological vector space is a vector space that is als ...

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. An abelian topological ...

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 (algebra), group and a topological space, the group operations are Continuous (topology), continuous, and the g ...

* Properly discontinuous * Sheaf space