|
This is a list of general topology topics, by Wikipedia page. See also: In mathematics, general topology or point set topology is that branch of topology which studies elementary properties of topological spaces and structures defined on them. ...
This is a glossary of some terms used in the branch of mathematics known as topology. ...
This is a list of geometric topology topics, by Wikipedia page, organized roughly by dimension. ...
This is a list of algebraic topology topics, by Wikipedia page. ...
Basic concepts
Topological spaces are structures that allow one to formalize concepts such as convergence, connectedness and continuity. ...
In topology and related fields of mathematics, a set U is called open if, intuitively speaking, you can wiggle or change any point x in U by a small amount in any direction and still be inside U. In other words, if x is surrounded only by elements of U...
In topology and related branches of mathematics, a closed set is a set whose complement is open. ...
In topology, a clopen set (or closed-open set) in a topological space is a set which is both open and closed. ...
In mathematics, the closure of a set S consists of all points which are intuitively close to 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. ...
For a different notion of boundary related to manifolds, see that article. ...
The word Boundary has a variety of meanings. ...
In mathematics, the term dense has at least three different meanings. ...
In topology, X, a G-delta set (or Gδ set) is a countable intersection of open sets. ...
In mathematics, an Fσ set (said F-sigma set) is a countable union of closed sets. ...
In topology and related areas in mathematics closeness is one of the basic concepts in a topological space. ...
In topology and related areas of mathematics, a neighbourhood (or neighborhood) is one of the basic concepts in a topological space. ...
In topology, a continuous function is generally defined as one for which preimages of open sets are open. ...
This word should not be confused with homomorphism. ...
In topology, a local homeomorphism is a map from one topological space to another that respects locally the topological structure of the two spaces. ...
In topology, an open map is a function between two topological spaces which maps open sets to open sets. ...
In mathematics, a germ is an equivalence class of continuous functions from one topological space to another (often from the real line to itself), in which one point x0 in the domain has been singled out as privileged. ...
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 say that the base generates the topology T. Bases...
In topology, a subbase (or subbasis) 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. A slightly different definition is used by some authors, and there are other useful equivalent formulations...
In mathematics, a cover of a set X is a collection of subsets C of X whose union is X. In symbols, if C = {Uα : α ∈ A} is an indexed family of subsets of X, then C is a cover if More generally, if Y is a subset of X...
In topology, an atlas describes how a complicated space is glued together from simpler pieces. ...
In mathematics, the concept of a limit is used to describe the behavior of a function, as its argument gets close to either some point, or infinity; or the behavior of a sequences elements, as their index approaches infinity. ...
In mathematics, informally speaking, a limit point (or cluster point) of a set S in a topological space X is a point x in X that can be approximated by points of S other than x as well as one pleases. ...
In mathematics the term net has at least two meanings. ...
In mathematics, a filter is a special subset of a partially ordered set. ...
In mathematics, especially in order theory, an ultrafilter is a subset of a partially ordered set (a poset) which is maximal among all proper filters. ...
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. ...
In mathematics, the Baire category theorem is an important tool in the study of complete spaces, such as Banach spaces and Hilbert spaces, that arise in topology and functional analysis. ...
In topology, a subset A of a topological space X is called nowhere dense if the interior of the closure of A is empty. ...
In topology and related branches of mathematics, a Baire space is a topological space in which, intuitively, there are enough points for certain limit processes. ...
In mathematics, in particular in general topology and set theory, a Banach-Mazur game is a game played between two players, trying to pin down elements in a set (space). ...
Compactness and countability In mathematics, a compact space is a space that resembles a closed and bounded subset of Euclidean space Rn in that it is small in a certain sense and contains all its limit points. The modern general definition calls a topological space compact if every open cover of it has...
In mathematics, a relatively compact subspace (or relatively compact subset) Y of a topological space X is a subset whose closure is compact. ...
In mathematical analysis, the Heine-Borel theorem states: A subset of the real numbers R is compact iff it is closed and bounded. ...
In mathematics, Tychonoffs theorem states that the product of any collection of compact topological spaces is compact. ...
In topology, 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. ...
In mathematics, compactification is applied to topological spaces to make them compact spaces. ...
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. ...
In mathematics, a paracompact space is a topological space in which every open cover admits an open locally finite refinement. ...
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. ...
In topology, a compactly generated space is a topological space X satisfying the following condition: a subspace A is closed in X if and only if A ∩ K is closed in K for all compact subspaces K ⊆ X. Equivalently, one can replace closed with open in this definition. ...
In mathematics, an axiom of countability is a property of certain mathematical objects (usually in a category) that requires the existence of a countable set with certain properties, while without it such sets might not exist. ...
In topology, 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 local base. ...
In topology, a second-countable space is a topological space satisfying the second axiom of countability. Specifically, a space is said to be second-countable if its topology has a countable base. ...
In topology and related areas of mathematics a topological space is called separable if it contains a countable dense subset; that is, a set with a countable number of elements whose closure is the entire space. ...
In mathematics, a Lindelöf space is a topological space in which every open cover has a countable subcover. ...
In topology, a σ-compact space is a topological space that is the union of countably many compact subsets. ...
Connectedness In topology and related branches of mathematics, a topological space X is said to be disconnected if it is the union of two disjoint nonempty open sets. ...
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. ...
In topology and related branches of mathematics, the T0 spaces or Kolmogorov spaces form a broad class of well behaved topological spaces. ...
The title given to this article is incorrect due to technical limitations. ...
In topology and related branches of mathematics, a Hausdorff space is a topological space in which points can be separated by neighbourhoods. ...
In topology and related fields of mathematics, a completely Hausdorff space is a type of Hausdorff space satisfying a slightly stronger separation axiom. ...
In topology and related fields of mathematics, regular spaces and T3 spaces are particularly nice kinds of topological spaces. ...
In topology and related branches of mathematics, Tychonoff spaces and completely regular spaces are particularly nice kinds of topological spaces. ...
In topology and related branches of mathematics, normal spaces, T4 spaces, and T5 spaces are particularly nice kinds of topological spaces. ...
Urysohns lemma in topology states that if X is a normal topological space and A and B are disjoint closed subsets of X, then there exists a continuous function from X into the unit interval [0, 1], f : X → [0, 1], such that f(a) = 0 for all a...
The Tietze extension theorem in topology states that, if X is a normal topological space and f : A → R is a continuous map from a closed subset A of X into the real numbers carrying the standard topology, then there exists a continuous map F : X → R with F(a...
In mathematics, a paracompact space is a topological space in which every open cover admits an open locally finite refinement. ...
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 In topology and related areas of mathematics, the disjoint union (also called the direct sum, free union, 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. ...
Topological spaces are structures that allow one to formalize concepts such as convergence, connectedness and continuity. ...
For quotient spaces in linear algebra, see quotient space (linear algebra). ...
Examples See also: List of examples in general topology. This is a list of useful examples in general topology, a field of mathematics. ...
In topology and related fields of mathematics, 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. ...
In mathematics, a function f from a topological space A to a set B is called locally constant, iff 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. ...
In topology, a topological space with the trivial topology is one where the only open sets are the empty set and the entire space. ...
In mathematics, a cofinite subset of a set X is a subset Y whose complement in X is a finite set. ...
In mathematics, the possible topologies on a given set X form a partially ordered set: if a collection τ1 of subsets of X contains each subset in a collection τ2, and these are both topologies on X, we say that τ1 is a finer (alt. ...
In topology, the cartesian product of topological spaces is turned into a topological space in the following way. ...
The restricted product is a construction in the theory of topological groups. ...
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 space. ...
In mathematics, the unit interval is the interval [0,1], that is the set of all real numbers x such that zero is less than or equal to x and x is less than or equal to one. ...
In mathematics, the word continuum sometimes denotes the real line. ...
The extended real number line is obtained from the real number line R by adding two elements: +∞ and −∞ (which are not considered to be real numbers). ...
In topology, the long line is a topological space analogous to the real line, but much longer. ...
In topology, Sierpiński space S is the simplest example of a topological space that does not satisfy the T1 axiom. ...
The Cantor set, introduced by German mathematician Georg Cantor, is a remarkable construction involving only the real numbers between zero and one. ...
In mathematics, the term Cantor space is sometimes used to denote the topological abstraction of the classical Cantor set: A topological space is a Cantor space if it is homeomorphic to the Cantor set. ...
Space-filling curves or Peano curves are curves, first described by Giuseppe Peano, whose ranges contain the entire 2-dimensional unit square (or the 3-dimensional unit cube). ...
In the branch of mathematics known as topology, the topologists sine curve is an example that has several interesting properties. ...
In mathematical analysis, the uniform norm assigns to real- or complex-valued functions f the nonnegative number This norm is also called the supremum norm or the Chebyshev norm. ...
In mathematics, the weak topology on a set, with respect to a collection of functions from that set into topological spaces, is the weakest (that is, smallest or coarsest) topology on the set which makes all the functions continuous. ...
In mathematics, a strong topology is a topology which is stronger than some other default topology. ...
In mathematics, the Hilbert cube is a topological space that provides an instructive example of some ideas in 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. ...
In topology, the Sorgenfrey plane is a frequently-cited counterexample to many otherwise plausible-sounding conjectures. ...
A real tree or R-tree is a metric space (M,d) such that for any x, y in M there is a unique arc from x to y, i. ...
In mathematics, the compact-open topology is a topology defined on the set of continuous maps between two topological spaces. ...
This article needs to be cleaned up to conform to a higher standard of quality. ...
In topology and related branches of mathematics, the Kuratowski closure axioms is a set of axioms that allows one to define a topology on a set. ...
A topological space is said to be unicoherent if it is connected and the following property holds: For any partition of into closed sets , the intersection is connected. ...
In mathematics, p-adic solenoid is the inverse limit of the inverse system (Si, q i) (i runs over natural numbers), where each Si is a circle, and q i wraps the circle Si+1 p times around the circle Si. ...
In topology, one defines uniform spaces in order to study concepts such as uniform continuity, completeness and uniform convergence. ...
In mathematical analysis, a function f(x) is called uniformly continuous if, roughly speaking, small changes in the input x effect small changes in the output f(x) (continuity), and furthermore the size of the changes in f(x) depends only on the size of the changes in x but...
In mathematics, a function f : M → N between metric spaces M and N is called Lipschitz continuous (or is said to satisfy a Lipschitz condition) if there exists a constant K > 0 such that d(f(x), f(y)) ≤ K d(x, y) for all x and y in M...
In the mathematical field of topology a uniform isomorphism or uniform homeomorphism is a special isomorphism between uniform spaces which respects uniform properties. ...
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. ...
In topology and related areas of mathematics a uniformly connected space or Cantor connected space is a uniform space U so that every uniformly continuous functions from U to a discrete uniform space is constant. ...
In mathematics, a metric space is a set where a notion of distance between elements of the set is defined. ...
In mathematics, a metric space is a set (or space) where a distance between points is defined. ...
Taxicab geometry, considered by Hermann Minkowski in the 19th century, is a form of geometry in which the usual 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 coordinates. ...
In mathematics, an ultrametric space is a special kind of metric space. ...
The title given to this article is incorrect due to technical limitations. ...
P-adic analysis (p-adic analysis) is a branch of mathematics that deals with functions of p-adic numbers. ...
A synonym for ball (in geometry or topology, and in any dimension) is disk (or disc Geometry In metric geometry, a ball is a set containing all points within a specified distance of a given point. ...
In mathematical analysis and related areas of mathematics, a set is called bounded, if it is, in a certain sense, of finite size. ...
Suppose { fn } is a sequence of functions sharing the same domain in common (for the moment, we defer making precise the nature of the values of these functions, but the reader may take them to be real numbers if that makes anyone feel good). ...
A metrizable space is a topological space that is homeomorphic to a metric space. ...
In mathematical analysis, a metric space M is said to be complete (or Cauchy) if every Cauchy sequence of points in M has a limit that is also in M. Intuitively, a space is complete if it doesnt have any holes, if there arent any points missing. For...
In mathematical analysis, a Cauchy sequence, named after Augustin Cauchy, is a sequence whose elements become close as the sequence progresses. ...
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. ...
This is a glossary of some terms used in the branch of mathematics known as topology. ...
Hausdorff distance measures how far two compact subsets of a metric space are from each other. ...
If two objects are at a distance one mile from each other, it should be possible to construct a road of length one mile between them. ...
The category Met has metric spaces as objects and short maps as morphisms. ...
Order theory is a branch of mathematics that studies various kinds of binary relations that capture the intuitive notion of a mathematical ordering. ...
In mathematics, especially in topology and order theory, there is an ample supply of categorical dualities between certain categories of topological spaces and categories of partially ordered sets. ...
In mathematics, Stones representation theorem for Boolean algebras, named in honor of Marshall H. Stone, is the duality between the category of Boolean algebras and the category of Stone spaces, i. ...
In the branch of mathematics known as topology the specialization (or canonical) preorder defines a preorder on the set of the points of a topological space. ...
In mathematics, particularly in topology, a topological space X is sober if every irreducible closed subset of X is the closure of exactly one singleton of X. An irreducible closed subset of X is defined to be a nonempty closed subset of X which is not the union of two...
In mathematics, a topological space X is said to be spectral if 1) X is compact and T0; 2) The set C(X) of all compact-open subsets of (X,Ω) is a sublattice of Ω and a base for the topology; 3) X is sober, that is any nonempty...
In general topology the open sets of a topological space satisfy by definition the conditions: The union of arbitrarily many open sets is open. ...
In mathematics, the upper topology is the topology defined on a preordered set, in which the open sets are the up-sets. ...
This is a glossary of some terms used in various branches of mathematics that are related to the fields of order, lattice, and domain theory. ...
A monotone function f : P → Q between posets P and Q is Scott-continuous if, for every directed set D that has a supremum sup D in P, the set {fx | x in D} has the supremum f(sup D) in Q. Stated differently, a Scott-continuous function is one...
In mathematics, descriptive set theory is the study of certain classes of well-behaved sets of real numbers, e. ...
See also main article dimension Dimension (from Latin measured out) is, in essence, the number of degrees of freedom available for movement in a space. ...
Dimension (from Latin measured out) is, in essence, the number of degrees of freedom available for movement in a space. ...
In mathematics, the Lebesgue covering dimension of a topological space is defined to be the minimum value of n, such that any open cover has a refinement with no point included in more than n+1 elements. ...
In topology, Lebesgues number lemma states If the metric space X is compact and an open cover of X is given, then there exists a number δ > 0 such that every subset of X of diameter < δ is contained in some member of the cover. ...
Topological algebra In mathematics, a topological group G is a group that is also a topological space such that the group multiplication G × G → G and the inverse operation G → G are continuous maps. ...
In mathematics, a topological abelian group, or TAG, is a topological group that is also an abelian group. ...
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 R × R → R, where R × R carries the product topology. ...
In mathematics, a topological vector space X is a real or complex vector space which is endowed with a Hausdorff topology such that vector addition X × X → X and scalar multiplication K × X → X are continuous (where the product topologies are used and the base field K carries its standard...
In topology and related branches of mathematics, an action of a group G on a topological space X is called properly discontinuous if every element of X has a neighborhood that moves outside itself under the action of any group element but the trivial element. ...
In mathematics, a sheaf F on a given topological space X gives a set or richer structure F(U) for each open set U of X. The structures F(U) are compatible with the operations of restricting the open set to smaller subsets and gluing smaller open sets to obtain...
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 such as simplicial complexes. ...
In geometry polytope means, first, the generalization to any dimension of polygon in two dimensions, and polyhedron in three dimensions. ...
In geometry, a simplex or n-simplex is an n-dimensional analogue of a triangle. ...
In mathematics, a simplicial complex is a topological space of a particular kind, built up of points, line segments, triangles, and their n-dimensional counterparts. ...
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 is the current mathematics collaboration of the week! Please help improve it to featured article standard at manifold/rewrite. ...
The ship wants to know the distance d to the shore. ...
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. ...
In combinatorial mathematics, Sperners lemma states that every Sperner coloring of a triangulation of an n-dimensional simplex contains a cell colored with a complete set of colors. ...
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. ...
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. Given an index set I, and open sets Ui contained in X, the nerve N is the set of finite subsets of I...
|
Feeds |
Contact