Category: Topological spaces

Haefliger structure
In mathematics, a Haefliger structure on a topological space is a generalization of a foliation of a manifold, introduced by André Haefliger in 1970. Any foliation on a manifold induces a special kind
Order topology
In mathematics, an order topology is a certain topology that can be defined on any totally ordered set. It is a natural generalization of the topology of the real numbers to arbitrary totally ordered
Simplicial space
In mathematics, a simplicial space is a simplicial object in the category of topological spaces. In other words, it is a contravariant functor from the simplex category Δ to the category of topologica
Lower limit topology
In mathematics, the lower limit topology or right half-open interval topology is a topology defined on the set of real numbers; it is different from the standard topology on (generated by the open int
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 topologi
Sierpiński space
In mathematics, the Sierpiński space (or the connected two-point set) is a finite topological space with two points, only one of which is closed.It is the smallest example of a topological space which
Homotopy sphere
In algebraic topology, a branch of mathematics, a homotopy sphere is an n-manifold that is homotopy equivalent to the n-sphere. It thus has the same homotopy groups and the same homology groups as the
Knaster–Kuratowski fan
In topology, a branch of mathematics, the Knaster–Kuratowski fan (named after Polish mathematicians Bronisław Knaster and Kazimierz Kuratowski) is a specific connected topological space with the prope
In mathematics, a -space is a topological space that satisfies a certain a basic selection principle. An infinite cover of a topological space is an -cover if every finite subset of this space is cont
List of examples in general topology
This is a list of useful examples in general topology, a field of mathematics. * Alexandrov topology * Cantor space * Co-kappa topology * Cocountable topology * Cofinite topology * Compact-open
Moore plane
In mathematics, the Moore plane, also sometimes called Niemytzki plane (or Nemytskii plane, Nemytskii's tangent disk topology), is a topological space. It is a completely regular Hausdorff space (also
Split interval
In topology, the split interval, or double arrow space, is a topological space that results from splitting each point in a closed interval into two adjacent points and giving the resulting ordered set
Overlapping interval topology
In mathematics, the overlapping interval topology is a topology which is used to illustrate various topological principles.
Appert topology
In general topology, a branch of mathematics, the Appert topology, named for Antoine Appert, is a topology on the set X = {1, 2, 3, ...} of positive integers.In the Appert topology, the open sets are
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 r
Euclidean space
Euclidean space is the fundamental space of geometry, intended to represent physical space. Originally, that is, in Euclid's Elements, it was the three-dimensional space of Euclidean geometry, but in
Infinite broom
In topology, a branch of mathematics, the infinite broom is a subset of the Euclidean plane that is used as an example distinguishing various notions of connectedness. The closed infinite broom is the
Arithmetic progression topologies
In general topology and number theory, branches of mathematics, one can define various topologies on the set of integers or the set of positive integers by taking as a base a suitable collection of ar
Partition topology
In mathematics, the partition topology is a topology that can be induced on any set X by partitioning X into disjoint subsets P; these subsets form the basis for the topology. There are two important
The notion of a fibration generalizes the notion of a fiber bundle and plays an important role in algebraic topology, a branch of mathematics. Fibrations are used, for example, in postnikov-systems or
Excluded point topology
In mathematics, the excluded point topology is a topology where exclusion of a particular point defines openness. Formally, let X be any non-empty set and p ∈ X. The collection of subsets of X is then
Equivariant topology
In mathematics, equivariant topology is the study of topological spaces that possess certain symmetries. In studying topological spaces, one often considers continuous maps , and while equivariant top
Half-disk topology
In mathematics, and particularly general topology, the half-disk topology is an example of a topology given to the set , given by all points in the plane such that . The set can be termed the closed u
Priestley space
In mathematics, a Priestley space is an ordered topological space with special properties. Priestley spaces are named after Hilary Priestley who introduced and investigated them. Priestley spaces play
Simplicial complex
In mathematics, a simplicial complex is a set composed of points, line segments, triangles, and their n-dimensional counterparts (see illustration). Simplicial complexes should not be confused with th
Dogbone space
In geometric topology, the dogbone space, constructed by R. H. Bing, is a quotient space of three-dimensional Euclidean space such that all inverse images of points are points or , yet it is not homeo
Fixed-point space
In mathematics, a Hausdorff space X is called a fixed-point space if every continuous function has a fixed point. For example, any closed interval [a,b] in is a fixed point space, and it can be proved
Box topology
In topology, the cartesian product of topological spaces can be given several different topologies. One of the more obvious choices is the box topology, where a base is given by the Cartesian products
Linear flow on the torus
In mathematics, especially in the area of mathematical analysis known as dynamical systems theory, a linear flow on the torus is a flow on the n-dimensional torus which is represented by the following
Loop space
In topology, a branch of mathematics, the loop space ΩX of a pointed topological space X is the space of (based) loops in X, i.e. continuous pointed maps from the pointed circle S1 to X, equipped with
Metric space
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. Met
Geometric topology (object)
In mathematics, the geometric topology is a topology one can put on the set H of hyperbolic 3-manifolds of finite volume.
Rose (topology)
In mathematics, a rose (also known as a bouquet of n circles) is a topological space obtained by gluing together a collection of circles along a single point. The circles of the rose are called petals
Lexicographic order topology on the unit square
In general topology, the lexicographic ordering on the unit square (sometimes the dictionary order on the unit square) is a topology on the unit square S, i.e. on the set of points (x,y) in the plane
Cantor tree
In mathematical set theory, the Cantor tree is either the full binary tree of height ω + 1, or a topological space related to this by joining its points with intervals, that was introduced by Robert L
In mathematics, a pseudomanifold is a special type of topological space. It looks like a manifold at most of its points, but it may contain singularities. For example, the cone of solutions of forms a
Arens square
In mathematics, the Arens square is a topological space, named for Richard Friederich Arens. Its role is mainly to serve as a counterexample.
Locally constant sheaf
In algebraic topology, a locally constant sheaf on a topological space X is a sheaf on X such that for each x in X, there is an open neighborhood U of x such that the restriction is a constant sheaf o
Discontinuous group
A discontinuous group is a mathematical concept relating to mappings in topological space.
Number line
In elementary mathematics, a number line is a picture of a graduated straight line that serves as visual representation of the real numbers. Every point of a number line is assumed to correspond to a
First uncountable ordinal
In mathematics, the first uncountable ordinal, traditionally denoted by or sometimes by , is the smallest ordinal number that, considered as a set, is uncountable. It is the supremum (least upper boun
Erdős space
In mathematics, Erdős space is a topological space named after Paul Erdős, who described it in 1940. Erdős space is defined as a subspace of the Hilbert space of square summable sequences, consisting
Topological monoid
In topology, a branch of mathematics, a topological monoid is a monoid object in the category of topological spaces. In other words, it is a monoid with a topology with respect to which the monoid's b
Shrinking space
In mathematics, in the field of topology, a topological space is said to be a shrinking space if every open cover admits a shrinking. A shrinking of an open cover is another open cover indexed by the
Discrete two-point space
In topology, a branch of mathematics, a discrete two-point space is the simplest example of a totally disconnected discrete space. The points can be denoted by the symbols 0 and 1.
Comb space
In mathematics, particularly topology, a comb space is a particular subspace of that resembles a comb. The comb space has properties that serve as a number of counterexamples. The topologist's sine cu
Discrete space
In topology, a discrete space is a particularly simple example of a topological space or similar structure, one in which the points form a discontinuous sequence, meaning they are isolated from each o
Alexandroff plank
Alexandroff plank in topology, an area of mathematics, is a topological space that serves as an instructive example.
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
Prüfer manifold
In mathematics, the Prüfer manifold or Prüfer surface is a 2-dimensional Hausdorff real analytic manifold that is not paracompact. It was introduced by and named after Heinz Prüfer.
Dunce hat (topology)
In topology, the dunce hat is a compact topological space formed by taking a solid triangle and gluing all three sides together, with the orientation of one side reversed. Simply gluing two sides orie
Arens–Fort space
In mathematics, the Arens–Fort space is a special example in the theory of topological spaces, named for Richard Friederich Arens and M. K. Fort, Jr.
Sierpiński triangle
The Sierpiński triangle (sometimes spelled Sierpinski), also called the Sierpiński gasket or Sierpiński sieve, is a fractal attractive fixed set with the overall shape of an equilateral triangle, subd
CW complex
A CW complex (also called cellular complex or cell complex) is a kind of a topological space that is particularly important in algebraic topology. It was introduced by J. H. C. Whitehead to meet the n
Long line (topology)
In topology, the long line (or Alexandroff line) is a topological space somewhat similar to the real line, but in a certain way "longer". It behaves locally just like the real line, but has different
Particular point topology
In mathematics, the particular point topology (or included point topology) is a topology where a set is open if it contains a particular point of the topological space. Formally, let X be any non-empt
The pseudocircle is the finite topological space X consisting of four distinct points {a,b,c,d } with the following non-Hausdorff topology: . This topology corresponds to the partial order where open
Smith–Volterra–Cantor set
In mathematics, the Smith–Volterra–Cantor set (SVC), fat Cantor set, or ε-Cantor set is an example of a set of points on the real line that is nowhere dense (in particular it contains no intervals), y
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,
Cubical complex
In mathematics, a cubical complex (also called cubical set and Cartesian complex) is a set composed of points, line segments, squares, cubes, and their n-dimensional counterparts. They are used analog
Infinite loop space machine
In topology, a branch of mathematics, given a topological monoid X up to homotopy (in a nice way), an infinite loop space machine produces a group completion of X together with infinite loop space str
Hyperbolic space
In mathematics, hyperbolic space of dimension n is the unique simply connected, n-dimensional Riemannian manifold of constant sectional curvature equal to -1. It is homogeneous, and satisfies the stro
Compact convergence
In mathematics compact convergence (or uniform convergence on compact sets) is a type of convergence that generalizes the idea of uniform convergence. It is associated with the compact-open topology.
Extension topology
In topology, a branch of mathematics, an extension topology is a topology placed on the disjoint union of a topological space and another set. There are various types of extension topology, described
Nilpotent space
In topology, a branch of mathematics, a nilpotent space, first defined by Emmanuel Dror (1969), is a based topological space X such that * the fundamental group is a nilpotent group; * acts nilpoten
Cosmic space
In mathematics, particularly topology, a cosmic space is any topological space that is a continuous image of some separable metric space. Equivalently (for regular T1 spaces but not in general), a spa
Partially ordered space
In mathematics, a partially ordered space (or pospace) is a topological space equipped with a closed partial order , i.e. a partial order whose graph is a closed subset of . From pospaces, one can def
Divisor topology
In mathematics, more specifically general topology, the divisor topology is a specific topology on the set of positive integers greater than or equal to two. The divisor topology is the poset topology
Homology sphere
In algebraic topology, a homology sphere is an n-manifold X having the homology groups of an n-sphere, for some integer . That is, and for all other i. Therefore X is a connected space, with one non-z
Tychonoff plank
In topology, the Tychonoff plank is a topological space defined using ordinal spaces that is a counterexample to several plausible-sounding conjectures. It is defined as the topological product of the
Menger sponge
In mathematics, the Menger sponge (also known as the Menger cube, Menger universal curve, Sierpinski cube, or Sierpinski sponge) is a fractal curve. It is a three-dimensional generalization of the one
Pointwise convergence
In mathematics, pointwise convergence is one of various senses in which a sequence of functions can converge to a particular function. It is weaker than uniform convergence, to which it is often compa
Sierpiński carpet
The Sierpiński carpet is a plane fractal first described by Wacław Sierpiński in 1916. The carpet is a generalization of the Cantor set to two dimensions; another is Cantor dust. The technique of subd
Mosely snowflake
The Mosely snowflake (after Jeannine Mosely) is a Sierpiński–Menger type of fractal obtained in two variants either by the operation opposite to creating the Sierpiński-Menger snowflake or Cantor dust
Graph (topology)
In topology, a branch of mathematics, a graph is a topological space which arises from a usual graph by replacing vertices by points and each edge by a copy of the unit interval , where is identified
Topological space
In mathematics, a topological space is, roughly speaking, a geometrical space in which closeness is defined but cannot necessarily be measured by a numeric distance. More specifically, a topological s
Hedgehog space
In mathematics, a hedgehog space is a topological space consisting of a set of spines joined at a point. For any cardinal number , the -hedgehog space is formed by taking the disjoint union of real un
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
Ran space
In mathematics, the Ran space (or Ran's space) of a topological space X is a topological space whose underlying set is the set of all nonempty finite subsets of X: for a metric space X the topology is
In mathematics, particularly topology, the K-topology is a topology that one can impose on the set of all real numbers which has some interesting properties. Relative to the set of all real numbers ca
Projectively extended real line
In real analysis, the projectively extended real line (also called the one-point compactification of the real line), is the extension of the set of the real numbers, by a point denoted ∞. It is thus t
Fort space
In mathematics, there are a few topological spaces named after M. K. Fort, Jr.
Klein bottle
In topology, a branch of mathematics, the Klein bottle (/ˈklaɪn/) is an example of a non-orientable surface; it is a two-dimensional manifold against which a system for determining a normal vector can
Finite topological space
In mathematics, a finite topological space is a topological space for which the underlying point set is finite. That is, it is a topological space which has only finitely many elements. Finite topolog
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.
S and L spaces
In mathematics, S-space is a regular topological space that is hereditarily separable but is not a Lindelöf space. L-space is a regular topological space that is hereditarily Lindelöf but not separabl
Baire space (set theory)
In set theory, the Baire space is the set of all infinite sequences of natural numbers with a certain topology. This space is commonly used in descriptive set theory, to the extent that its elements a
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 o
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 r
Interlocking interval topology
In mathematics, and especially general topology, the interlocking interval topology is an example of a topology on the set S := R+ \ Z+, i.e. the set of all positive real numbers that are not positive
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 introduced by Ge
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
Hawaiian earring
In mathematics, the Hawaiian earring is the topological space defined by the union of circles in the Euclidean plane with center and radius for endowed with the subspace topology: The space is homeomo