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- 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

Γ-space

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

Fibration

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

Pseudomanifold

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

Pseudocircle

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

K-topology

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

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