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In topology
Topology

Topology is a major area of mathematics that has emerged through the development of concepts from geometry and set theory, such as those of space, dimension, shape, transformation and others....
 and related areas of mathematics
Mathematics

Mathematics is the study of quantity, structure, space, change, and related topics of pattern and form. Mathematicians seek out patterns whether found in numbers, space, natural science, computers, imaginary abstractions, or elsewhere....
, a quotient space (also called an identification space) is, intuitively speaking, the result of identifying or "gluing together" certain points of a given space. The points to be identified are specified by an equivalence relation
Equivalence relation

In mathematics, an equivalence relation is, loosely, a binary relation on a Set that specifies how to split up the set into subsets such that every element of the larger set is in exactly one of the subsets....
. This is commonly done in order to construct new spaces from given ones.

Definition
Suppose X is a topological space
Topological space

Topological spaces are mathematical structures that allow the formal definition of concepts such as convergence, connected space, and Continuous function ....
 and ~ is an equivalence relation
Equivalence relation

In mathematics, an equivalence relation is, loosely, a binary relation on a Set that specifies how to split up the set into subsets such that every element of the larger set is in exactly one of the subsets....
 on X.






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Encyclopedia


In topology
Topology

Topology is a major area of mathematics that has emerged through the development of concepts from geometry and set theory, such as those of space, dimension, shape, transformation and others....
 and related areas of mathematics
Mathematics

Mathematics is the study of quantity, structure, space, change, and related topics of pattern and form. Mathematicians seek out patterns whether found in numbers, space, natural science, computers, imaginary abstractions, or elsewhere....
, a quotient space (also called an identification space) is, intuitively speaking, the result of identifying or "gluing together" certain points of a given space. The points to be identified are specified by an equivalence relation
Equivalence relation

In mathematics, an equivalence relation is, loosely, a binary relation on a Set that specifies how to split up the set into subsets such that every element of the larger set is in exactly one of the subsets....
. This is commonly done in order to construct new spaces from given ones.

Definition


Suppose X is a topological space
Topological space

Topological spaces are mathematical structures that allow the formal definition of concepts such as convergence, connected space, and Continuous function ....
 and ~ is an equivalence relation
Equivalence relation

In mathematics, an equivalence relation is, loosely, a binary relation on a Set that specifies how to split up the set into subsets such that every element of the larger set is in exactly one of the subsets....
 on X. We define a topology on the quotient set X/~ (the set consisting of all equivalence class
Equivalence class

In mathematics, given a Set X and an equivalence relation ~ on X, the equivalence class of an element a in X is the subset of all elements in X which are equivalent to a:...
es of ~) as follows: a set of equivalence classes in X/~ is open
Open set

In metric topology and related fields of mathematics, a Set U is called open if, intuitively speaking, starting from any point x in U one can move by a small amount in any direction and still be in the set U....
 if and only if their union
Union (set theory)

In set theory, the term Union refers to a set operation used in the convergence of set elements to form a resultant set containing the elements of both sets....
 is open in X. This is the quotient topology on the quotient set X/~.

Equivalently, the quotient topology can be characterized in the following manner: Let q : X → X/~ be the projection map which sends each element of X to its equivalence class. Then the quotient topology on X/~ is the finest topology for which q is continuous
Continuous function (topology)

In topology and related areas of mathematics a continuous function is a morphism between topological spaces. Intuitively, this is a function f where a set of points near f always contain the of a set of points near x....
.

Given a surjective map f : X → Y from a topological space X to a set Y we can define the quotient topology on Y as the finest topology for which f is continuous. This is equivalent to saying that a subset V ⊆ Y is open in Y if and only if its preimage f−1(V) is open in X. The map f induces an equivalence relation on X by saying x1~x2 if and only if
If and only if

If and only if, in logic and fields that rely on it such as mathematics and philosophy, is a biconditional logical connective between statements....
 f(x1) = f(x2). The quotient space X/~ is then homeomorphic to Y (with its quotient topology) via the homeomorphism which sends the equivalence class of x to f(x).

In general, a surjective, continuous map f : X → Y is said to be a quotient map if Y has the quotient topology determined by f.

Examples

  • Gluing. Often, topologists talk of gluing points together. If X is a topological space and points are to be "glued", then what is meant is that we are to consider the quotient space obtained from the equivalence relation a ~ b if and only if a = b or a = x, b = y (or a = y, b = x). The two points are henceforth interpreted as one point.
  • Consider the unit square I2 = [0,1]×[0,1] and the equivalence relation ~ generated by the requirement that all boundary points be equivalent, thus identifying all boundary points to a single equivalence class. Then I2/~ is homeomorphic to the unit sphere
    Sphere

    A sphere is a symmetrical geometrical object. In non-mathematical usage, the term is used to refer either to a round ball or to its two-dimensional surface....
     S2.
  • Adjunction space
    Adjunction space

    In mathematics, an adjunction space is a common construction in topology where one topological space is attached or "glued" onto another. Specifically, let X and Y be a topological spaces with A a subspace of Y....
    . More generally, suppose X is a space and A is a subspace of X. One can identify all points in A to a single equivalence class and leave points outside of A equivalent only to themselves. The resulting quotient space is denoted X/A. The 2-sphere is then homeomorphic to the unit disc with its boundary identified to a single point: D2/∂D2.
  • Consider the set X = R of all real number
    Real number

    In mathematics, the real numbers may be described informally in several different ways. The real numbers include both rational numbers, such as 42 and −23/129, and irrational numbers, such as pi and the square root of two; or, a real number can be given by an infinite decimal representation, such as 2.4871773339...., where the digits co...
    s with the ordinary topology, and write x ~ y if and only if
    If and only if

    If and only if, in logic and fields that rely on it such as mathematics and philosophy, is a biconditional logical connective between statements....
     x−y is an integer
    Integer

    The integers are natural numbers including 0 and their negative and non-negative numberss . They are numbers that can be written without a fractional or decimal component, and fall within the set ....
    . Then the quotient space X/~ is homeomorphic to the unit circle
    Unit circle

    In mathematics, a unit circle is a circle with a 1 radius, i.e., a circle whose radius is 1. Frequently, especially in trigonometry, "the" unit circle is the circle of radius 1 centered at the origin in the Cartesian coordinate system in the Euclidean plane....
     S1 via the homeomorphism which sends the equivalence class of x to exp(2πix).
  • A vast generalization of the previous example is the following: Suppose a topological group
    Topological group

    In mathematics, a topological group is a group G together with a topological space on G such that the group's binary operation and the group's inverse function are continuous function ....
     G acts
    Group action

    In algebra and geometry, a group action is a way of describing symmetry of objects using group . The essential elements of the object are described by a Set and the symmetries of the object are described by the symmetry group of this set, which consists of bijective transformation of the set....
     continuously on a space X. One can form an equivalence relation on X by saying points are equivalent if and only if they lie in the same orbit. The quotient space under this relation is called the orbit space, denoted X/G. In the previous example G = Z acts on R by translation. The orbit space R/Z is homeomorphic to S1.


Warning: The notation R/Z is somewhat ambiguous. If Z is understood to be a group acting on R then the quotient is the circle. However, if Z is thought of as a subspace of R, then the quotient is an infinite bouquet of circles
Bouquet of circles

In mathematics, a rose is a topological space obtained by quotient space together a collection of circle along a single point. The circles of the rose are called petals....
 joined at a single point.

Properties


Quotient maps q : X → Y are characterized among surjective maps by the following property: if Z is any topological space and f : Y → Z is any function, then f is continuous if and only if f O q is continuous.

Quotientspace 01
The quotient space X/~ together with the quotient map q : X → X/~ is characterized by the following universal property
Universal property

In various branches of mathematics, a useful construction is often viewed as the ?most efficient solution? to a certain problem. The definition of a universal property uses the language of category theory to make this notion precise....
: if g : X → Z is a continuous map such that a~b implies g(a)=g(b) for all a and b in X, then there exists a unique continuous map f : X/~ → Z such that g = f O q. We say that g descends to the quotient.

The continuous maps defined on X/~ are therefore precisely those maps which arise from continuous maps defined on X that respect the equivalence relation (in the sense that they send equivalent elements to the same image). This criterion is constantly being used when studying quotient spaces.

Given a continuous surjection f : X → Y it is useful to have criteria by which one can determine if f is a quotient map. Two sufficient criteria are that f be open or closed. Note that these conditions are only sufficient, not necessary. It is easy to construct examples of quotient maps which are neither open nor closed.

Compatibility with other topological notions

  • Separation
    • In general, quotient spaces are ill-behaved with respect to separation axioms. The separation properties of X need not be inherited by X/~, and X/~ may have separation properties not shared by X.
    • X/~ is a T1 space
      T1 space

      In topology and related branches of mathematics, T1 spaces and R0 spaces are particular kinds of topological spaces....
       if and only if every equivalence class of ~ is closed in X.
    • If the quotient map is open then X/~ is a Hausdorff space
      Hausdorff space

      In topology and related branches of mathematics, a Hausdorff space, separated space or T2 space is a topological space in which distinct points have disjoint neighbourhood ....
       if and only if ~ is a closed subset of the product space X×X.
  • Connectedness
    Connectedness

    In mathematics, connectedness is used to refer to various properties meaning, in some sense, "all one piece". When a mathematical object has such a property, we say it is connected; otherwise it is disconnected....
    • If a space is connected or path connected, then so are all its quotient spaces.
    • A quotient space of a simply connected or contractible space need not share those properties.
  • Compactness
    Compact space

    In mathematics, a topological space is called compact if each of its open covers has a finite set subcover.Note: Some authors such as Nicolas Bourbaki use the term "quasi-compact" for this instead, and reserve the term "compact" for topological spaces that are both Hausdorff spaces and "quasi-compact"....
    • If a space is compact, then so are all its quotient spaces.
    • A quotient space of a locally compact space need not be locally compact.
  • Dimension
    Dimension

    In mathematics, the dimension of a space is roughly defined as the minimum number of coordinates needed to specify every point within it. For example: a point on the unit circle in the plane can be specified by two Cartesian coordinates but one can make do with a single coordinate , so the circle is 1-dimensional even though it exists in...
    • The topological dimension of a quotient space can be more (as well as less) than the dimension of the original space; space-filling curve
      Space-filling curve

      In mathematical analysis, a space-filling curve is a curve whose range contains the entire 2-dimensional unit square . Because Giuseppe Peano was the first to discover one, space-filling curves in the Plane are commonly called Peano curves....
      s provide such examples.


See also


Topology

  • Subspace (topology)
  • Product space
  • Disjoint union (topology)
    Disjoint union (topology)

    In general topology and related areas of mathematics, the disjoint union 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....
  • Final topology
    Final topology

    In general topology and related areas of mathematics, the final topology on a Set , with respect to a family of functions into , is the finest topology on X which makes those functions continuous function ....
  • Mapping cone
    Mapping cone

    In mathematics, especially homotopy theory, the mapping cone is a construction of topology, analogous to a quotient space. It is also called the homotopy cofiber, and also notated ....


Algebra

  • Quotient group
    Quotient group

    In mathematics, given a group G and a normal subgroup N of G, the quotient group, or factor group, of G over N is intuitively a group that "collapses" the normal subgroup N to the identity element....
  • Quotient space (linear algebra)
    Quotient space (linear algebra)

    In linear algebra, the quotient of a vector space V by a linear subspace N is a vector space obtained by "collapsing" N to zero. The space obtained is called a quotient space and is denoted V/N ....
  • Quotient category
    Quotient category

    In mathematics, a quotient category is a category obtained from another one by identifying sets of morphisms. The notion is similar to that of a quotient group or quotient space, but in the categorical setting....
  • Mapping cone (homological algebra)
    Mapping cone (homological algebra)

    In homological algebra, the mapping cone is a construction on a map of chain complexes inspired by the mapping cone. In the theory of triangulated categories it is a kind of combined kernel and cokernel: if the chain complexes take their terms in an abelian category, so that we can talk about cohomology, then the cone of a map f being a...