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. The points to be identified are specified by an equivalence relation. This is commonly done in order to construct new spaces from given ones. A Möbius strip, a surface with only one side and one edge; such shapes are an object of study in topology. ... For other meanings of mathematics or math, see mathematics (disambiguation). ... In mathematics, an equivalence relation, denoted by an infix ~, is a binary relation on a set X that is reflexive, symmetric, and transitive. ...

Definition

Suppose X is a topological space and ~ is an equivalence relation on X. We define a topology on the quotient set X/~ (the set consisting of all equivalence classes of ~) as follows: a set of equivalence classes in X/~ is open if and only if their union is open in X. This is the quotient topology on the quotient set X/~. Topological spaces are structures that allow one to formalize concepts such as convergence, connectedness and continuity. ... In mathematics, an equivalence relation, denoted by an infix ~, is a binary relation on a set X that is reflexive, symmetric, and transitive. ... 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: [a] = { x in X | x ~ a } The notion of equivalence classes is useful for constructing sets... 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: [a] = { x ∈ X | x ~ a } The notion of equivalence classes is useful for constructing sets out... 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 set theory and other branches of mathematics, the union of a collection of sets is the set that contains everything that belongs to any of the sets, but nothing else. ...

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. 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, a continuous function is generally defined as one for which preimages of open sets are open. ...

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 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 mathematics, a surjective function (or onto function or surjection) is a function with the property that all possible output values of the function are generated when the input ranges over all the values in the domain. ... In mathematics, a set can be thought of as any collection of distinct things considered as a whole. ... In mathematics, the image of an element x in a set X under the function f : X → Y, denoted by f(x), is the unique y in Y that is associated with x. ... It has been suggested that this article or section be merged with Logical biconditional. ... This word should not be confused with homomorphism. ...

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 S².
• Adjunction space. 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 numbers with the ordinary topology, and write x ~ y if and only if x−y is an integer. Then the quotient space X/~ is homeomorphic to the unit circle 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 G acts 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 (a Hawaiian earring) joined at a single point. A sphere (< Greek σφαίρα) is a perfectly symmetrical geometrical object. ... An adjunction space is a common construction in topology where one topological space is attached or glued onto another. ... Topological spaces are structures that allow one to formalize concepts such as convergence, connectedness and continuity. ... In mathematics, the real numbers may be described informally in several different ways. ... It has been suggested that this article or section be merged with Logical biconditional. ... The integers are commonly denoted by the above symbol. ... This word should not be confused with homomorphism. ... Illustration of a unit circle. ... 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 symmetry group describes all symmetries of objects. ... In mathematics, groups are often used to describe symmetries of objects. ... In mathematics, a bouquet of circles or rose is a construction in topology occurring when some number of circles are glued to each other so that they share a single common point. ... In mathematics, the Hawaiian earring is the topological space that arises by considering the one-point compactification of a countably infinite family of open intervals. ...

Properties

Quotient maps q : X → Y are characterized 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.

The quotient space X/~ together with the quotient map q : X → X/~ is characterized by the following universal property: 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. Wikipedia does not have an article with this exact name. ... In various branches of mathematics, certain constructions are frequently defined or characterised by an abstract property which requires the existence of a unique morphism under certain conditions. ...

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. In topology, an open map is a function between two topological spaces which maps open sets to open sets. ... In topology, an open map is a function between two topological spaces which maps open sets to open sets. ... In logic, the words necessary and sufficient describe relations that hold between propositions or states of affairs, if one is conditional on the other. ... In logic, the words necessary and sufficient describe relations that hold between propositions or states of affairs, if one is conditional on the other. ...

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 if and only if every equivalence class of ~ is closed in X.
  • If the quotient map is open then X/~ is a Hausdorff space if and only if ~ is a closed subset of the product space X×X.
• Connectedness
  • 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
  • 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
  • The topological dimension of a quotient space can be more (as well as less) than the dimension of the original space; space-filling curves provide such examples.

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. ... The title given to this article is incorrect due to technical limitations. ... In topology, an open map is a function between two topological spaces which maps open sets to open sets. ... In topology and related branches of mathematics, a Hausdorff space is a topological space in which points can be separated by neighbourhoods. ... In topology, the cartesian product of topological spaces is turned into a topological space in the following way. ... 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 branches of mathematics, a topological space X is said to be disconnected if it is the union of two disjoint nonempty open sets. ... A geometrical object is called simply connected if it consists of one piece and doesnt have any circle-shaped holes or handles. Higher-dimensional holes are allowed. ... This is a glossary of some terms used in the branch of mathematics known as topology. ... In mathematics, a subset of Euclidean space Rn is called compact if it is closed and bounded. ... 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. ... :For other senses of this word, see dimension (disambiguation). ... 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. ... 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). ...

See also

In topology:

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

In algebra: Topological spaces are structures that allow one to formalize concepts such as convergence, connectedness and continuity. ... In topology, the cartesian product of topological spaces is turned into a topological space in the following way. ... In the mathematical field of topology a direct sum, direct disjoint sum or coproduct is an important universal construction for topological spaces. ... In topology and related areas of mathematics, the final topology on a set is the strongest topology to make a family of functions into continuous. ...

• Quotient group
• Quotient space (linear algebra)

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. ... In linear algebra, the quotient of a vector space V by a subspace N is a vector space obtained by collapsing N to zero. ...

References

• Stephen Willard, General Topology, (1970) Addison-Wesley Publishing Company, Reading Massachusetts.
• Quotient space on PlanetMath