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topology In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities ...

topology
and related areas of
mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and their changes (cal ...
, the quotient space of a
topological space In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no gener ...
under a given
equivalence relation In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities a ...
is a new topological space constructed by endowing the
quotient set In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities an ...
of the original topological space with the quotient topology, that is, with the finest topology that makes
continuous Continuity or continuous may refer to: Mathematics * Continuity (mathematics), the opposing concept to discreteness; common examples include ** Continuous probability distribution or random variable in probability and statistics ** Continuous ga ...
the canonical projection map (the function that maps points to their equivalence classes). In other words, a subset of a quotient space is Open set, open if and only if its preimage under the canonical projection map is open in the original topological space. Intuitively speaking, the points of each equivalence class are or "glued together" for forming a new topological space. For example, identifying the points of a sphere that belong to the same diameter produces the projective plane as a quotient space.


Definition

Let \left(X, \tau_X\right) be a
topological space In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no gener ...
, and let \,\sim\, be an
equivalence relation In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities a ...
on X. The
quotient set In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities an ...
, Y = X / \sim\, is the set of equivalence classes of elements of X. The equivalence class of x \in X is denoted [x]. The , , associated with \,\sim\, refers to the following Surjection, surjective map: \begin q :\;&& X &&~\to & ~X/ \\[0.3ex] && x &&~\mapsto& ~[x]. \end For any subset S \subseteq X / (so in particular, s \subseteq X for every s \in S) the following holds: \ = \bigcup_ s = q^(S). The quotient space under \,\sim\, is the quotient set Y equipped with the quotient topology, which is the topology whose open sets are the all those subsets U \subseteq Y = X / such that \ = \cup_ u is an Open set, open subset of \left(X, \tau_X\right); that is, U \subseteq X / is open in the quotient topology on X / if and only if \ \in \tau_X. Thus, \tau_Y = \left\. Equivalently, the open sets of the quotient topology are the subsets of Y that have an open preimage under the canonical map q : X \to X / (which is defined by q(x) = [x]). Similarly, a subset S \subseteq X / is Closed set, closed in X / if and only if \ is a closed subset of \left(X, \tau_X\right). The quotient topology is the final topology on the quotient set, with respect to the map x \mapsto [x].


Quotient map

A map f : X \to Y is a quotient map (sometimes called an identification map) if it is Surjective function, surjective, and a subset V \subseteq Y is open if and only if f^(V) is open. Equivalently, a surjection f : X \to Y is a quotient map if and only if for every subset D \subseteq Y, D is closed in Y if and only if f^(D) is closed in X. Final topology definition Alternatively, f is a quotient map if it is onto and Y is equipped with the final topology with respect to f. Saturated sets and quotient maps A subset S of X is called Saturated set, saturated (with respect to f) if it is of the form S = f^(T) for some set T, which is true if and only if f^(f(S)) = S (although f^(f(S)) \supseteq S always holds for every subset S \subseteq X, equality is in general not guaranteed; and a non-saturated set exists if and only if f is not Injective map, injective). The assignment T \mapsto f^(T) establishes a Bijection, one-to-one correspondence (whose inverse is S \mapsto f(S)) between subsets T of Y = f(X) and saturated subsets of X. With this terminology, a surjection f : X \to Y is a quotient map if and only if for every subset S of X, S is open in X if and only if f(S) is open in Y. In particular, open subsets of X that are saturated have no impact on whether or not the function f is a quotient map; non-saturated subsets are irrelevant to the definition of "quotient map" just as they are irrelevant to the open-set definition of Continuous function, continuity (because a function f : X \to Y is continuous if and only if for every subset S of X, f(S) being open in f(X) implies S is open in X). Every quotient map is Continuous function, continuous but not every continuous map is a quotient map. A continuous surjection f : X \to Y to be a quotient map if and only if X has some open subset S such that f(S) is open in Y (this statement remains true if both instances of the word "open" are replaced with "closed"). Quotient space of fibers characterization Given an
equivalence relation In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities a ...
\,\sim\, on X, the canonical map q : X \to X / that sends x \in X to its equivalence class [x] := \ (that is, q(x) := [x]) is a quotient map that satisfies q(x) = q^(q(x)) for all x \in X; moreover, for all a, b \in X, a \,\sim\, b if and only if q(a) = q(b). In fact, let \pi : X \to Y be a surjection between topological spaces (not yet assumed to be continuous or a quotient map) and declare for all a, b \in X that a \,\sim\, b if and only if \pi(a) = \pi(b). Then \,\sim\, is an equivalence relation on X such that for every x \in X, [x] = \pi^(\pi(x)) so that \pi([x]) = \ \subseteq Y is a singleton set, which thus induces a bijection \hat : X / \;\;\to\; Y defined by \hat([x]) := \pi(x) (this is well defined because \pi([x]) is a singleton set and \hat([x]) is just its unique element; that is, \pi([x]) = \ for every x). Define the map q : X \to X / as above (by q(x) := [x]) and give X / \sim the quotient topology induced by q (which makes q a quotient map). These maps are related by: \pi = \hat \circ q \quad \text \quad q = \hat^ \circ \pi. From this and the fact that q : X \to X / \sim is a quotient map, it follows that \pi : X \to Y is continuous if and only if this is true of \hat : X / \sim \;\;\to\; Y. Furthermore, \pi : X \to Y is a quotient map if and only if \hat : X / \sim \;\;\to\; Y is a homeomorphism (or equivalently, if and only if both \hat and its inverse are continuous).


Related definitions

A is a surjective map f : X \to Y with the property that for every subset T \subseteq Y, the restriction f\big\vert_ ~:~ f^(T) \to T is also a quotient map. There exist quotient maps that are not hereditarily quotient.


Examples

* Gluing. Topologists talk of gluing points together. If X is a topological space, gluing the points x and y in X means considering the quotient space obtained from the equivalence relation a \sim b if and only if a = b or a = x, b = y (or a = y, b = x). * Consider the unit square I^2 = [0, 1] \times [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 I^2 / \sim is homeomorphic to the sphereS^2. *Adjunction space. More generally, suppose X is a space and A is a Subspace (topology), 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 a Unit disc, closed disc with its boundary identified to a single point: D^2 / \partial. * Consider the set \R of real numbers with the ordinary topology, and write x \sim y if and only if x - y is an integer. Then the quotient space X / \sim is homeomorphic to the unit circle S^1 via the homeomorphism which sends the equivalence class of x to \exp(2 \pi i x). * A generalization of the previous example is the following: Suppose a topological group G Group action (mathematics), 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 (group theory), 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 S^1. **''Note'': The notation \R / \Z is somewhat ambiguous. If \Z is understood to be a group acting on \R via addition, then the Quotient group, quotient is the circle. However, if \Z is thought of as a topological subspace of \R (that is identified as a single point) then the quotient \ \cup \ (which is Bijection, identifiable with the set \ \cup (\R \setminus \Z)) is a countably infinite bouquet of circles joined at a single point \Z. * This next example shows that it is in general true that if q : X \to Y is a quotient map then every convergent sequence (respectively, every Convergent net, convergent Net (mathematics), net) in Y has a Lift (mathematics), lift (by q) to a convergent sequence (or convergent net) in X. Let X = [0, 1] and \,\sim ~=~ \ ~\cup~ \left\. Let Y := X /\,\sim\, and let q : X \to X / \sim\, be the quotient map q(x) := [x], so that q(0) = q(1) = \ and q(x) = \ for every x \in (0, 1). The map h : X / \,\sim\, \to S^1 \subseteq \Complex defined by h([x]) := e^ is well-defined (because e^ = 1 = e^) and a homeomorphism. Let I = \N and let a_ := \left(a_i\right)_ \text b_ := \left(b_i\right)_ be any sequences (or more generally, any nets) valued in (0, 1) such that a_ \to 0 \text b_ \to 1 in X = [0, 1]. Then the sequence y_1 := q\left(a_1\right), y_2 := q\left(b_1\right), y_3 := q\left(a_2\right), y_4 := q\left(b_2\right), \ldots converges to [0] = [1] in X / \sim\, but there does not exist any convergent lift of this sequence by the quotient map q (that is, there is no sequence s_ = \left(s_i\right)_ in X that both converges to some x \in X and satisfies y_i = q\left(s_i\right) for every i \in I). This counterexample can be generalized to Net (mathematics), nets by letting (A, \leq) be any directed set, and making I := A \times \ into a net by declaring that for any (a, m), (b, n) \in I, (m, a) \; \leq \; (n, b) holds if and only if both (1) a \leq b, and (2) if a = b \text m \leq n; then the A-indexed net defined by letting y_ equal a_i \text m = 1 and equal to b_i \text m = 2 has no lift (by q) to a convergent A-indexed net in X = [0, 1].


Properties

Quotient maps q : X \to Y are characterized among surjective maps by the following property: if Z is any topological space and f : Y \to Z is any function, then f is continuous if and only if f \circ q is continuous. The quotient space X / \sim together with the quotient map q : X \to X / \sim is characterized by the following universal property: if g : X \to Z is a continuous map such that a \sim b implies g(a) = g(b) for all a, b \in X, then there exists a unique continuous map f : X / \sim \to Z such that g = f \circ q. In other words, the following diagram commutes: One says that g ''descends to the quotient'' for expressing this, that is that it factorizes through the quotient space. The continuous maps defined on X / \sim 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 copiously used when studying quotient spaces. Given a continuous surjection q : X \to Y it is useful to have criteria by which one can determine if q is a quotient map. Two sufficient criteria are that q be Open map, open or Closed map, closed. Note that these conditions are only Sufficient condition, sufficient, not Necessary condition, necessary. It is easy to construct examples of quotient maps that are neither open nor closed. For topological groups, the quotient map is open.


Compatibility with other topological notions

Separation axioms, Separation * In general, quotient spaces are ill-behaved with respect to separation axioms. The separation properties of X need not be inherited by X / \sim, and X / \sim may have separation properties not shared by X. * X / \sim is a T1 space if and only if every equivalence class of \,\sim\, is closed in X. * If the quotient map is Open map, open, then X / \sim is a Hausdorff space if and only if ~ is a closed subset of the product space X \times 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. Compact space, 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.


See also

Topology * * * * * * * Algebra * * * *


References

* * * * * * * * {{reflist, group=proof Continuous mappings General topology Group actions (mathematics) Quotient objects, Space (topology) Topology