In

topology
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and related areas of mathematics
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, the quotient space of a topological space
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under a given equivalence relation
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is a new topological space constructed by endowing the quotient set
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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 $\backslash left(X,\; \backslash tau\_X\backslash right)$ be atopological space
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, and let $\backslash ,\backslash sim\backslash ,$ be an equivalence relation
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on $X.$ The quotient set
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, $Y\; =\; X\; /\; \backslash sim\backslash ,$ is the set of equivalence classes of elements of $X.$ The equivalence class of $x\; \backslash in\; X$ is denoted $[x].$
The , , associated with $\backslash ,\backslash sim\backslash ,$ refers to the following Surjection, surjective map:
$$\backslash begin\; q\; :\backslash ;\&\&\; X\; \&\&~\backslash to\; \&\; ~X/\; \backslash \backslash [0.3ex]\; \&\&\; x\; \&\&~\backslash mapsto\&\; ~[x].\; \backslash end$$
For any subset $S\; \backslash subseteq\; X\; /$ (so in particular, $s\; \backslash subseteq\; X$ for every $s\; \backslash in\; S$) the following holds:
$$\backslash \; =\; \backslash bigcup\_\; s\; =\; q^(S).$$
The quotient space under $\backslash ,\backslash sim\backslash ,$ is the quotient set $Y$ equipped with the quotient topology, which is the topology whose open sets are the all those subsets $U\; \backslash subseteq\; Y\; =\; X\; /$ such that $\backslash \; =\; \backslash cup\_\; u$ is an Open set, open subset of $\backslash left(X,\; \backslash tau\_X\backslash right);$ that is, $U\; \backslash subseteq\; X\; /$ is open in the quotient topology on $X\; /$ if and only if $\backslash \; \backslash in\; \backslash tau\_X.$
Thus,
$$\backslash tau\_Y\; =\; \backslash left\backslash .$$
Equivalently, the open sets of the quotient topology are the subsets of $Y$ that have an open preimage under the canonical map $q\; :\; X\; \backslash to\; X\; /$ (which is defined by $q(x)\; =\; [x]$).
Similarly, a subset $S\; \backslash subseteq\; X\; /$ is Closed set, closed in $X\; /$ if and only if $\backslash $ is a closed subset of $\backslash left(X,\; \backslash tau\_X\backslash right).$
The quotient topology is the final topology on the quotient set, with respect to the map $x\; \backslash mapsto\; [x].$
Quotient map

A map $f\; :\; X\; \backslash to\; Y$ is a quotient map (sometimes called an identification map) if it is Surjective function, surjective, and a subset $V\; \backslash subseteq\; Y$ is open if and only if $f^(V)$ is open. Equivalently, a surjection $f\; :\; X\; \backslash to\; Y$ is a quotient map if and only if for every subset $D\; \backslash subseteq\; Y,$ $D$ 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 $f^(f(S))\; \backslash supseteq\; S$ always holds for every subset $S\; \backslash subseteq\; X,$ equality is in general not guaranteed; and a non-saturated set exists if and only if $f$ is not Injective map, injective). The assignment $T\; \backslash mapsto\; f^(T)$ establishes a Bijection, one-to-one correspondence (whose inverse is $S\; \backslash mapsto\; f(S)$) between subsets $T$ of $Y\; =\; f(X)$ and saturated subsets of $X.$ With this terminology, a surjection $f\; :\; X\; \backslash to\; Y$ 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 $f\; :\; X\; \backslash to\; Y$ 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 $f\; :\; X\; \backslash to\; Y$ 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 anequivalence relation
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$\backslash ,\backslash sim\backslash ,$ on $X,$ the canonical map $q\; :\; X\; \backslash to\; X\; /$ that sends $x\; \backslash in\; X$ to its equivalence class $[x]\; :=\; \backslash $ (that is, $q(x)\; :=\; [x]$) is a quotient map that satisfies $q(x)\; =\; q^(q(x))$ for all $x\; \backslash in\; X$; moreover, for all $a,\; b\; \backslash in\; X,$ $a\; \backslash ,\backslash sim\backslash ,\; b$ if and only if $q(a)\; =\; q(b).$
In fact, let $\backslash pi\; :\; X\; \backslash to\; Y$ be a surjection between topological spaces (not yet assumed to be continuous or a quotient map) and declare for all $a,\; b\; \backslash in\; X$ that $a\; \backslash ,\backslash sim\backslash ,\; b$ if and only if $\backslash pi(a)\; =\; \backslash pi(b).$ Then $\backslash ,\backslash sim\backslash ,$ is an equivalence relation on $X$ such that for every $x\; \backslash in\; X,$ $[x]\; =\; \backslash pi^(\backslash pi(x))$ so that $\backslash pi([x])\; =\; \backslash \; \backslash subseteq\; Y$ is a singleton set, which thus induces a bijection $\backslash hat\; :\; X\; /\; \backslash ;\backslash ;\backslash to\backslash ;\; Y$ defined by $\backslash hat([x])\; :=\; \backslash pi(x)$ (this is well defined because $\backslash pi([x])$ is a singleton set and $\backslash hat([x])$ is just its unique element; that is, $\backslash pi([x])\; =\; \backslash $ for every $x$).
Define the map $q\; :\; X\; \backslash to\; X\; /$ as above (by $q(x)\; :=\; [x]$) and give $X\; /\; \backslash sim$ the quotient topology induced by $q$ (which makes $q$ a quotient map). These maps are related by: $$\backslash pi\; =\; \backslash hat\; \backslash circ\; q\; \backslash quad\; \backslash text\; \backslash quad\; q\; =\; \backslash hat^\; \backslash circ\; \backslash pi.$$
From this and the fact that $q\; :\; X\; \backslash to\; X\; /\; \backslash sim$ is a quotient map, it follows that $\backslash pi\; :\; X\; \backslash to\; Y$ is continuous if and only if this is true of $\backslash hat\; :\; X\; /\; \backslash sim\; \backslash ;\backslash ;\backslash to\backslash ;\; Y.$ Furthermore, $\backslash pi\; :\; X\; \backslash to\; Y$ is a quotient map if and only if $\backslash hat\; :\; X\; /\; \backslash sim\; \backslash ;\backslash ;\backslash to\backslash ;\; Y$ is a homeomorphism (or equivalently, if and only if both $\backslash hat$ and its inverse are continuous).
Related definitions

A is a surjective map $f\; :\; X\; \backslash to\; Y$ with the property that for every subset $T\; \backslash subseteq\; Y,$ the restriction $f\backslash big\backslash vert\_\; ~:~\; f^(T)\; \backslash to\; T$ 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 $a\; \backslash sim\; b$ if and only if $a\; =\; b$ or $a\; =\; x,\; b\; =\; y$ (or $a\; =\; y,\; b\; =\; x$). * Consider the unit square $I^2\; =\; [0,\; 1]\; \backslash times\; [0,\; 1]$ 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 $I^2\; /\; \backslash sim$ is homeomorphic to the sphere$S^2.$ *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: $D^2\; /\; \backslash partial.$ * Consider the set $\backslash R$ of real numbers with the ordinary topology, and write $x\; \backslash sim\; y$ if and only if $x\; -\; y$ is an integer. Then the quotient space $X\; /\; \backslash sim$ is homeomorphic to the unit circle $S^1$ via the homeomorphism which sends the equivalence class of $x$ to $\backslash exp(2\; \backslash pi\; i\; x).$ * 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 $G\; =\; \backslash Z$ acts on $\backslash R$ by translation. The orbit space $\backslash R\; /\; \backslash Z$ is homeomorphic to $S^1.$ **''Note'': The notation $\backslash R\; /\; \backslash Z$ is somewhat ambiguous. If $\backslash Z$ is understood to be a group acting on $\backslash R$ via addition, then the Quotient group, quotient is the circle. However, if $\backslash Z$ is thought of as a topological subspace of $\backslash R$ (that is identified as a single point) then the quotient $\backslash \; \backslash cup\; \backslash $ (which is Bijection, identifiable with the set $\backslash \; \backslash cup\; (\backslash R\; \backslash setminus\; \backslash Z)$) is a countably infinite bouquet of circles joined at a single point $\backslash Z.$ * This next example shows that it is in general true that if $q\; :\; X\; \backslash to\; Y$ 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 $\backslash ,\backslash sim\; ~=~\; \backslash \; ~\backslash cup~\; \backslash left\backslash .$ Let $Y\; :=\; X\; /\backslash ,\backslash sim\backslash ,$ and let $q\; :\; X\; \backslash to\; X\; /\; \backslash sim\backslash ,$ be the quotient map $q(x)\; :=\; [x],$ so that $q(0)\; =\; q(1)\; =\; \backslash $ and $q(x)\; =\; \backslash $ for every $x\; \backslash in\; (0,\; 1).$ The map $h\; :\; X\; /\; \backslash ,\backslash sim\backslash ,\; \backslash to\; S^1\; \backslash subseteq\; \backslash Complex$ defined by $h([x])\; :=\; e^$ is well-defined (because $e^\; =\; 1\; =\; e^$) and a homeomorphism. Let $I\; =\; \backslash N$ and let $a\_\; :=\; \backslash left(a\_i\backslash right)\_\; \backslash text\; b\_\; :=\; \backslash left(b\_i\backslash right)\_$ be any sequences (or more generally, any nets) valued in $(0,\; 1)$ such that $a\_\; \backslash to\; 0\; \backslash text\; b\_\; \backslash to\; 1$ in $X\; =\; [0,\; 1].$ Then the sequence $$y\_1\; :=\; q\backslash left(a\_1\backslash right),\; y\_2\; :=\; q\backslash left(b\_1\backslash right),\; y\_3\; :=\; q\backslash left(a\_2\backslash right),\; y\_4\; :=\; q\backslash left(b\_2\backslash right),\; \backslash ldots$$ converges to $[0]\; =\; [1]$ in $X\; /\; \backslash sim\backslash ,$ but there does not exist any convergent lift of this sequence by the quotient map $q$ (that is, there is no sequence $s\_\; =\; \backslash left(s\_i\backslash right)\_$ in $X$ that both converges to some $x\; \backslash in\; X$ and satisfies $y\_i\; =\; q\backslash left(s\_i\backslash right)$ for every $i\; \backslash in\; I$). This counterexample can be generalized to Net (mathematics), nets by letting $(A,\; \backslash leq)$ be any directed set, and making $I\; :=\; A\; \backslash times\; \backslash $ into a net by declaring that for any $(a,\; m),\; (b,\; n)\; \backslash in\; I,$ $(m,\; a)\; \backslash ;\; \backslash leq\; \backslash ;\; (n,\; b)$ holds if and only if both (1) $a\; \backslash leq\; b,$ and (2) if $a\; =\; b\; \backslash text\; m\; \backslash leq\; n;$ then the $A$-indexed net defined by letting $y\_$ equal $a\_i\; \backslash text\; m\; =\; 1$ and equal to $b\_i\; \backslash text\; m\; =\; 2$ has no lift (by $q$) to a convergent $A$-indexed net in $X\; =\; [0,\; 1].$Properties

Quotient maps $q\; :\; X\; \backslash to\; Y$ are characterized among surjective maps by the following property: if $Z$ is any topological space and $f\; :\; Y\; \backslash to\; Z$ is any function, then $f$ is continuous if and only if $f\; \backslash circ\; q$ is continuous. The quotient space $X\; /\; \backslash sim$ together with the quotient map $q\; :\; X\; \backslash to\; X\; /\; \backslash sim$ is characterized by the following universal property: if $g\; :\; X\; \backslash to\; Z$ is a continuous map such that $a\; \backslash sim\; b$ implies $g(a)\; =\; g(b)$ for all $a,\; b\; \backslash in\; X,$ then there exists a unique continuous map $f\; :\; X\; /\; \backslash sim\; \backslash to\; Z$ such that $g\; =\; f\; \backslash circ\; q.$ 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 $X\; /\; \backslash sim$ 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 $q\; :\; X\; \backslash to\; Y$ 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 $X\; /\; \backslash sim,$ and $X\; /\; \backslash sim$ may have separation properties not shared by $X.$ * $X\; /\; \backslash sim$ is a T1 space if and only if every equivalence class of $\backslash ,\backslash sim\backslash ,$ is closed in $X.$ * If the quotient map is Open map, open, then $X\; /\; \backslash sim$ is a Hausdorff space if and only if ~ is a closed subset of the product space $X\; \backslash times\; X.$ 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