Quotient space (topology)

In topology and related areas of mathematics, the quotient space of a topological space under a given equivalence relation is a new topological space constructed by endowing the quotient set of the original topological space with the quotient topology, that is, with the finest topology that makes continuous the canonical projection map (the function that maps points to their equivalence classes). In other words, a subset of a quotient space is 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 identified 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 X {\displaystyle X} be a topological space, and let ∼ {\displaystyle \sim } be an equivalence relation on X . {\displaystyle X.} The quotient set Y = X / ∼ {\displaystyle Y=X/{\sim }} is the set of equivalence classes of elements of X . {\displaystyle X.} The equivalence class of x ∈ X {\displaystyle x\in X} is denoted [ x ] . {\displaystyle [x].} The construction of Y {\displaystyle Y} defines a canonical surjection q : X → Y , x ↦ [ x ] . {\displaystyle q:X\to Y,x\mapsto [x].}

As discussed below, q {\displaystyle q} is a quotient mapping, commonly called the canonical quotient map, or canonical projection map, associated to X / ∼ . {\displaystyle X/{\sim }.}

The quotient space under ∼ {\displaystyle \sim } is the set Y {\displaystyle Y} equipped with the quotient topology, whose open sets are those subsets U ⊆ Y {\textstyle U\subseteq Y} whose preimage q − 1 ( U ) {\displaystyle q^{-1}(U)} is open. In other words, U {\displaystyle U} is open in the quotient topology on X / ∼ {\displaystyle X/{\sim }} if and only if { x ∈ X : [ x ] ∈ U } {\textstyle \{x\in X:[x]\in U\}} is open in X . {\displaystyle X.} Similarly, a subset S ⊆ Y {\displaystyle S\subseteq Y} is closed if and only if { x ∈ X : [ x ] ∈ S } {\displaystyle \{x\in X:[x]\in S\}} is closed in X . {\displaystyle X.}

The quotient topology is the final topology on the quotient set, with respect to the map x ↦ [ x ] . {\displaystyle x\mapsto [x].}

Quotient map A map f : X → Y {\displaystyle f:X\to Y} is a quotient map (sometimes called an identification map) if it is surjective and Y {\displaystyle Y} is equipped with the final topology induced by f . {\displaystyle f.} The latter condition admits two more-elementary formulations: a subset V ⊆ Y {\displaystyle V\subseteq Y} is open (closed) if and only if f − 1 ( V ) {\displaystyle f^{-1}(V)} is open (resp. closed). Every quotient map is continuous but not every continuous map is a quotient map. Saturated sets A subset S {\displaystyle S} of X {\displaystyle X} is called saturated (with respect to f {\displaystyle f} ) if it is of the form S = f − 1 ( T ) {\displaystyle S=f^{-1}(T)} for some set T , {\displaystyle T,} which is true if and only if f − 1 ( f ( S ) ) = S . {\displaystyle f^{-1}(f(S))=S.}

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