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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.
数学、特に位相幾何学や関連する分野においての商空間 (quotient space, identification space とも呼ばれる) は、直感的には、ある空間に対して、特定の点 (一点とは限らない) 同士を互いに「貼り合わせる」ことにより得られる空間である。貼り合わせる点同士はある同値関係によって指定される。ある空間から別の空間を生成する方法として、一般的に用いられる。
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/~.
X を位相空間, ~ を X 上の同値関係とする。商集合 X/~ は、X の ~ についての同値類を点とする集合であるが、この集合について、位相を以下のように定める。
これを、商集合 X/~ 上の商位相と呼ぶ。
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.
商位相を以下のように特徴づけることもできる。q: X → X/~ を、X上の各点について、その点が属する同値類を対応付ける射影とする。このとき、X/~ 上の商位相は、q を連続にする位相のうち、最も強い位相である。これは上の定義と同値である。
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).
位相空間 X から (位相が導入されていない) 集合 Y への全射 f: X → Y について、f が連続となる位相のうち最も強い位相として Y 上の商位相を定めることができる。これは、V ⊆ Y が開集合であることを f の逆像 f−1(V) が開集合であることで定義することによって定まる位相と同じである。x1, x2 ∈ X について、f(x1) = f(x2) のとき x1 と x2 が同値である、とする同値関係 ~ を定めることができるが、 x を f(x) の同値類へ写す写像により、商空間 X/~ は (f についての商位相が導入された) Y と同相 である。
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.
一般に、位相空間 X, Y 間の連続な全射 f: X → Y は、Y が f による商位相を持つときに 商写像 と呼ばれる。
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 joined at a single point.
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 ∘ 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.
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.
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