English Dictionary

quotient map

名詞

(topology) A surjective, continuous function from one topological space to another one, such that the latter one's topology has the property that if the inverse image (under the said function) of some subset of it is open in the function's domain, then the subset is open in the target space.

日本語の意味

商写像: 位相空間間の全射かつ連続な写像であり、目標空間の位相は、写像の逆像が開集合となる部分集合について開集合と定義される。

During the seminar, the professor proved that the natural projection from X onto X/~ is a quotient map, meaning the codomain is equipped with the quotient topology so that a subset of it is open exactly when its preimage in X is open.

セミナーで教授は、X から X/∼ への自然な射影が商写像であることを示し、つまり像側の位相は商位相であり、ある部分集合が像側で開であるのはその逆像が X で開であるときに限る、と説明した。

意味（1）

(topology) A surjective, continuous function from one topological space to another one, such that the latter one's topology has the property that if the inverse image (under the said function) of some subset of it is open in the function's domain, then the subset is open in the target space.

If the square [0,1]⨯0,1 (with the subspace topology) in ℝ² (with the standard topology) is mapped onto the set [0,1)⨯[0,1) with the quotient map f(x,y)=(⌊x⌋,⌊y⌋), then the target space (with its induced quotient topology) is topologically equivalent to a torus.

quotient maps （ plural ）