quotient map

English

Noun

quotient map (plural quotient maps)

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 $[0,1]\times [0,1]$ (with the subspace topology) in $\mathbb {R} ^{2}$ (with the standard topology) is mapped onto the set $[0,1)\times [0,1)$ with the quotient map $f(x,y)=(\lfloor x\rfloor ,\lfloor y\rfloor )$, then the target space (with its induced quotient topology) is topologically equivalent to a torus.

Synonyms

• identification map

Related terms

• quotient space
• quotient topology