PlanetMath: quotient space

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quotient space

(Definition)

Let be a topological space, and let $\sim$ be an equivalence relation on . Write for the set of equivalence classes of under $\sim$ . The quotient topology on is the topology whose open sets are the subsets $U \subset X^*$ such that

$\displaystyle \bigcup U \subset X$

is an open subset of

. The space

is called the quotient space of the space

with respect to $\sim$ . It is often written $X/\sim$ .

The projection map $\pi: X \longrightarrow X^*$ which sends each element of to its equivalence class is always a continuous map. In fact, the map $\pi$ satisfies the stronger property that a subset of is open if and only if the subset $\pi^{-1}(U)$ of is open. In general, any surjective map $p: X \longrightarrow Y$ that satisfies this stronger property is called a quotient map, and given such a quotient map, the space is always homeomorphic to the quotient space of under the equivalence relation

$\displaystyle x \sim x' \iff p(x) = p(x').$

As a set, the construction of a quotient space collapses each of the equivalence classes of $\sim$ to a single point. The topology on the quotient space is then chosen to be the strongest topology such that the projection map $\pi$ is continuous.

For $A \subset X$ , one often writes for the quotient space obtained by identifying all the points of with each other.

"quotient space" is owned by djao.

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