= Solution
The <quotient set> $X/{\sim}$ consists of the <equivalence classes> $[x]=\{y\in X:y\sim x\}$. Its collapsing map is the <canonical projection of a quotient set>
$$
p:X\longrightarrow X/{\sim},\qquad p(x)=[x].
$$
Equip this <set> with the <quotient topology>:
$$
U\subseteq X/{\sim}\text{ is open}\quad\Longleftrightarrow\quad p^{-1}(U)\text{ is open in }X.
$$
This is a <topology> because <preimages> preserve arbitrary <unions>, finite <intersections>, the <empty set>, and the whole <set>. The resulting <topological space> is the <quotient space>. The displayed rule immediately implies that the <preimage> of every <open set> is <open>, so $p$ is <continuous>; it is also <surjective>.
Back to article page
