Definition 3

Let $X$ be a topological space, $R$ an equivalence relation on $X$. The quotient space of $X$ by $R$ is the quotient set $X/R$ with the topology which is the quotient of the topology of $X$ by the relation $R$ (§ 2, no. 4, Example 1).

Unless the contrary is expressly stated, whenever we speak of $X/R$ as a topological space, it is to be understood that we mean the quotient space of $X$ by $R$. We shall often say that this topological space is the space obtained by identifying the points of $X$ which belong to the same equivalence class mod $R$.

Let $\varphi$ be the canonical mapping $X \to X/R$. By definition (§ 2, no. 4, Proposition 6 and its corollary) the open (resp. closed) sets in $X/R$ are the sets $A$ such that $\varphi^{-1}(A)$ is open (resp. closed) in $X$; in other words, the open (resp. closed) sets in $X/R$ are in one-to-one correspondence with the open (resp. closed) subsets of $X$ which are saturated with respect to $R$ and are the canonical images of these subsets.