The quotient of a topological space $X$ by a subspace $Y\subseteq X$ is the quotient by the equivalence relation $\sim$ given by $y\sim y’$ if $y$ and $y’\in Y$.
Let $q: X\to X/Y$ be given by $x\mapsto [x]$. This function is (well-defined and) surjective. The topology on the quotient $X/Y$ is the collection of $U\subset X/Y$ such that the inverse image $q^{-1}(U)$ is open in $X$.
Wikidata ID: Q1139111