Definition. Let $X$ be a topological space, and let $X^*$ be a partition of $X$ into disjoint subsets whose union is $X$. Let $p\colon X\to X^*$ be the surjective map that carries each point of $X$ to the element of $X^*$ containing it. In the quotient topology induced by $p$, the space $X^*$ is called a quotient space of $X$.
Let $X=\{1,2,3\}$ with the topology $\tau=\{\{3\}, \{2, 3\},X,\emptyset\}$.
Let $X^*=\{\{1,2\},\{3\}\}$ be a partition of $X$.
Define a function $p:X\to X^*$ by the rule that $p(x) = \{1,2\}$ if $x=1,2$, and $p(x) = \{3\}$ if $x=3$.
Then the topology induced by $p$ is $\tau'=\{\{\{3\}\},\emptyset, X^*\}$.
Is $(X^*,\tau')$ a quotient space?
Yes, I think so. The topology on $X^*$ is the coarsest topology such that $p$ is a continuous map. This is definition of quotient topology. So, a subset $S$ in $X^*$ is open, iff $p^{-1}(S)$ is open in $X$.