Here are the definitions that were given in Munkres's general topology:
Definition 1: A topological space satisifes the T1 axiom if every finite point set is closed.
Definition 2: Let $X$ be a topological space, and let $X^*$ be a partition of $X$ into disjoint subsets whose union is $X$. Let $p:X \to X^*$ 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 $X$.
Question: I'm currently stuck on one statement that says in order for $X^*$ to satisfy the T1 axiom, one simply requires that each element of the partition $X^*$ be a closed subset of $X$. I'm not quite sure how is this true. Can anyone explain this to me? Thank you!
By definition the quotient topology consists of all $U\subset X^*$ such that $p(U)^{-1}$ is open in $X$, so also for every closed $C\subset X^*$, we have that its preimmage is closed.
Knowing this, if we get an element $x^*\in X^*$ we have that it is a class of elements of $X$, in other terms it means $x^*=A\subset X$, so if we show that $A$ is closed, then by definition $\{ x^* \}$ is closed in $X^*$.
This is what we wanted, if fact a finite union of closed subset is closed.