Given a projective surface $S$, and a smooth projective curve $C\subset S$ over $\mathbb{C}$. Furthermore we have a locally free sheaf $E$ of rank $r$ on $S$.

Then for any $l\geq 1$, the projective scheme $\operatorname{Quot}(E,l)$ classifies quotients $E\to T$, such that $\operatorname{dim}_{\mathbb{C}}(H^0(S,T))=l$.

Now for fixed $E$ and $l$, i am interested in the subset of quotients which have their support $\operatorname{supp}(T)=\lbrace p_1,\ldots,p_n\rbrace$ in the given curve $C$, i.e.

$X:=\lbrace E\rightarrow T | \operatorname{supp}(T)\subset C\rbrace \subset \operatorname{Quot}(E,l).$

Is $X$ a closed subscheme of the $\operatorname{Quot}$-scheme, i.e. is restricting the support of the quotient sheaf $T$ a closed condition?

I always get confused, when a subset is defined by an open, closed or locally closed condition. In many text one can read something like "this is obviously an open condition". But i find it hard to see by what condition a subset is defined. Are there some "simple" rules which one can check to see this rather quick?