I want to determine if the two statements below are true or false:
In the discrete topology $\overline{Y} = Y$, for all $Y ⊆ X$.
In the trivial topology $\overline{Y} = X$, for all $Y ⊆ X$.
How does knowing the topology help us here? Doesn't it only tell us what subsets are open, but how does that relate to the closure?
If you know the open stes, then you know the closed sets, since they are the complements of the open sets.
a) It is true, since every set is open and therefore every set is closed.
b) It is false (unless $X$ turns out to be the empty set), since $\overline\emptyset=\emptyset$.