Please help me to prove or disprove the following:
1.$\text{}$ If $\{F_i\mid i=1,2,\ldots\}$ is a collection of closed sets in $\mathbb{R}^2$, then the union of the $F_i$ is also closed in $\mathbb{R}^2$.
By the properties of closed sets, we can say that finite union of closed sets is closed. But how can we construct a counterexample to show that the arbitrary union is not closed in $\mathbb{R}^2$?
2.$\text{}$ In a topological space, the intersection of $\mathrm{int}(A)$ with the derived set of $A$ ($A^d$) is empty.
3.$\text{}$ $\{1-\frac{1}{n}\mid n=1,2,\ldots\}$ is closed in $\mathbb{R}$.
A set is closed iff it contains all of its limit points. Is zero a limit point of this? If so this set is closed. Am I correct?