Sequence of closed set in topological space

Question

How do I find an example for a sequence of closed sets in a topological space $X$ whose union is not closed? Would $\{{1\over n}\} n\to \infty$ work?

Answer 1

Yes. This would work. Other example may include $\{q\}$ for $q\in\Bbb Q$. It is a countable collection so we can make it into a sequence.

As well the classical example: for $n\in\Bbb N$ let $F_n=[\frac1n,\frac n{n+1}]$, then $\bigcup_{n=1}^\infty F_n=(0,1)$ which is open and not closed.

To see that this union equals $(0,1)$ note that given $x\in(0,1)$ there is some $n$ such that $\frac1n<x<\frac n{n+1}$, therefore $x\in F_n$ and so $x\in\bigcup F_n$; and on the other hand if $x\leq 0$ or $1\leq x$ then $x\notin F_n$ for all $n$ so it cannot be in the union.