Infinite class of closed sets whose union is not closed

Give an example of an infinite class of closed sets whose union is not closed.

Solution 1:

Every subset $S\subset X$ of a Hausdorff space is the union of its singleton subsets, which are closed : $$S=\bigcup_{s\in S} \lbrace s\rbrace $$

Solution 2:

I think probably the most instructive example is considering $\displaystyle A_n=\left[\frac{1}{n},\infty\right)$.

Solution 3:

Can you express $(0,1)$ as an increasing union of closed sets? Maybe find a pair of sequences $a_n$ and $b_n$ with $a_n$ decreasing to $0$ and $b_n$ increasing to $1$? Then you can try taking $[a_n,b_n]$ and see if that works.

Solution 4:

The union of intervals of the form $\left[\frac{1}{n} ,1- \frac{1}{n}\right]$ will be one example: $$ \bigcup_{n=2}^\infty \left[\frac{1}{n} ,1- \frac{1}{n}\right] = (0,1) $$ The behaviour of the interval is already stated above.