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

Question

give an example of an infinite class of closed sets whose union is not closed. Thanks for your help

Answer 1

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

Answer 2

the union of intervals of the form $(1/n ,1- 1/n ) = (0,1) $ will be a example. the behaviour of the interval already stated above.

Answer 3

As another example, let $X$ be any infinite set, and consider the cofinite topology on $X$ (ie all open sets are either the empty set or sets whose complement is finite). Every proper closed subset of $X$ is finite. So, fixing an element $x_0\in X$, we have the union closed sets equaling an open set: $$X\setminus\{x_0\}=\bigcup\limits_{x\not=x_0} \{x\}$$

Answer 4

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

Answer 5

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.