Give an example for a sequence $A_{n}$ of closed sets $A_{n} \subset \mathbb{R}$
such that the union $\bigcup_{n=0}^{\infty} A_{n}$is not closed. Give an example for a sequence
$U_{n}$of open sets $U_{n} \subset \mathbb{R}$, such that the intersection $\bigcap_{n=0}^{\infty} U_{n}$is not open

Question

Give an example for a sequence   $A_{n}$ of closed sets  $A_{n} \subset \mathbb{R}$
such that the union  $\bigcup_{n=0}^{\infty} A_{n}$is not closed. Give an example for a sequence
  $U_{n}$of open sets  $U_{n} \subset  \mathbb{R}$, such that the intersection  $\bigcap_{n=0}^{\infty} U_{n}$is not open

anonymous · Answer

$$
A_n=\{  \frac 1 n\}
$$
is closed

anonymous · Answer

$$
\cup A_n
$$
is not closed. Why?

anonymous · Answer

$$
U_n =(-\frac 1 n , \frac 1 n)
$$
is open.

anonymous · Answer

ok..

anonymous · Answer

$$
\cap U_n =\{0\}
$$
is not open.

anonymous · Answer

hmm

anonymous · Answer

can it be a solution for our question or is it like a hint what u wrote there?

anonymous · Answer

No, it is the solution.

anonymous · Answer

cool so short, thank you very much... i hope i can understand it soon by myself how it done..

anonymous · Answer

yw