Question

Q/ The union of infinite collcction of closed set may not be closcd. Give an example

Answer 1

There are many examples and one will be

$$\bigcup_{n=2}^\infty \left[\frac{1}{n} ,1- \frac{1}{n}\right] = (0,1)$$

Bonus:

$$\displaystyle U_n=\left[\frac{1}{n},\infty\right)\implies \bigcup_{n\in \N}U_n=(0,\infty)$$