Answer to Question #256560 in Real Analysis for Rashmi

Question

Prove that the union of two closed sets is a closed set. Give an example to show that union of an infinite number of closed sets need not be a closed set

Answer 1

Let A₁ and A₂ be two closed sets.

(we know that if a set is closed then its complement is open)

therefore A₁^c and A₂^c both are open sets.

let S = A₁ $\bigcup$ A₂

taking complement both side we get: S^c = (A₁ $\bigcup$ A₂)^c

using De Morgan's Law { which states that if A and B are two sets then

(A $\bigcup$ B)^c = A^c $\bigcap$ B^c }

therefore using De Morgan's Law we get: S^c = A₁^c $\bigcap$ A₂^c

$\because$ the Intersection of a finite collection of open sets is open.

as A₁^c and A₂^c both are open sets. therefore A₁^c $\bigcap$ A₂^c is an open set.

$\implies$ S^c is an open set

$\implies$ S is closed set

$\implies$ Hence, the union of two closed sets is a closed set

In the second part, we have to give an example to show that the union of an infinite number of closed sets need not be a closed set.

Let A_n = [$\frac{1}{n}$ , 1-$\frac{1}{n}$ ]

we can clearly see that A_n is closed for each n

as n $\to$ $\infty$ , $\frac{1}{n}$ $\to$ 0 & (1-$\frac{1}{n}$) $\to$ 1

$\cup^{\infty}_{n=1}$ A_n = (0,1) which is an open set.

Hence, the union of an infinite number of closed sets need not be a closed set.