# Answer on Real Analysis Question for Akhtar Rasool

Question #5738

Find an example in which the infinite union of closed sets

1) is not closed

2) is closed.

1) is not closed

2) is closed.

Expert's answer

Find an example in which the infinite union of closed sets

1) is not closed

For each positive integer n, let Cn = [1 - 1/n, 2]. Clearly, each En is a closed set.

It can be checked that union (n=1 to infinity) Cn = (1, 2], which is not closed.

2) is closed

Let Cn = [-n, n] for all positive integers n. Each set is closed and the union is the set of all real numbers which is closed.

1) is not closed

For each positive integer n, let Cn = [1 - 1/n, 2]. Clearly, each En is a closed set.

It can be checked that union (n=1 to infinity) Cn = (1, 2], which is not closed.

2) is closed

Let Cn = [-n, n] for all positive integers n. Each set is closed and the union is the set of all real numbers which is closed.

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