# Finite Union of Closed Sets

Prove a finite union of closed sets is closed

Proof: Suppose $A_{\alpha_i}$, $i = 1,2,\ldots,n$ are closed and we want to prove $\bigcup_{i = 1}^n A_{\alpha_i}$ is closed, we know from De Morgans Law that $(\bigcup_{i = 1}^n A_{\alpha_i})^c = \bigcap_{i = 1}^n A_{\alpha_i}^c$ also from the previous problem we have proved that a finite intersection of open sets is open. Therefore, $\bigcup_{i = 1}^n A_{\alpha_i}$ is closed.

I am not sure if am right, any suggestions would be greatly appreciated. For brevity's sake just assume that I proved that a finite intersection of open sets is open.

• It is standard to use "$\cap$" in things like $A\cap B$ and $A_1\cap\cdots\cap A_n$, and "$\bigcap$" in things like $\bigcap_{i=1}^n A_i$. I have edited accordingly. ${}\qquad{}$ – Michael Hardy Feb 22 '15 at 14:45