Originally Posted by

**aliceinwonderland** I am trying to find an error in the below proof. Any advice will be highly appreciated.

Proof: $\displaystyle \overline{\cup A_{\alpha}} \subset \cup\bar{A_{\alpha}}$

If $\displaystyle \{A_{\alpha}\}$ is a collection of sets in X and if $\displaystyle x \in \overline{\cup A_{\alpha}} $, then every neighborhood $\displaystyle U$ of x intersects $\displaystyle \cup A_{\alpha}$. Then $\displaystyle U$ must intersect some $\displaystyle A_{\alpha}$, so that x must belong to the closure of some $\displaystyle A_{\alpha}$. Therefore, $\displaystyle x \in \cup\bar{A_{\alpha}}$.