Question

Prove that the union of a finite number of open sets is open

Answer 1

Solution:

Suppose $\left\{A_{i}: i \in I\right\}$  is a collection of open sets, indexed by I , and let $A=\bigcup_{i \in I} A_{i}$. Let $x \in A$ be arbitrary.

Then x belongs to at least one of the sets $A_{i}$ .

Since this set is open, it contains an open ball about x ; clearly, this ball lies in A .

But $x \in A$  was chosen arbitrarily, and so A meets the definition of an open set.