Hi

The two sets $\displaystyle V_{1}, V_{2}$ are subsets of the Euclidian space $\displaystyle \mathbb{R}^n$.

I need to show that

(1) $\displaystyle V_{1} \cup V_{2} $ is open.

(2) $\displaystyle V_{1} \cap V_{2} $ is open.

Definion:

I know according to the definition that a subset $\displaystyle V$ of $\displaystyle \mathbb{R}^n$ is said to be open, if there for every point $\displaystyle x \in V$ exist a real number $\displaystyle \epsilon > 0$ given every point $\displaystyle y \in \mathbb{R}^n$. Then $\displaystyle |x - y| < \epsilon$ and $\displaystyle y \in V$

Solution:

(1) and (2)

Then if $\displaystyle V_{1}, V_{2}$ both are open subsets of $\displaystyle \mathbb{R}^n$ by definition above. Then if there exist a point x and y in both subsets, then if there still exist a epsilon > 0, then their respective Union and Intersection is also open according to the definition above.

How does that sound? Or do I need to above something more concrete?

Or do I need to add something about the union of two open sets are closed? Bu if the union of two subsets are closed, then how possible can it be open?

Best Regards.

Billy