hi.help with a proof.
prove : Any union of open sets is open

Question

zehanz · Answer

What is the definition of an open set?

zehanz · Answer

It is important to have the right definition here. Are you working with open intervals on the real line, or the more generalised open sets found in metric spaces?

anonymous · Answer

a set whose points are all interior points

zehanz · Answer

Now we need to know what defines an interior point.

anonymous · Answer

x in a set A is an interior point of A when there exist r>0 such that $$B(x,r) \subset A$$

zehanz · Answer

OK, we're going to use that!

Suppose $V=V_1 \cup V_2 \cup ... \cup V_n$, for any positive integer n.

zehanz · Answer

We have to prove for any x in V, that x is an interior point.

Take any x from V.
Because V is the union of n open sets, there must be at least one i, with i from 1, ..., n, such that $x \in V_i$. 
Now $V_i$ is an open set, so there exist an r>0 such that $B_{x,r}\subset V_i$.
This means, because $V_i\subset V$, that $B_{x,r}\subset V$.

We now have proven that any x is an interior point of V, hence V is open.

anonymous · Answer

Thank u very much

zehanz · Answer

YW! I used to find this kind of problems very hard, until I realized they mostly don't go far beyond writing down the definitions...