# Help Me Understand: Proof that Finite Intersection of Open Sets is Open

I don't see how the third line (starting with Thus: $\exists \epsilon_i$...) is justified. That is: just because $x \in U_i$, for all $i$, how do I know that a neighborhood of $x$ is in $U_i$, for all $i$?

In the context of a metric space, a subset $U$ is open when for any $x$ in $U$, one can find an $\epsilon$-ball centered at $x$ and contained in $U$.

The third line applies this definition to find a collection of open balls centered at $x$ and contained in the respective $U_i$.

This is just by definition: the $U_i$ are open, so there is an $\epsilon_i$-ball around $x$ that is contained in $U_i$.

• Okay: I understand it now. There is some neighborhood in each $U_i$ for that x, thus we just pick the smallest one, and then it HAS to be in all the others. Thanks. – user156349 Jun 10 '14 at 17:14
• @user156349 Yup! You got it. – user98602 Jun 10 '14 at 17:15