Prove that the intersection of a finite number of open sets is open.

A set $A$ is open if every $x \in A$, there exists some $\epsilon > 0$ such that $B(x,\epsilon) \equiv \{y : y \in C,\, d(x,y)<\epsilon \} \subset A$ i.e. $x$ is in some open ball that is in $A$, i.e. every $x \in A$ is an interior point.

Let $X = O_1 \bigcap O_2$. Then any $x \in X$ implies $x \in O_1$ and $x \in O_2$, implying that you should always have an open ball around $x$.

We prove via contradiction. Now assume $X$ is not open. Since $X$ is not open, there exists $x \in X$ such that $x$ is on the boundary of $X$. This implies that $x$ is on the boundary of either $O_1$ or $O_2$. But this is a contradiction, since $x$ has to be an interior point of both $O_1$ and $O_2$. Therefore, all elements of $X$ are interior points. Therefore, $X$ must be open.

For your second question, for any positive integer $n$, let $A_n=(-1/n,1/n)$.

The intersection of all the $A_n$ is $\{0\}$.

Or else let $B_n=(0,1/n)$. The intersection of all the $B_n$ is the empty set, which is closed (and open).

The answer to the first question depends on the details of your definition of open. Let us define a set $A$ of reals to be open if for any $x\in A$, there is a positive $\epsilon$ (which usually depends on $x$) such that the interval $(x-\epsilon,x+\epsilon)$ is a subset of $A$.

Now let $x$ be in the intersection of your $O_i$. Then for every $i$, there is a positive $\epsilon_i$ such that $(x-\epsilon_i,x+\epsilon_i)$ is a subset of $O_i$. Let $\epsilon$ be the smallest of the $\epsilon_i$. Then the interval $(x-\epsilon,x+\epsilon)$ is a subset of $O_1\cap O_2\cap\cdots\cap O_n$.

If you're using a topological space, by definition the intersection of any two open sets gives an open set, so by repeating that finitely many times you end with an open set. For example, let $O_1$, $O_2$, and $O_3$ be open sets. $O_1 \cap O_2$ is open, so $(O_1\cap O_2)\cap O_3$ is also open. If there is also an $O_4$, then $\left(\left(O_1\cap O_2\right)\cap O_3\right)\cap O_4$ is open. This process can be carried on for any finite collection of open sets.