I am presently reading about the concept of a topological basis, and I seem to be encountering conflicts between various sources. The definition of basis I am working with is the following:

If $\displaystyle X$ is a set, a basis for a topology on $\displaystyle X$ is a collection $\displaystyle \mathcal{B}$ of subsets of $\displaystyle X$ such that

(1) For each $\displaystyle x \in X$, there exists a basis element $\displaystyle B$ such that $\displaystyle x \in B$

(2) If $\displaystyle x \in B_1 \cap B_2$, where $\displaystyle B_1$ and $\displaystyle B_2$ are basis elements, then there is a basis element $\displaystyle B_3$ such that $\displaystyle x \in B_3 \subseteq B_1 \cap B_2$.

Here is where the confusion begins. Some sources (I can't reference any in particular, because I have looked through too many) simply define the topology $\displaystyle \tau$ generated from the basis $\displaystyle \mathcal{B}$ as that set containing every arbitrary union of elements in $\displaystyle \mathcal{B}$, while some (such as Munkres) define it as $\displaystyle \tau = \{ \mathcal{O}~|~ \forall x \in \mathcal{O}, \exists B \in \mathcal{B} ~s.t.~ x \in B \}$. Are these definitions equivalent or are they two different notions with a similar name?

I have another question, in regards to the first of the two definitions. If we are only taking unions, how could one obtain the empty set through these unions?