Questions about bases in topology I have some questions regarding bases in topology. There are different ways of defining a topology and I want to distinguish between them. 

Let $X$ be a topological space. A collection $\mathcal{B}\subseteq 2^{X}$ is called a basis for the topology on X if the following are satisfied:

1) Every element of $\mathcal{B}$ is an open subset of $X$.
2) Every open subset of $X$ is the union of some collection of elements of $\mathcal{B}$.

Let $X$ be a set, and suppose $\mathcal{B}\subseteq 2^{X}$. Then $\mathcal{B}$ is a basis for some topology on $X$ iff it satisfies the following:

1) $\bigcup\limits_{B\in\mathcal{B}}B=X$
2) If $B_1,B_2\in\mathcal{B}$ and $x\in B_1\cap B_2$, then there exists an element $B_3\in\mathcal{B}$ such that $x\in B_3 \subseteq B_1\cap B_2$.
When asked to prove that a set is a basis for the topology on a top. space $X$, we should use the first definition. However, in this case, do we need to also show the "basis criterion?"

(Basis Criterion) Suppose $X$ is a topological space, and $\mathcal{B}$ is a basis for its topology. Show that a subset $U\subseteq X$ is open iff it satisfies the following condition: For every $x\in U$, there exists $B\in\mathcal{B}$ such that $x\in B\subseteq U$.

When should we used the second definition (proposition)? 
 A: If you're given a topological space, meaning you're given the topology, you certainly need to use the first definition, as you state.
You should use the second statement if you are given a set $X$ but not told its topology. I can imagine a question like:
"Here is a set $X$, and here is a subset of $2^X$. Is this subset of $2^X$ a basis for a topology?"
Here you can't use the first definition at all, because you don't have any notion of what the open subsets are. You're being asked "If I give you this basis, can you give me a topology?" as opposed to "Here's a topology, is this a basis for the topology I have given you?".
Your "Basis criterion" appears to be talking about sets $U$ (which you haven't defined - I'm guessing these are your open sets in the topological space $X$?), which means you already have a topology, and so you are not in the situation to use your second definition. You need this criterion to hold in order that your thing is a basis for the topology you've been given.
A: That proposition will only prove that your set $\mathcal{B}$ is the basis for some topology. This doesn't mean it will actually be a basis for the topology you are in fact considering. 
The basis criterion along with all sets in $\mathcal{B}$ being open is equivalent to your first definition. So you have the following proposition which is easy to prove.

Let $(X, \tau)$ be a topological space and a set $\mathcal{B} \subset \tau$. Then $\mathcal{B}$ is a basis for the topology $\tau$ on $X$ if and only if for every open $U$ and every $x \in U$, there exists a $B \in \mathcal{B}$ such that $x \in B \subset U$.

