Topology, Basis of a given topology. We defined basis for a topology, and there is something that I do not understand. Here is how we defined the basis.
Given a topological space $\left(X,\mathscr T\right)$
  we defined basis for the topology to be the set $\mathscr B$
 , consisting of subsets of $X$
  if it satisfies 2 conditions. 
First, for all $x\in X$
  there exists $B\in\mathscr B$
  such that $x\in B$
 . Secondly if $x\in B_{1}\cap B_{2}$
  , for $B_{1},B_{2}\in\mathscr B$
 , then there exists $B_{3}$
  such that $x\in B_{3}\subseteq B_{1}\cap B_{2}$
 . 
So,I am studying from the book Topology, by Munkres. And it stated that the basis is a subset of the topology. But, if I choose $X=\left\{ a,b\right\}$ 
  and $\tau=\left\{ \emptyset,X\right\}$ 
 , and I can define $\mathbb{B}=\left\{ \left\{ a\right\} ,\left\{ b\right\} \right\} $
 . The set $\mathbb{B}$
  satisfies the conditions of the definitions. However it's not a subset of the topology. What am I missing here? 

If we already have a topology given on a set, then a basis of the topological space must be a subset of the topology given on the set and it must generate ( i.e. by taking arbitrary unions, finite intersections of the basis elements) the whole topological space. So, if you choose any topology on a set and try to find a basis of it, you need to keep in mind that the subset of the topology you choose must satisfy the conditions you mentioned as the definition of a basis from Munkres and this subset must generate the entire topology.
Now, if we have X just as a set and we take a collection of its subsets such that they satisfy the definition you mentioned of a basis, then this subset generates a topology on X which might be different from any other topology on X generated by some other basis.
