The motivation for the definition of a basis just learning about topology and I am a little confused as to why we define a basis as below.

A base is a collection B of subsets of X satisfying these two
  properties:
  
  
*
  
*The base elements cover X. 
  
*Let B1, B2 be base elements and let I be
  their intersection. Then for each x in I, there is a base element B3
  containing x and contained in I.
  

I am confused about;


*

*the first clause of the definition of a basis. So if the base covers the set X then isnt it pretty much a topology???

*the second clause of the definition of a basis. Why is B3 necessary, I dont see how this helps or what it does. 


Really, I just dont get how a basis 'generates' a topology. And how it is the minimum requirement to generate the topology.
 A: You should know that topology raised as generalization of metric spaces. Thus, elements of topology (open sets) have "nice" properties such as open sets have in metric spaces. And thus elements of basis of topology are in a way equivalent to open balls in metric spaces (because every open set in metric space can be written as union of open balls).
$(1)$ It is obvious that collection of set that cover $X$ do not in general form a topology on $X$. For instance $\{X\}$ covers $X$, but it is not a topology since it does not contain empty set.
$(2)$ To get idea why this is necessary, look at definition of topology and try to prove three conditions that family of sets should satisfy to be a topology on $X$ using these one (and also first condition). 
A: You have two real questions here, they come after two bullet points.
The first is in which way the basis generates the topology. Stated differently: Given only the basis, can we reconstruct the topology, and how? It turns out that the open sets are exactly those sets that are unions of elements of the basis. If you prove this, then the meaning of (1) and (2) will become clearer to you.
The second is, in which way such a basis is minimal. The answer to this is easy: it is not. Indeed the topology itself is a basis. Now, it is understandable that you find this confusing, if you encountered the word “basis” first in linear algebra, but terminology is not always completely consistent across branches of mathematics.
