Understanding a topology I have these two diagrams. 
A is a topology whereas B is not but I do not understand why B is not also a topology. 
The definition of a topology is: 
Given a non empty set X, we say that $\sigma$ is a topology on X if and only if:
1)"$\phi$ and X are both to be in $\sigma$."-This is true for B.
2) "The intersection of two members of $\sigma$ is again in $\sigma$".- b and a intersect in B
3) "The union of any collection of elements of $\sigma$ is again in $\sigma$."-This is true because b is in both sets. 
Why is B not a topology? 


A topology is a collection of subsets of the space. 
The (sub)sets, the elements of the topology, in in your example $B$ are $X= \{a,b,c\}$ (the full space), $S_1= \{a,b\}$, $S_2 = \{b,c\}$ and $ \emptyset = \{\}$. 
Your properties need to apply to the collection/set consisting of these four. 
You clearly have 1. But you do not have 2., as the intersection of $S_1$ and $S_2$ is not any of the sets listed. 
And, while you do have 3. your argument is not correct as the point is that the union of any of these four sets is still a member of these four. Note that the union of $S_1$ and $S_2$ is the full space. 
Contrast with $A$. There the intersection of the two sets is the empty set and it is thus fine. 
