What do the different topologies of the set {a,b} mean? Let us consider the set $\{a,b\}$ containing only two elements. I can think of four different topological spaces associated to this set:


*

*$\{ \{\emptyset\}, \{a,b\} \}$.

*$\{ \{\emptyset\}, \{a\}, \{a,b\} \}$.

*$\{ \{\emptyset\}, \{b\}, \{a,b\} \}$.

*$\{ \{\emptyset\}, \{a\}, \{b\}, \{a,b\} \}$.


My question is: Geometrically, what do these different topological spaces mean? 
For case 1, I am supposing that a and b are so close to each other that there is no open sets containing only $a$ or $b$. The points are lumped together. But if this reasoning is correct then I do not know what case 2 means. If there is an open set containing only $a$ it means that the points $a$ and $b$ are not not lumped together. Then why is not there an open set containing only $b$? Similarly, for case 3, why is not there an open set containing only $a$?
I would appreciate a simple or intuitive answers since I do not have a formal background in topology.
 A: You have the right idea for how to think about (1).  For (2) and (3), here is an intuition which I find very helpful.  Think of (2) as just the ordinary real line $\mathbb{R}$, except that all the nonzero numbers have been collapsed down to a single point $a$ and $b$ is $0$.  So $\{a\}$ is open, since it corresponds to the entire set $\mathbb{R}\setminus\{0\}$ in $\mathbb{R}$, which is open.  But $\{b\}$ is not open: any open interval around $0$ must contain nonzero points, so any open set containing $b$ needs to also contain $a$.
To put it another way, the set $\{a\}$ is open because the point $a$ is a really "fat" point--it secretly represents the entire set $\mathbb{R}\setminus\{0\}$.  On the other hand, the point $b$ really is just a single skinny point, and any open set around it will have to touch the fat point $a$ which surrounds it.
A: The long and short of it is that for non-Hausdorff topologies, like this one, there isn't going to be a nice, intuitive definition. What I'd probably say for case 2 is that $a$ is arbitrarily close to $b$, but they're not the same, since $b$ is not arbitrarily close to $a$. Obviously, this isn't super intuitive, since our usual, everyday definition for "close" is implicitly symmetric. Take this as a cautionary tale, that the definition of a topological space is so loose, you can come up with a lot of really strange stuff.
What I think the moral of this is though, is that you shouldn't be looking at finite topologies to build your intuition. Usually, when one begins studying some field of math, one begins by looking at finite cases to build intuition, such as with Groups or Sets, but here, it's better to take $\mathbb{R}^2$ to be your primitive case, and to build up from that.
A: *

*a and b are (topologically) indistinct.  

*b sticks to a (b is in closure of {a}).     

*a sticks to b (a is in closure of {b}).     

*a and b are distinct and separate.

