Let be $M$ a set of three elements, $M=\{a,b,c\}$. I should find at least six distinct topologies.
Distinct means for all $a,b,c\in M: a\neq b, b\neq c, c\neq a$.
How can I find the topologies?
My first propable topologies are
$$\{\emptyset,M\}, \{\emptyset,\{a\},\{b,c\}, M\}\ 3\text{ times}. $$
Let us find all topologies $T$ on $M=\{a,b,c\}$. Note that $T$ contains $\emptyset$ and $M$ by definition. We now distinguish cases dependent on the number of singletons that $T$ contains.
If $T$ contains all singletons, then $T = 2^M$ since $T$ is closed under unions.
If $T$ contains two singletons $\{a\}$ and $\{b\}$, then $T$ contains $\{a,b\}$ and can then also contain $\{b,c\}$ or $\{a,c\}$ (but not both). So we find the topologies $\{\emptyset,\{a\},\{b\},\{a,b\},M\}$, $\{\emptyset,\{a\},\{b\},\{a,b\},\{a,c\},M\}$ and $\{\emptyset,\{a\},\{b\},\{a,b\},\{b,c\},M\}$. In total there are $9$ topologies with two singletons.
If $T$ contains only the singleton $\{a\}$, then $T$ may contain $\{a,b\}$ and $\{a,c\}$ as well. We find as possible topologies $\{\emptyset, \{a\}, M\}$, $\{\emptyset,\{a\},\{a,b\},M\}$, $\{\emptyset,\{a\},\{a,c\},M\}$ and $\{\emptyset, \{a\},\{a,b\},\{a,c\},M\}$. If $T$ also contains $\{b,c\}$ it is easily seen that $T=\{\emptyset, \{a\},\{b,c\},M\}$. We find that there are $5$ topologies containing $\{a\}$ as its only singleton, so $15$ topologies with just one singleton.
If $T$ contains no singletons it can contain at most one doubleton. So we find the topologies $\{\emptyset,M\}$, $\{\emptyset,\{a,b\},M\}$ and two similar topologies. These are $4$ topologies.
In total, there are $1+9+15+4=29$ topologies on $\{a,b,c\}$.
There are 29 topologies on 3 point sets, and 9 up to homeomorphism. Those 9 are, omitting the empty set and the whole set: $\{\},\{\{a\}\},\{\{a\},\{b,c\}\},\{\{a\},\{a,b\}\},\{\{a\},\{a,b\},\{a,c\}\},\{\{a\},\{b\},\{a,b\}\},\{\{a\},\{b\},\{a,b\},\{a,c\}\},\{\{a,b\}\},\{\{a\},\{b\},\{c\},\{a,b\},\{a,c\},\{a,b\}\}$
