What is the topology here with three elements in sets? What is $\tau$ for my paint drawn picture below?
I can't decide between these two interpretations:
$$1)\, \tau = \{a,b,c,\{b\},\{a,b,\{b\}\},\{b,c,\{b\}\}$$
and the simpler
$$2)\,\tau=\{\{a,b\},\{b,c\},\{b\}\}$$
Perhaps both are incorrect even.



Don't know what to title it as, since the title would normally include the topology :P

Both are incorrect. The first one because topology is a collection of subsets, and your suggested collection is not. The second one is not a topology because $\emptyset$ and $\{a,b,c\}$ are missing from it. That being said, once you include those two you get the correct set. Which is 
\begin{align*}
\tau=\{\emptyset,\{a,b\},\{b\},\{b,c\},\{a,b,c\}\}.
\end{align*}
