I just had the following question on an exam:
Is the topology given by the subbase $\{a,c,d\}, \{a,c,b\}$ path connected?
I wasn't sure about this, my guess is that its not. I sketched the following:
The topology is $\big\{\{a,c,d\}, \{a,c,b\},\{a,c\}, \{a,c,b,d\}\big\}$. Then noted the limit of a constant sequence $(c)$ can have two limits - $a$ or $c$. (Indeed check every open set containing $a$ and see that it also contains $c$). I think this will force a path to be not uniquely defined. I'm not sure though. Thoughts?
Finite spaces are pretty weird in some ways. In this introductory paper by May it is shown (proposition 4.8) that a connected finite space is in fact path-connected and your space is certainly connected as all non-empty open sets intersects (in $a$ and $c$). The latter also implies that the constant $c$ (or $a$ or alternating if you like) sequence actually converges to all $4$ points $a,b,c,d$.