I was thinking about partial orders as directed graphs. I then started to think about inducing a total order from a given partial order. This led me to think about 'minimal' elements in the partial order. I wanted to then think of what it would mean to have an initial vertex in a graph, that is, a vertex that has a directed edge leading out of it, to every other vertex in the set we have taken a relation over.
If I consider open sets of a noetherian topological space to be the vertices of the graph, and inclusions to be the directed edges, there will be a minimal object, the empty set. But with a nice enough topological space, one could see that removing the empty set will give a large collection of minimal elements. (although in the case of say $[0,1]\subset \Bbb R$, there won't be any minimal object, because arbitrary intersections are open, and so we can't deal with things that shrink, like $A_i=(1-1/i,1)$ ).
But, if I do have an artinian topological space here, this doesn't tell me that there is a unique minimal vertex in my graph.
When does a partial order have a unique minimal element?