Q:

How are accumulation points related to topologies?

Is it possible to have two topologies, one strictly finer than the other, yet all of the accumulation points are the same?

A:

Since the closure of a set is the union of the set with the set of its accumulation points, if all accumulation points of all sets are the same, the closures of all sets are the same, and therefore the set of closed sets is the same. Because the open sets are the complements of the closed sets, then also the set of open sets is the same. But the set of open sets is the topology, therefore the topologies are the same.

