# 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?

• @SolutionExists: No, an accumulation point $x$ of a set is a point where in every neighbourhood of $x$ there is a point of the set other than $x$. In certain topological spaces, this implies that you can construct a sequence of such points that converges to $x$, but that's not its defining property. – celtschk Jun 10 '20 at 5:57