Example of $x$ being adherent point but not accumulation point? So I was just reading Apostol and I see that if $x$ is an accumulation point of set $S$, it has to be an adherent point as well. I guess it's possible for $x$ to be an adherent point only, not an accumulation point. But I couldn't come up with a concrete example. Any idea would be appreciated.

I’m assuming that Apostol defines an adherent point of $S$ to be a point $p$ such that every nbhd of $p$ contains a point of $S$. Let $S=\{0\}$ in $\Bbb R$: $0$ is an adherent point of $S$, but it’s not a limit point of $S$. For a less trivial example let $S=\{0\}\cup[1,2]$: again $0$ is an adherent point of $S$ — every nbhd of $0$ obviously contains a point of $S$ — but not a limit point, since the nbhd $(-1,1)$ contains no point of $S$ other than $0$.
More generally, every point of a set $S$ is automatically an adherent point of $S$, but if the point has a nbhd that does not contain any other point of $S$, it won’t be a limit point of $S$.
