  Let $X$ be a first countable space which admits a non-trivial convergent
sequence and let $\mathcal{I}$ be an analytic P-ideal. First, it is shown that
the sets of $\mathcal{I}$-limit points of all sequences in $X$ are closed if
and only if $\mathcal{I}$ is also an $F_\sigma$-ideal.
  Moreover, let $(x_n)$ be a sequence taking values in a Polish space without
isolated points. It is known that the set $A$ of its statistical limit points
is an $F_\sigma$-set, the set $B$ of its statistical cluster points is closed,
and that the set $C$ of its ordinary limit points is closed, with $A\subseteq
B\subseteq C$. It is proved the sets $A$ and $B$ own some additional
relationship: indeed, the set $S$ of isolated points of $B$ is contained also
in $A$. Conversely, if $A$ is an $F_\sigma$-set, $B$ is a closed set with a
subset $S$ of isolated points such that $B\setminus S\neq \emptyset$ is regular
closed, and $C$ is a closed set with $S\subseteq A\subseteq B\subseteq C$, then
there exists a sequence $(x_n)$ for which: $A$ is the set of its statistical
limit points, $B$ is the set of its statistical cluster points, and $C$ is the
set of its ordinary limit points.
  Lastly, we discuss topological nature of the set of $\mathcal{I}$-limit
points when $\mathcal{I}$ is neither $F_\sigma$- nor analytic P-ideal.
