"limit" vs. "limit point" of a sequence in a topological space

Question

Let $(E,\tau)$ be a topological space and $(x_n)_{n\in\mathbb N}\subseteq E$.

I'm highly confused by the notion of a limit point $x\in E$ of $(x_n)_{n\in\mathbb N}$.

If $\tau$ is induced by a metric $d$, then I usually mean that $d(x_n,x)\xrightarrow{n\to\infty}0$, when I say that $x$ is a "limit point" of $(x_n)_{n\in\mathbb N}$. It's actually the limit point, since $x$ must be unique. This terminology differs from $x$ being an "accumulation point", which usually means that there is an increasing $(n_k)_{k\in\mathbb N}\subseteq\mathbb N$ with $d(x_{n_k},x)\xrightarrow{k\to\infty}0$.

Now, in the context of general topology, I've seen statements like "$x$ is a limit point of $(x_n)_{n\in\mathbb N}$ if and only if there is a subnet $(y_i)_{i\in I}$ of $(x_n)_{n\in\mathbb N}$ which converges to $x$". But this sounds more like a generalization of the characterization of an accumulation point.

Moreover, it seems like one distinguishes between "limit points" and "limits" of $(x_n)_{n\in\mathbb N}$. If I'm not missing something, the "limit" of $(x_n)_{n\in\mathbb N}$ is unique in Hausdorff spaces. So, at least in the Hausdorff case, it wouldn't make much sense to talk about "a" limit point, since it must be unique.

So, how are these terms precisely defined and related?

If it matters, I'm mostly interested in the case where $E$ is a $\mathbb R$-Banach space and $\tau$ is the weak topology on $E$.

Answer 1

A limit point (or accumulation point) $p$ for a sequence $(x_n)_n$ in a space $X$, is a point that the sequence gets close to infinitely often:

For all open sets $O_p$ containing $x$ (or epsilon ball around $p$ if you prefer in a metric space) and every index $n$ there is an index $m > n$ such that $x_m \in U$.

This notion can be defined for any net defined on some directed index set $I$, of course. If $X$ is first countable (as in a metric space) this means that we can define a subsequence of $(x_n)$ that converges to $p$: just take the countable local nested base $U_n(p)$ around $p$ and recursively choose increasing $n_k$ so that $x_{n_k} \in U_k(p)$ for $k \in \Bbb N$. More generally, in any space and a net $(x_i)_{i \in I}$ we can choose a subnet of the net that converges to $p$ when $p$ is a limit point of the net. This sort of explains the name: it's a limit of a subnet (or subsequence in metric context).

A limit of the sequence is much stronger: there all points of the tail of a sequence have to lie in any neighbourhood of the limit, not just infinitely many. Limits in Hausdorff spaces are unique, while a sequence can have more limit points. A trivial example of this is an alternating sequence like $1,-1,1,-1,1,-1,\ldots$, e.g.: no limit but two limit points.

There are also notions of limit point of a set, and accumulation point of a set, which do not need to have a link with sequences. So mind the context in the terminology..