Is it possible for a limit point not to be a sequential limit point? I'm studying elementary topology, and I'm trying to understand the difference between limit points and sequential limit points. So far, I have only studied several basic topologies on subsets of the natural numbers, such as the discrete topology and the cofinite topology. In order to develop some intuition, I'm attempting to come up with an example of a space composed strictly of natural numbers, with a subset such that some point x is a limit point of the subset, but not a sequential limit point of it. Is this possible on the natural numbers, or will every limit point also be a sequential limit point?
 A: It is possible to construct such a space, but it’s not trivial. I’ll give you two examples. One is easily described directly in terms of $\Bbb N$, but you have to know something about ultrafilters; the other is a space most easily defined on the set $\Bbb N\times\Bbb N$, not on $\Bbb N$, but you can use a bijection between the two sets to transfer the topology to $\Bbb N$ itself, if you really want to do so.
Note: My $\Bbb N$ includes $0$.
For the first example let $\mathscr{U}$ be a free (i.e., non-principal) ultrafilter on $\Bbb N$. Points of $\Bbb Z^+$ are isolated, and the open nbhds of $0$ are precisely the members of $\mathscr{U}$ that contain $0$. Every member of $\mathscr{U}$ is infinite, so $0$ is clearly a limit point of $\Bbb Z^+$.
Suppose, now, that $\sigma=\langle a_n:n\in\Bbb N\rangle$ is any sequence of positive integers, and let $A=\{a_n:n\in\Bbb N\}$. If $A\notin\mathscr{U}$, then $\Bbb N\setminus A\in\mathscr{U}$ by a basic property of ultrafilters, so $\Bbb N\setminus A$ is an open nbhd of $0$ that contains no point of $\sigma$, and hence $\sigma$ certainly does not converge to $0$. If $A\in\mathscr{U}$, then $A$ is infinite, so we can partition it into two infinite subsets $A_0$ and $A_1$. It is a basic fact about ultrafilters that exactly one of $A_0$ and $A_1$ belongs to $\mathscr{U}$; assume without loss of generality that $A_0\in\mathscr{U}$. Let $U=\{0\}\cup A_0$; then $U\in\mathscr{U}$, so $U$ is an open nbhd of $0$. And $U\cap A_1=\varnothing$, so $U$ does not contain a tail of $\sigma$: there is no $n_0\in\Bbb N$ such that $a_n\in U$ for all $n\ge n_0$. Thus, $\sigma$ does not converge to $0$ in this case, either, and it follows that $0$ is not the limit of any sequence in $\Bbb Z^+$.
My second example is the Arens-Fort space; it’s similar in spirit but more elementary. Let $X=\Bbb N\times\Bbb N$, and let $p=\langle 0,0\rangle\in X$. Points of $X\setminus\{p\}$ are isolated. For $n\in\Bbb N$ let $C_n=\{n\}\times\Bbb N$. A set $U\subseteq X$ is an open nbhd of $p$ if and only if $p\in U$, and the set $\{n\in\Bbb N:C_n\setminus U\text{ is infinite}\}$ is finite.

If you sketch $\Bbb N\times\Bbb N$, you’ll see that the set $C_n$ is the column of points with first coordinate $n$. Say that a set $S$ almost contains $C_n$ if it contains all but at most finitely many points of $C_n$, i.e., if $C_n\setminus S$ is finite. Then a set containing $p$ is open if and only if it almost contains all but finitely many of the columns $C_n$, i.e., if and only if there are only finitely many columns that it does not almost contain.

Clearly $p$ is a limit point of the set $X\setminus\{p\}$. Let $\sigma=\big\langle\langle n_k,m_k\rangle:k\in\Bbb N\big\rangle$ be a sequence in $X\setminus\{p\}$. Suppose that for each $n\in\Bbb N$ the set $\{k\in\Bbb N:n_k=n\}$ is finite. Then for each $n\in\Bbb N$ there is an $\ell_n\in\Bbb N$ such that $m_k\le\ell_n$ whenever $n_k=n$. Let $$U=\{p\}\cup\{\langle n,m\rangle\in X:m>\ell_n\}\,;$$ then $U$ is an open nbhd of $p$ that contains no point of $\sigma$.
Otherwise there is an $n\in\Bbb N$ such that $N=\{k\in\Bbb N:n_k=n\}$ is infinite. Let $$U=\{p\}\cup\{\langle\ell,m\rangle\in X:\ell\ne n\}\,;$$ then $U$ is an open nbhd of $p$ that contains no point of the infinite subsequence $\big\langle\langle n_k,m_k\rangle:k\in N\big\rangle$ of $\sigma$. In either case $\sigma$ does not converge to $p$, so $p$ is not the limit of a sequence in $X\setminus\{p\}$.
