Ordered sets that are like sequences If $f$ is a function on a segment of the real line containing the point $0$ with values in a topological space and $f(x)\to f(0)$ as $x\to 0$, then there exists a sequence $(x_n)$ such that $f(x_n)\to f(0)$ (actually, one can take any sequence tending to $0$). If we consider the weak topology, then some elements from the closure of a set cannot be achieved as a limit of a sequence. Thus, although the set of real numbers from a neighborhood of $0$ is not a sequence, it is "like a sequence". Does there exist a terminology that allows one to characterize the sets of indices, convergence along which implies the existence of a convergent sequence?
 A: As Qiaochu Yuan mentioned in his comment, there are two tools used to describe convergence in topological spaces - nets and filters.
On of them is the concept of the net, which is a map from a directed set to a topological space. Your example with real numbers seems to me to be very similar to the net on $\mathbb R\setminus\{0\}$ ordered by
$$a\prec b \Leftrightarrow |a|\ge|b|.$$ 
If we consider $f:{\mathbb R\setminus\{0\}}\to X$ as a net on this directed set, then we get that 
$$f(x)\to L \Leftrightarrow (\forall U\in\mathcal N_L)(\exists \varepsilon>0) [|x|<\varepsilon \Rightarrow f(x)\in U],$$
where $\mathcal N_L$ denotes the set of all neighborhoods of $L$ in the space $X$.

I guess that what you call "like a sequence" is the fact, that a given directed set contains a cofinal subset of order type $(\mathbb N,\le)$.
For a directed set $(D,\le)$, we call a subset $A\subseteq D$ cofinal if for every $d_0\in D$ there exists $a\in A$ such that $d_0\le a$.
It is relatively easy to see that if the net $(x_d)_{d\in D}$ converges to $L$, then so does $(x_d)_{d\in A}$ for any cofinal subset. So if there is a cofinal subset which is order-isomorphic to $\mathbb N$, you get a sequence.
(I believe that, for a directed set, having a cofinal subset of order type $\mathbb N$ is equivalent to having infinite countable cofinal subset.) 

I do not know about a similar condition for filters. (Your example with reals could be considered as a filter convergence for the filter generated by the base $f[\mathcal N_0]=\{f[A]; A\in\mathcal N_0\}$ given by the neighborhood filter of zero in $\mathbb R$.) 
(Perhaps the main problem being here is that I am not sure how to rephrase "being like a sequence" from your example in the language of filters and filter convergence. But it could be something close to the notion of P-ideals for sequences. However, this is probably not in the direction you intended. Also, I might be biased, since I worked with this notion quite a bit.)
