# 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?

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$f$ needs to have a particular kind of domain for the first sentence to be true. I'm not really sure I understand your last question; could you clarify? Are you looking for something like a sequence indexed by ordinals (en.wikipedia.org/wiki/Ordinal_number) or are you looking for a notion of generalized sequence such that provided by nets (en.wikipedia.org/wiki/Net_(mathematics) ) or filters (en.wikipedia.org/wiki/Filter_(mathematics) )? – Qiaochu Yuan May 27 '11 at 11:19
You did not specify, what is the domain of your function $f$. If it is arbitrary then the first sentence is not true for arbitrary topological spaces (if I understand it correctly). It is true in sequential spaces. en.wikipedia.org/wiki/Sequential_space – Martin Sleziak May 27 '11 at 11:19
Thank you, correction made. – peno May 27 '11 at 11:22
My question is not about the spaces, but about the set of indices. A segment of the real line is nice, in contrast with the net of weakly open sets. – peno May 27 '11 at 11:31
I am not sure about the tags, but if the question you had in mind was really about nets, perhaps general topology would be better instead of (or in addition to?) elementary set theory and functional analysis. – Martin Sleziak May 27 '11 at 12:16

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.)

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> I guess that what you call "like a sequence" is the fact, that a given directed set contains a cofinal subset of order type (N,≤). -- Yes!! – peno May 27 '11 at 11:58
you wrote "I call a subset A⊆D cofinal if for every d0∈D there exists a∈A such that d0∈A." Please check and correct if necessary. I understood this in the sense that $d_0\leq a$. – peno May 27 '11 at 13:08