# Nicest Way to Put a Topology on a Directed Set so that Definition of Limit Remains Unchanged?

What is the nicest way ot put a topology on an arbitrary directed set, $$Z$$, so that the following definition of limit remains unchanged?

Suppose $$f:X \rightarrow Y$$ where $$X$$and $$Y$$ are topological spaces.

$$\textbf{Definition:}$$ We say ''the limit of $$f$$ as $$x$$ approaches $$p$$ is $$L$$'' and write $$\lim\limits_{x \rightarrow p}f(x)=L$$ provided:

For every neighborhood of $$L$$ (in $$Y$$), $$N(L)$$, there is some punctured neighborhood of $$p$$, $$\overset{\cdot}{U}(p)$$, such that $$f[\overset{\cdot}{U}(p)] \subseteq N(L)$$. (See here.)

This definition makes rigorous all the kinds of limits one encounters in a typical college algebra or calculus course (i.e. limits of functions and limits of sequences -- provided one extends the domain/range to include $$\pm \infty$$ and puts the order topology on them).

Is there a nice way to put a topology on an arbitrary directed set, $$Z$$, so that the above definition encompasses the limit of a net as well?

Note: Special cases of the above definition that come up in college algebra and calculus courses include:

Let $$Z$$ be a directed set and let $$Z^*=Z\cup\{\infty\}$$ where $$\infty\not\in Z$$. Topologize $$Z^*$$ by saying that a set $$U$$ is open iff either $$\infty\not\in U$$ or there exists $$i\in Z$$ such that $$j\in U$$ for all $$j\geq i$$. (Such sets are closed under finite intersections since $$Z$$ is directed.) Then given a map $$f:Z^*\to X$$ for some other topological space $$X$$, the following are equivalent:

1. $$f$$ is continuous
2. $$\lim_{z\to\infty} f(z)=f(\infty)$$ in your sense.*
3. The net $$(f(z))_{z\in Z}$$ converges to $$f(\infty)$$.

To prove this, note that $$1\to 2$$ is immediate since $$\lim_{z\to\infty} f(z)=f(\infty)$$ is just another way of saying that $$f$$ is continuous at $$\infty$$. The equivalence of $$2$$ and $$3$$ is immediate from the definitions, since the deleted neighborhoods of $$\infty$$ in $$Z^*$$ are exactly sets which contain all sufficiently large elements of $$Z$$. Finally, $$2\to 1$$ since $$Z$$ is an open discrete subset of $$Z^*$$ and so any map on $$Z^*$$ is automatically continuous at every point of $$Z$$, so to be continuous you just have to check continuity at $$\infty$$.

*Beware that you should be very careful with this notation in general, since there can be different values $$L$$ and $$L'$$ such that both $$\lim_{z\to\infty} f(z)=L$$ and $$\lim_{z\to\infty} f(z)=L'$$. So unless you know the limit is unique, you should consider "$$\lim_{z\to\infty} f(z)=L$$" as an atomic predicate, rather than an actual statement of equality about some object "$$\lim_{z\to\infty} f(z)$$".

• Thanks! I debated whether to specify that $Y$ be Hausdorff so that the limit, as you pointed out, would be unique (if it exists). – Selrach Dunbar Aug 14 at 22:20
• This is a very large topology on $Z$. After thinking some more about your anser, it occurrs to me that a much smaller topology on $Z$ (that would also allow the given definition of limit to encompass the concept of limit of a net) would be the topology that takes just the upsets, i.e. $[a,\infty]:=\{z \in Z | a \leq z \} \cup \{\infty\}$, as a basis. – Selrach Dunbar Aug 15 at 21:36
• True, if that's all you care about. In practice, though, I find that the characterization in terms of condition (1) is actually more useful than than the characterization using the limit at $\infty$, and for condition (1) you need the topology on $Z$ to be discrete. – Eric Wofsey Aug 15 at 21:49
• Another way to think about it is that there are two ways of measuring the "size" of a topology: by how many open sets there are, and by how many nets converge. My topology on $Z^*$ is very large in open sets but dually it is very small in nets (essentially no nontrivial nets converge except that the identity net on $Z$ converges to $\infty$). – Eric Wofsey Aug 15 at 21:50
• With the original question answered, now I am wondering if there is another way to topologize an arbitrary directed set $Z$ so that the limit of a net is as described by the definition above and reduces to the usual (order) topology when applied to the specific directed set $[0,1)$ (in the standard simple order). – Selrach Dunbar Aug 15 at 21:50