If we have any directed set $(D,\le)$ then we can add a point $\infty\notin D$ and then consider the topology on $C(D)=D\cup\{\infty\}$ such that all points of $D$ are isolated and the local base at $\infty$ consists of all "upper sets" $\langle d,\infty\rangle=\{\infty\}\cup\{d'\in D; d'\ge d\}$. (I.e., the topology is given by the base1 $\mathcal B=\{\{d\}; d\in D\}\cup\{\langle d,\infty\rangle; d\in D\}$.)
Then we can get the following observation:2
Fact: Let $X$ be a topological space, $\ell\in X$ and $x\colon D\to X$ be a net. Then the net $(x_d)_{d\in D}$ converges to $\ell$ if and only if the map $\overline x\colon C(D)\to X$ given by $$\overline x(d)= \begin{cases} x_d,&\text{if }x\in D,\\ \ell,&\text{if }x=\infty.\\ \end{cases}$$ is continuous.
In the other words, we have characterization of the convergence $x_d\to\ell$ using the map given by $\overline x|_D=x$ and $x(\infty)=\ell$.
Question. Are there some textbooks which explicitly mention this observation? I.e., where this topology on $D\cup\{\infty\}$ is defined and the relation between convergence of nets continuity of functions $C(D)\to X$ is mentioned?
Although this observation is rather trivial, I consider it worth mentioning when I am introducing convergence of nets to students.
- I have an example of topological space which is tightly linked with convergence of nets.
- It is often (if not always) useful to have several possible viewpoints of the same notion.
- When proving various results about nets I can point to corresponding results on continuity. (Characterization of convergence in product corresponds to characterization of continuity of a function into a product space - in both cases it suffices to consider each coordinate separately. If we have a convergent net $x\colon D\to X$ then for a continuous function $f\colon X\to Y$ we have a convergent net $f\circ x\colon D\to Y$. This corresponds to composition of continuous maps $\overline{f\circ x}=f\circ\overline x$.)
- When defining subnet, I have another directed net $E$ and a cofinal map $h\colon E\to D$. Then I can explain cofinality also in the way: This means that $h(e)\to\infty$, if we consider $(h(e))_{e\in E}$ as a net in the topological space $C(D)$.
Since this observation is rather trivial, it must have been noticed by many people. That's the reason why I was a bit surprised that I did not notice this observation mentioned in various textbooks and introductory text that I checked.
Remark: We could get a similar result for $\mathcal F$-limits. I.e., if we have a function $f\colon M\to X$ and a filter $\mathcal F$ on $M$, then we can define when $\ell\in X$ is an $\mathcal F$-limit of the function $f$. And if we choose appropriate topology on the set $C(\mathcal F)$ on the set $M\cup\{\infty\}$ then we get analogous observation. (In this case the topology is $\mathcal T=\{\{x\}; x\in M\}\cup \{\{\infty\}\cup F; F\in\mathcal F\}$. If you take a filterbase instead of a filter, you get a base for the topology.) However, there are only a few books in general topology that actually consider convergence of filters in this generality - some references can be found here: Where has this common generalization of nets and filters been written down?.
I am mentioning this here mainly because the observation for nets which I mentioned above is a special case of the observation for filters. (For every directed sets, we have a filterbase $\{\langle d,\infty); d\in D\}$ on the set $D$. For the filter obtained from this filterbase, the notion of $\mathcal F$-limit coincides with the usual definition of a limit of a net.)
1It is easy to check that this system indeed forms a base. Whenever we have $\infty\in\langle d_1,\infty\rangle \cap \langle d_2,\infty\rangle$ then there is an $d\in D$ with $d\ge d_1,d_2$, which means that $\infty\in\langle d,\infty\rangle\subseteq \langle d_1,\infty\rangle \cap \langle d_2,\infty\rangle$. And the analogous condition is trivially true for isolated points.
2Proof of this fact is basically just rewriting the definitions.
