function as a net?

Question

One can see net as a generalization of a sequence.

As done in http://en.wikipedia.org/wiki/Net_(mathematics), in the special case where $f: M\backslash \{a\} \rightarrow X$ where M is a metric space and X is a topological space, one can see $f$ as a net from the directed set $M\backslash \{a\}$ with partial order given by the distance to $a\in M$. The closer a point is to $a$, the "greater" it is.

Question: It is possible to see an arbitrary continuous function from a topological space to another as a net?

(it's a question for free, i was not looking for any application of it)

Edit: actually one can define $f:M \rightarrow X$. it's just when we want to have a directed set that we take a point away.

Answer 1

If you assume the Well-ordering theorem then yes! In that case, if we have $f:X\rightarrow Y$ then the Well-ordering theorem gives us a well-ordering of $X$ hence it becomes a directed set.

It is still interesting to see if one can use the topological structure on $X$ to turn it into a directed set.

Answer 2

The conventional way (to treat a topological limit as a net limit) is like this. (Taken from Kelley's paper of around 1950). [Kelley, J. L. Convergence in topology. Duke Math. J. 17, (1950). 277–283.]

Let $M$, $X$ be topological spaces. Let $f : M \to X$ be a function. We are interested in $\lim_{m \to a} f(m)$. Define a directed set $D$ as follows. $D$ consists of pairs $(E,p)$, where $E$ is a neighborhood of $a$ and $p \in E.$ For ordering: $(E_1,p_1) \ge (E_2,p_2)$ iff $E_1 \subseteq E_2$. Check it is directed. Define a net $T_f : D \to X$ by $T_f(E,p) = f(p)$. Then $x \in X$ is a value of $\lim_{m\to a} f(m)$ iff it is a limit of the net $T_f$.

The usual modification can be made when $f$ is defined on a subset $N$ of $M$ and $a$ is in the closure of $N$. Take pairs $(E,p)$ where $E$ is a neighborhood of $a$ and $p \in E \cap N$.

REMARK. by taking all points of every neighborhood, we avoid requiring the Axiom of Choice.