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How can we think of nets as a generalization of sequences?
Basically, we can see sequences as a way of enumerating elements of a set: in facty a sequence is a function $x_n:\mathbb{N}\rightarrow X$.
Now, directed set abstract from naturals the following property: $$\alpha,\beta\in\ D\rightarrow\; \exists \gamma\ st\ \alpha\prec\gamma,\ \beta\prec\gamma$$ Cam we then see nets as a generalization of the ordinary idea of counting, in some sense? What is the intuition behind them and behind directed sets?

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    $\begingroup$ Directed sets made a bit more sense to me (in the sense of why they are useful in topology) when I realised that, given a point $x$ in a topological space $X$, the set of neighbourhoods of $x$ form a directed set (with $U \preceq V \iff U \supseteq V$). So, while a sequence can be "too short" to approach a point, having a point for every neighbourhood of $x$ is plenty of points to properly approach $x$... any more would be overkill! $\endgroup$ – Theo Bendit Sep 2 at 10:39
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    $\begingroup$ @Theo Bendit: So, while a sequence can be "too short" to approach a point --- The first few years after I knew about nets (mostly from the topology texts by Munkres and Willard) I had the impression that they were basically a complicated way of using arbitrary transfinite ordinal sequences that would be needed when the spaces are not first countable, and I always wondered why authors didn't just give this more intuitive description. Then, sometime in spring 1982, (continued) $\endgroup$ – Dave L. Renfro Sep 2 at 13:16
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    $\begingroup$ my (print) copy of the May 1982 issue of American Mathematical Monthly arrived and I saw the paper On the inadequacy of cofinal subnets and transfinite sequences, and I suddenly realized that I had been wrong for years, and since then I've instead wondered why texts don't say that nets are NOT just transfinitely long sequences for spaces that are not first countable. Also, see the Mathematics StackExchange question/answers is a net stronger than a transfinite sequence for characterizing topology? $\endgroup$ – Dave L. Renfro Sep 2 at 13:23
  • $\begingroup$ Nets used to be called "generalis(z)ed sequences" by some authors.. If you use $(\Bbb N, \le)$ as the directed set you get exactly sequences. $\endgroup$ – Henno Brandsma Sep 2 at 21:56
  • $\begingroup$ @Dave L. Renfro, thanks I was precisely searching for this kind of motivation $\endgroup$ – Francesco Bilotta Sep 3 at 11:23
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To begin a net over the directed set $\mathbb{N}$ is just a sequence. So nets are in a way a generalization of sequences. Let me motivate the analogy further.

Why do we need nets? Let $X$ and $Y$ be topological spaces. When $X$ is first countable the following conditions on a function $f: X \rightarrow Y$ are equivalent:

  1. $f$ is continuous,
  2. For every sequence $x_n \rightarrow x$ in $X$ we have $f(x_n) \rightarrow f(x)$.

When the first countability assumption is no longer true, the statement $2. \Rightarrow 1.$
is no longer valid in general. However the following equivalence holds in any case

  1. $f$ is continuous,
  2. For every net $x_\alpha \rightarrow x$ in $X$ we have $f(x_\alpha) \rightarrow f(x)$.

More generally, almost every property about the topology of metric spaces (where the space is first countable) which has an equivalent formulation using sequences, remains true in general topological spaces but one has to replace sequences by nets.

Let me explain why the property $\forall \alpha, \beta \in D: \exists \gamma \in D: \alpha, \beta \leq \gamma$ is crucial for nets to have any application. In topology it happens a lot that we have a sequence $(x_n)$ such that

  1. $\exists N_1 \in \mathbb{N}: \forall n \geq N_1: x_n$ satisfies some property A,
  2. $\exists N_2 \in \mathbb{N}: \forall n \geq N_2: x_n$ satisfies some property B.

If we choose $N = \max \{N_1,N_2\}$ then for all $n \geq N: x_n$ satisfies both property $A$ and $B$. This strategy is applied very often when $A$ and $B$ together with the triangle inequality give something meaningful.

The same strategy can be applied for nets precisely because of the property that I mentioned.

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    $\begingroup$ Do you mean first countable, rather than second? Not all metric spaces are second countable. (Take some non-separable metric space.) $\endgroup$ – Theoretical Economist Sep 2 at 10:54
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    $\begingroup$ Thanks for the comment. I changed my terminology. $\endgroup$ – abcdef Sep 2 at 10:59

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