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 fact a sequence is a function $x_n:\mathbb{N}\rightarrow X$.
Now, a directed set can be seen as an abstraction from the natural numbers, with the following property:$$\alpha,\beta\in\ D\rightarrow\; \exists \gamma\ st\ \alpha\prec\gamma,\ \beta\prec\gamma$$Can 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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Intuition behind nets
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