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The following is the definition of subnets given in Folland's Real Analysis (page 126)

A subnet of a net $\langle x_{\alpha}\rangle_{\alpha\in A}$ is a net $\langle y_{\beta}\rangle_{\beta\in B}$ together with a map $\beta\mapsto \alpha_\beta$ from $B$ to $A$ such that

  • for every $\alpha_0\in A$ there exists $\beta_0\in B$ such that $\alpha_\beta\gtrsim\alpha_0$ whenever $\beta\gtrsim \beta_0$;
  • $y_\beta=x_{\alpha_\beta}$.

I have a hard time developing an intuition regarding the first condition.

How is it useful? What is the main use of it?

Would anyone come with a simple example of a net $\langle y_\beta\rangle_{\beta\in B}$ such that the first condition is not satisfied?

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    http://math.stackexchange.com/a/1126950/4280 and the links in it, give some more motivation. – Henno Brandsma Mar 16 '17 at 19:28
  • @HennoBrandsma: Thanks for the link. I'm now reading on Handbook of Analysis and its Foundation which the main reference in the linked pdf in that answer. One should note that Folland's definition corresponds to the Kelly subnets. –  Mar 16 '17 at 19:46

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The condition says that we move further along $A$, the subnet will go there too. It is needed to see that for a convergent net, every subnet converges to that same limit, which is a property known from sequences that we like to keep.

If we leave it out, we could take a constant map to any fixed $\alpha_0$ and get a total trivialisation of the subnet which would converge to that $x_{\alpha_0}$.

Henno Brandsma
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It is nothing but the definition of a subsequence .

Note that if $(x_k)$ is a sequence and $(x_{n_k})$ is a subsequence of $(x_k)$ then $n_k$ is an increasing function such that $n_k\ge k$.

Here also if we choose $\alpha_0$ then I can choose some $\beta_0$ satisfying the given property since the above holds.

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  • Thanks for your attention. As it is said in Folland, while subnets perform much the same functions as subsequences, it should not be taken too literally. In particular, the index set $B$ may well have larger cardinality than the index set $A$, and a subnet of a sequence need not be a subsequence. The question in OP asks about the first condition in the definition of subnets and in particular looks for an example when it fails. –  Mar 15 '17 at 18:44