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I'm trying to find simple examples where nets are necessary to describe the space instead of sequences. I know for example that if a space is first countable, then convergence can be described by sequences. However, I think that most examples of spaces which are not first countable are usually pathological, at least the ones I thought of.

Can anyone give me the simple examples, hopefully used in reality and not just given as counter-examples, such that their properties need nets or filters to be described and can not be described by sequences?

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    $\begingroup$ I think the Stone-Cech compactification $\beta \mathbb{N}$ is an example, and it certainly occurs "in practice," but I don't know much about the details: mathoverflow.net/questions/138161/… $\endgroup$ – Qiaochu Yuan Jan 19 at 9:31
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$[0,1]^{\Bbb R}$ is a classic example where nets or filters are needed: it's the set of functions from the reals to $[0,1]$ in the pointwise (product) topology. Such topologies are common in functional analysis. It's compact but not sequentially compact. The Čech-Stone compactification $\beta\Bbb N$ of $\Bbb N$, an important object in many branches of maths, is another example.

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Take $\Bbb R^{\Bbb R}$, that is, the space of all functions from $\Bbb R$ into $\Bbb R$. Consider on it the product topology. This space is not first countable. However, a sequence $(f_n)_{n\in\Bbb N}$ of elements of $\Bbb R^{\Bbb R}$ converges to some $f\in\Bbb R^{\Bbb R}$ with respect to this topology if and only if $(f_n)_{n\in\Bbb N}$ converges pointwise to $f$. So, I doubt that you will see this example as a pathological one.

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