# Non Pathological examples of non sequential spaces

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?

• 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/… – Qiaochu Yuan Jan 19 at 9:31

$$[0,1]^{\Bbb R}$$ 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.
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.