I've just learned about a filter converging in a topological space, but I just can't understand what's the motivation to define such a thing...
I get that it is a generalization of a sequence converging, but still, why?
Thanks.
EDIT: I've found this great presentation which can answer my question: http://www.math.tamu.edu/~wrobel/GIGEM2018/Talks/jgriffin-amsgigem.pdf
It's not really inspired by sequence convergence, there is a generalisation of convergence in terms of nets which is really inspired on that.
Both filters and nets offer a way to prove generalisations of well-known metric theorems about sequences, so that we can characterise continuity and closedness in terms of convergence notions that work in all spaces, not just ones with extra structure like metrics.
Filters are inspired on neighbourhood filters, in any space $X$ and for $x \in X$ we define the neighbourhood filter of $x$ as $$\mathcal{N}(x)= \{N \subseteq X: \exists O \subseteq X \text{ open }: x \in O \subseteq N\}$$
One can check that this obeys the filter axioms, and the idea is that $\mathcal{N}(x)$ should converge to $x$ (everything "close to" $x$ is in it).
The actual definition is simply that a filter $\mathcal{F}$ of subsets of $X$ converges to $x$, i.e. $\mathcal{F} \to x$ iff $\mathcal{N}(x) \subseteq \mathcal{F}$; anything larger than a filter converging to $x$ also converges to $x$ (superfilter roughly corresponds to the idea of subsequence, say).
