TITLE: Filters vs nets in topology
QUESTION [24 upvotes]: Nets are a natural generalization of sequences in arbitrary topological spaces. Using the language of nets we can extend intuitive, classical sequential notions (compactness, convergence, etc.) to arbitrary spaces. Example of using: Reed, Simon "Methods of Modern Mathematical Physics: Functional Analysis".
There is an alternative (but essentially equivalent) language of filters. For example, Bourbaki use it a lot in his "General Topology".
IMHO, filters are completely unintuitive compared to nets, but many authors besides Bourbaki still uses filters to explain things. So, what are pros and cons of filters versus nets.

REPLY [4 votes]: I think of filters as defining a "flow" or a "direction" on a set. And to say that a function (into a topological space) converges along this filter means that as you go in this "direction", the function "tends" to a particular value.
I think filters are a natural generalization of sequences as well, if you reinterpret what it means for a sequence to converge. Namely, define $A_n=\{m\in\mathbb{N}:m\geq n\}$. Then the collection of all $\{A_n\}$ defines a filterbase. We usually focus on the points in the image of a sequence, which is exemplified by the notation $\{x_n\}$. Instead of focusing on the image points of a sequence, let's actually give it a name. Let $f$ be a sequence, that is a function from $\mathbb{N}$ into (say) a topological space. Then the sequence $f$ converges (in the usual sense) to $x$ if and only if the filterbase $\{f(A_n)\}$ converges to $x$.
In a parallel way, say we have a set $X$. And we have a function $f$ from $X$ into a topological space. And suppose we have a filter(base) $\{A_\alpha\}$. Then we say that $f$ converges to $x$ along the filter(base) $\{A_\alpha\}$ if the filterbase $\{f(A_\alpha)\}$ converges to $x$.
I like filters because they more readily allow us to think of "convergence in a direction" rather than "convergence around a point". As an example, the filter base $\{A_n\}$ can be said to "flow to infinity". And when we define a function $g$ on $\mathbb{N}$ which converges along this filterbase, we can think of extending $g$ in "this direction" instead of just extending $g$ to the singular point $\infty$.
Some of the more useful filterbases are
$$\{(x,\infty)\}_{x\in\mathbb{R}}\qquad \{z\in\mathbb{C}:|z|\geq r\}_{r\in[0,\infty)}\qquad \{(x_0-\epsilon)\cup(x_0+\epsilon)\}_{\epsilon\in [0,\infty)}$$
The first one induces a "flow" along the real line which tends to infinity.
The second one induces a "flow" on the complex plane which tends further and further away from 0. The third induces a "flow" on the real line which "sinks in" on the point $x_0$. Convergence along these filters (when a function does converge) are usually denoted by
$$\lim_{x\rightarrow\infty}f(x)\quad \lim_{|z|\rightarrow\infty}f(z)\quad \lim_{x\rightarrow x_0}f(x)$$
respectively. Thus convergence along a filterbase does have relatively immediate examples. But the power of filter(base)s comes along when you want to talk about convergence in a non-canonical "direction".
Perhaps the most readily available example of a non-canonical direction, which still comes up some times, is the filterbase
$$\{z\in\mathbb{C}: |\Re(z)|\leq \epsilon\,\text{ and }\,\Im(z)\geq 1/\epsilon\}_{\epsilon\in(0,\delta)}$$
Convergence along this filterbase is usually denoted by
$$\lim_{z\rightarrow i\infty}f(z)$$
But now you can imagine many more "directions" on many other sets.