Filters vs nets in topology

Question

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.

Answer 1

what are (dis)advantages of the net vs filter languages.

nets:

• Some statements are easier with nets. e.g.

If $X$ is a topological space and $A\subset X$ then $a\in \overline A$ iff some net on $A$ converges to $a$.

with filters one has to define what convergence of a filter on $A$ in $X$ means.

• Because nets are more intuitive and we can think of a net as a collection of points (somehow ordered), some natural questions may be inspired; for example:

Suppose $X$ is a topological space and every net on $A\subseteq X$ has a convergent subnet. Is $\overline A$ compact?

(the answer is yes in Tychonoff spaces)

• Almost all statements about sequences in analysis, can be translated to nets on topological or uniform spaces. e.g. this or this or even this with nets.

• Net theorems will stick in mind, especially if you have studied analysis, because they can be imagined. But filters are more abstract. This is why I prefer nets.

filters:

• With filters some proofs about compactness are easier. Even Tychonoff Theorem can be proved with filters.

• Any (diagonal) uniformity is a filter. Before studying uniform spaces one should study filters.

• Filter has something to do with Bornology.

• Convergence of a filter controls the convergence of all nets which correspond to that filter. This says filters only have the necessary features for convergence while nets have features that are hardly pertinent to convergence.

• Unlike superfilters, there are several definitions for subnets. So before using the word subnet you should clarify what you mean by that.

More about the so-called equivalence of filters and nets can be found in last pages of this pdf.

Answer 2

Nets involve a partial order relation on the indexing set, and only a part of the information contained in that relation is relevant for topological purposes. The relevant part is just what is retained when one passes from the net to the associated filter. So, in a sense, the use of filters discards irrelevant information that is present in nets.

I believe I learned about nets before filters, so my preference for filters is probably not based on timing. It's more likely to have resulted from a (congenital?) preference for simplicity and for discarding or at least ignoring irrelevant information. I agree, though, that after one learns basic notions in the context of sequences, nets, being rather similar to sequences, will be more intuitive, until one encounters subnets.

Answer 3

Filters are very natural. Filters tell you when something happens "almost everywhere", that is on a "big" set.

Convergence is something that needs to happen "almost everywhere", that is, $x_i\to x$ (where $x_i$ is a net) if every open set contains "almost all" the $x_i$'s. That's a very obvious use of "almost everywhere". In some sense, almost all the net is almost everywhere around $x$.

This is why filters are great for convergence.

Answer 4

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.