# Why were filters and nets in topology named filters and nets?

I am wondering why do mathematicians categorizes some structures and called them filters , Nets?

In English, filter means: A porous material through which a liquid or gas is passed in order to separate the fluid from suspended particulate matter.

Is this meaning of filter correspond to that of mathematics? What is the reason behind calling filter as filter in maths? Is their any reason for naming Nets (the structure) as Nets?

Please, may be my question is somehow useless but I believe in understanding of how and why our mathematical structures are named that way.

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An interesting fact, mentioned in Megginson's An introduction to Banach space theory p.143: The term net was actually first used by J. L. Keeley in a 1950 paper on topological convergence The terminology was not Kelley’s invention, though. Kelley had wanted to call such an object a way. However, nets have subnets, which Kelley would have dubbed subways. Norman Steenrod talked him out of it. After some prodding by Kelley, Steenrod suggested the term net as a substitute for way. –  Martin Sleziak Mar 20 '12 at 7:04
The title seems to ask why filters and nets are used, rather than why they are called like that. –  wildildildlife Mar 20 '12 at 16:49
Are the real numbers called like that because they are "real"? Are the irrational numbers really "irrational"? These are just names, words. Syntactic symbols for objects defined in a particular way. –  Asaf Karagila Mar 24 '12 at 20:37
"Filter" is taken from the French "Filtre" so you will probably have to get your answer from French speakers. –  GEdgar Mar 24 '12 at 23:58

## 3 Answers

The primary reason we call filters "filters" is the following aspect of the definition:

If $A \in \mathcal{F}$ and $A \subset B$ then $B \in \mathcal{F}$

That is to say, a filter is not just an ordinary subset of $P(X)$ (the power set of $X$). Roughly speaking, it catches subsets of a particular size. Thus, if $A$ is caught in the filter, and $B$ is larger than $A$, $B$ is also caught in the filter.

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A very nice reason, but I want more of this. –  Hassan Muhammad Mar 20 '12 at 6:57
@Hassan: There really isn’t any more. Mathematical terminology often has very little to do with the non-mathematical senses of the same words. When there is some connection, it’s often pretty tenuous, as is the case with filter. –  Brian M. Scott Mar 20 '12 at 7:00
The other aspect of the definition of filter – that the intersection (meet) of two things in a filter is also in the filter – also goes in line with the suggestion that it axiomatises a notion of "largeness". After all, how could the intersection of two large things fail to be large? –  Zhen Lin Mar 20 '12 at 8:30
@ZhenLin: Certainly you forgot the smiley at the end of your comment. –  Marc van Leeuwen Mar 20 '12 at 10:23

A generic sequence is often visualized in a linear fashion, something like an arrow. A generic net has a non-linear structure that splits and comes back together like the cords of a net, because that’s the kind of structure possessed by general directed sets. A typical example is the directed set of finite subsets of $\Bbb N$, ordered by $\subseteq$: the sets $\{0\},\{0,1\},\{0,2\}$, and $\{0,1,2\}$ are related as shown as the central diamond in the picture below, which shows a very small part of the whole directed set:

{0,1,4} {0,1,2} {0,2,3}
*     *     *
/ \   / \   / \
\ /   \ /   \ /   \ /
{1,4}*{0,1}*{0,2}*{2,3}*
/ \   / \   / \   / \
\ /   \ /   \ /   \ /   \ /
*     *     *     *     *
{4}   {1}   {0}   {2}   {3}
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Can you provide a simple example of filter geometrically? I will love to imagine that. –  Hassan Muhammad Mar 20 '12 at 7:13
@Hassan: Useful filters are generally pretty complicated. Still, perhaps this one will help. Let $E$ be the set of even integers, and let $\mathscr{F}=\{F\subseteq\Bbb Z:F\supseteq E\}$, the collection of all sets of integers that contain every even integer; then $\mathscr{F}$ is a filter on $\Bbb Z$. In terms of Isaac’s explanation, it only ‘catches’ every set big enough to contain $E$, so it’s a pretty coarse filter. $\mathscr{P}=\{A\subseteq\Bbb Z:0\in A\}$ is also a filter on $\Bbb Z$, but it’s a very fine filter: it ‘catches’ every set that contains $0$. –  Brian M. Scott Mar 20 '12 at 7:20

I am also very new to this concept, but this is how I think about filters. It is really an informal idea, so you don't have to agree with it.

Let $X$ be a topological space, and consider the collection of all the neighborhoods of $p \in X$, say $O(p)$. This $O(p)$ is clearly a filter, but it is intuitive to think that the point $p$ "filters" open sets. Think of $X$ as a plane and you drop open sets on it. The open sets that are "filtered by $p$" are precisely the neighborhoods.

The definition of filters is a generalization of this idea (at least in my opinion). Empty set cannot be filtered because it is too small. The whole set is filtered because it is too big. If there are two sets that are filtered, you may want to guess that there is something "in the middle" that filters the both. If there is a set that is filtered any other set containing it must be filtered because what filtered the smaller set must be in the bigger set. (Thus I think only "nontrivial" axiom is the intersection axiom.)

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