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. 
 A: 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}

A: 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.
A: 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.)
