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.

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 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}