Q:

Etymology of the term "filter"

According to this article, filters were introduced in general topology by Henri Cartan in 1937.
I wonder why he called them filters.

A:

It may not have been Cartan who named them. According to Michèle Audin, ‘Henri Cartan & André Weil du vingtième siècle et de la topologie’, p. $8$, the concept was originated at a meeting of Bourbaki in $1937$. After a morning of discussion most of the group went for a walk, but Cartan remained behind and came up with filters as a way of removing the countability restriction imposed by use of sequences. He came up with the idea, but not with the name (La chose était inventée mais pas son nom.) 
I don’t know what the original motivation for the name was was, but it is certainly possible to see filters on a topological space as a way of filtering out points of a space. For example, if $\mathscr{N}(x)$ is the neighborhood filter of a point $x$ in a space $X$, each $N\in\mathscr{N}(x)$ can be thought of as filtering out the points of $X\setminus N$, keeping only those points that are ‘$N$-close’ to $x$. It does seem that the original motivation for filters was to get a general notion of convergence in topological spaces, and that’s precisely the setting in which this metaphor works best. Subsequently, of course, the notion was generalized and many other applications were found, but it wouldn’t surprise me if this or some similar metaphor were the reason for the name.

