I propose the following way to introduce filters on sets to beginning students. (I am writing a book which contains this intro now.) You are welcomed to comment whether this is a good exposition and how to make it even better.
We sometimes want to define something resembling an infinitely small (or infinitely big) set, for example the infinitely small interval near
on the real line. Of course there are no such set, just like as there are no natural number which is the difference
. To overcome this shortcoming we introduce whole numbers, and
becomes well defined. In the same way to consider things which are like infinitely small (or infinitely big) sets we introduce filters.
An example of a filter is the infinitely small interval near
on the real line. To come to infinitely small, we consider all intervals
for all
. This filter consists of all intervals
for all
and also all subsets of
containing such intervals as subsets. Informally speaking, this is the greatest filter contained in every interval
for all
.
[A formal definition of a filter on a set goes here.]