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A filter is defined as follows:

Let $X$ be a non-empty set. A non-empty family $\mathcal{F}\subset \mathcal{P}(X)$ is a filter on $X$ if:

  1. $A\neq \emptyset$ for all $A\in \mathcal{F}$,
  2. If $A,B\in \mathcal{F}$ then $A\cap B\in \mathcal{F}$,
  3. If $A\in \mathcal{F}$ and $A\subset B\subset X$ then $B\in \mathcal{F}$

This idea of filters is somehow important in topology, and people use it to talk about convergence.

I wanted to have some intuition about this. What a filter is supposed to represent? What is the intuition about the meaning of filters? How should we think about them?

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Filters are ubiquitious in logic, where they are used to construct a single model from a bunch of models (a form of product) and in topology, where they can be used to speak of convergence in all topological spaces in a way similar to how sequences are used in metric spaces to capture topological properties.

A filter, as the name suggests, should be thought of as a devise that allows only certain sets to filter through it. The axioms say $\emptyset$ never filters through, if something filters through, then anything larger then it also filters through, and the intersection axiom, which is crucial, guarantees that whatever caused two sets to filter through will make their intersection filter through. You can think of the sets in the filter as being large (in some undefined sense). The intersection axiom should is very strong. For instance, you can think of makority sets in democratic systems. The axioms 1 and 3 hold, but axiom 2 typically fails. A democratic system in which axiom 2 holds as well is desirable, but not really attainable (if one wishes decisions to actually be taken that way) unless the system is a dictatorship.

Back to mathematics, and particularly topology, it is well known that sequences are not strong enough to capture the topology of all spaces. Intuitively, the topology may be far too intricate for a sequence, which is a very small (only $\aleph_0$ long) and highly structured (i.e., it is linearly ordered), to be able to detect all the features of the topology. Filters are much more intricate and they do suffice for the purposes of topology.

For a short discussion of filters at an elementary level, and for a construction of the real numbers given directly in terms of filters of rational numbers, you may be interested in the reals as rational Cauchy filters (New-Zealand Journal of Mathematics). Intuitively, the real numbers are defined as those filters of rational numbers which are point-like (the 'large sets' within each such filter get very small and concentrated) and minimal (there is no superfluous information).

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A filter is similar to a sequence, which is used to approach a point, which might or might not exist (depending on the completeness of the space). The differences are:

  1. Instead of using single points you use neighborhoods of the point to be approached

  2. You are not limited to countable many approximations

The limit of a filter might be empty, in which case you should think of the filter similar to a non convergent cauchy sequence (I. E. The limit is simply missing from the space). On the other hand the limit might be more than one point which means the filter is not "specific" enough. You should look up the concept of "ultrafilter" for this.

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    $\begingroup$ The set of limits of a filter is not the intersection of all its sets (that doesn't even involve the topology on your space!). Rather, it is the set of points such that every neighborhood of the point is in the filter. $\endgroup$ – Eric Wofsey Nov 20 '16 at 1:35
  • $\begingroup$ True, thanks. I edited it, but feel free to write a new answer, I'm not really convinced of the usefulness of mine. (but I can't find the delete button in the mobile app) $\endgroup$ – Simon Nov 20 '16 at 1:43

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