I saw the definition of a filter for the first time and am confused. Filter is supposed to be a collection F of subsets of a set X with the following three properties:
1. Empty set is not in F. No problem with this.
2. Any finite intersection of sets in F belongs to F. Also no problem.
3. Any subset of X containing a set of F belongs to F.
The #3 seems circular to me. If we start with nothing, how do we know which subset contains a set of F? I understand, that inuitively, the filter describes sets "large enough" to do something.
I have heard about principal filters (all subsets of X that contain given point) or neighborhood filters (all subsets of X having given point as their inferior point). These definitions seem okay to me, since we have something to start with. But generally, I don´t see it that way.
Where do we start when "choosing" the sets of the filter? Given a set X, how do we choose some sets belonging to F? Or does filter definition always come with further specification? Could you provide some examples?
Thank you in advance.
Best Answer
This would be a problem if the author was trying to define a specific filter, but this is not what is being done.
The author is defining what a filter is in general.
Once you have a specific filter you should be able to verify if these three axioms are satisfied.
Let's work with the most common filter I have seen:
$X = \mathbb R$ and $F=\{ A \subseteq \mathbb R | 0 \in A\}$.
$1$. The empty set is not in $F$ as $0 \not \in \varnothing$.
$2$. If $A\in F$ and $B\in F$ then $0\in A$ and $0\in B$ and so $0\in A\cap B$ and so $A\cap B \in F$.
$3$. if $A\in F$ and $S \supseteq A$ we have $0\in S$ and so $S\in F$.