Meaning behind Filter in Set Theory In a course in logic and set theory, we studied the concept of a Filter. We defined a filter $F \in P(S)$ on $S$ an equivalent of the following definition from Jech's Introduction to Set Theory:
(a) $S \in F$ and $\emptyset \notin F.$
(b) If $X\in F$ and $Y \in F$ then $X \cap Y \in F$.
(c) If $X \in F$ and $X \subseteq Y \subseteq S$, then $Y \in F$.
I am having trouble grasping this concept and it's meaning.
My question is, what is the intuition behind this definition, and why are these kinds of sets called filters?
Thanks
 A: It's similar to the concept of "almost everywhere". Suppose to every subset $T\subseteq S,$ you write
$$
\mu(T) \begin{cases} =1 & \text{if } T\in F, \\ = 0 & \text{if } S\smallsetminus T\in F, \\ \text{is undefined} & \text{otherwise.} \end{cases}
$$
Then, according to the definition of "filter", you have
\begin{align}
& \mu(\varnothing) = 0 \\[6pt]
& \mu(S) = 1 \\[6pt]
& \text{If } \mu(T_1), \mu(T_2) \text{ both exist, and } T_1\cap T_2 = \varnothing, \\
& \text{then } \mu(T_1\cup T_2) = \mu(T_1) + \mu(T_2).
\end{align}
Saying $\{x\in S: P(x)\} \in F$ is the same as saying $P(x)$ for almost all $x\in S.$
A: When you put a filter in your sink, the idea is that you filter out the big chunks of food, and let the water and the smaller chunks (which can—in principle—be washed through the pipes) go through.
You filter out the larger parts.
A filter filters out the larger sets. It is a way to say "these sets are 'large'", and the sort of make sense. The set of "everything" is definitely large, and "nothing" is definitely not; if something is larger than a large set, then it is also large; and two large sets intersect on a large set.
From a mathematical point of view you can think about this as being co-finite, or being of measure $1$ on the unit interval, or having a dense open subset (again on the unit interval). These are examples of ways where a set can be thought of as "almost everything". And that is the idea behind a filter.
