In
topology, a
filter E on a
set X is a
collection of
subsets of X satisfying the following
axioms:
- ∅ (the empty set) ∉ E
- If B ∈ E, and A is a subset of X containing B (B ⊂ A ⊂ X), then also A ∈ E
- If A, B ∈ E, then A ∩ B ∈ E.
If X is a
topological space, then a
filter E is said to
converge to a
point p of X if E contains every
neighborhood of p.
The concept was formulated by Henri Cartan as a response to the problem of convergence in general topological spaces. In spaces which are not as topologically nice as our familiar Euclidean space and the like, the sequence may not be a fine enough tool to describe all limit constructions: a point may be in the closure of a set and yet have no sequence from that set which converges to it. Describing convergence via filters solves this problem.
A complementary approach uses objects called nets, which generalize sequences in a more obvious way.