Let $X$ be a set. A **filter** $\mathcal F$ on $X$ is a set of subsets of $X$ fulfilling the following axioms:

- $F$ contains with every set also all of its supersets, formally $$O\in F\wedge O\subseteq S\Rightarrow S\in F.$$
- $F$ contains for each pair of set elements also their intersection, formally $$O_1,O_2\in\mathcal F\Rightarrow O_1\cap O_2\in\mathcal F.$$
- $F$ is not the empty set and it does not contain the empty set $$\mathcal F\neq\emptyset\wedge \emptyset\not\in\mathcal F.$$

If $X$ has a filter $\mathcal F,$ then it is called **filtered by** $\mathcal F.$

- A filter can be defined for any set, not only for topological spaces.
- Any set of neighborhoods of a point is a filter according to its properties.

Definitions: 1 2

Proofs: 3

Propositions: 4

**Steen, L.A.;Seebach J.A.Jr.**: "Counterexamples in Topology", Dover Publications, Inc, 1970**Jänich, Klaus**: "Topologie", Springer, 2001, 7th Edition