Proposition: Filter Base

Let $(X,\mathcal O)$ be a topological space. If $B$ is a non-empty set of subsets $O\subseteq X$ that does not contain the empty set $\emptyset,$ then the set $F$ of all subsets of $X$ that contain some element of $B$ is a filter if and only if the intersection of any two sets of $B$ contains a set in $B.$

In this case, the set $B$ is called a filter base of $F$ and $F$ is said to be generated by $B.$

Proofs: 1

Proofs: 1
Propositions: 2

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