Any collection \(X\) consisting of subsets of \(\omega\) can be viewed as a subspace of \(2\omega\) by identifying \(X\) with the set of characteristic functions of its elements. This way, it is possible to study the interaction between the set-theoretic and the topological properties of \(X\). I will give a survey of results and open questions on this topic, in the case when \(X=F\) is a filter on \(\omega\).