I am now trying to prove or disprove this innocently looking but somehow surprisingly hard conjecture:

**Conjecture** If $latex S$ is a generalized filter base then $latex \left\langle f \right\rangle \bigcap{\nobreak}^{\mathfrak{F}} S = \bigcap {\nobreak}^{\mathfrak{F}} \left\langle\left\langle f \right\rangle \right\rangle S$ for every funcoid $latex f$.

Oh well, it is not that hard. I just proved the conjecture. (I’ll put its solution online soon.)

Maybe this theorem will be of help solving other open problems I formulated.