Proposition 3

A point x is a cluster point of a filter base B if and only if every set of a fundamental system of neighbourhoods of x meets every set of B.

This follows immediately from the definitions.

This proposition and Corollary 2 to Proposition 1 of § 6, no. 2 show that the property "x is a cluster point of the filter $\mathfrak{F}$" is equivalent to the property "there is a filter which is finer than both $\mathfrak{F}$ and the neighbourhood filter of x". In other words: