Let Y be a nonempty topological space and let G be a filter

Let Y be a nonempty topological space and let G be a filter on Y . We say that the filter F converges to y Y , and we write F y, if Ny F, where Ny is the collection of all the neighborhoods of y.

(1) Prove that for all y Y , Ny is a filter.

(2) Prove that for all y Y , Ny y.

(3) Assume that Y is Hausdorff. Show that the limit of a filter on Y is unique.

Solution