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 2 Y , and we write F ! y, if Ny µ F, where Ny is the collection of all the neighborhoods of y.

(1) Prove that for all y 2 Y , Ny is a filter.

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

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

Solution