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.

1. 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