Answer to Question #36066 in Differential Geometry | Topology for Hans

Question

Let X = {1,2,3} and a function N on X given by N(1) = {1,2},{1,2,3} , N(2) = {2,3},{1,2,3} , N(3) = {2,3},{1,2,3} .
(a) Prove that N does not define a neighbourhood topology on X. (b) Add a minimal number of neighbourhoods to the above definition of N to indeed obtain a neighbourhood topology, called ˜ N.