Concept guide
Neighborhood: searching for open witnesses and listing all neighborhoods in a four-point topology
Let X={1,2,3,4} and τ={∅,{1},{1,2},{3,4},{1,3,4},X}. Decide whether N1={1,2,3} and N2={2,3,4} are neighborhoods of the point 2; prove that in a finite topology the neighborhoods of x are exactly the subsets containing O_x (the intersection of all open sets containing x), and use this to list all neighborhoods of the points 1, 2, 3 and 4; prove that neighborhoods are closed upwards; and show that the verdict for the same set can change under the discrete topology.