As I understand it a topology on a set X is a collection of subset that satisfy three conditions 1) The collection contains X and the null set 2) It is closed under unions (perhaps a better way to say this is any union sets in this collection is again in the collection). 3) The intersection of any finite number of sets is in the collection. Also I understand the Basis for a topology again to be a collection of subsets in X such that: 1) if x is in X than there is at least one basis element containing x. 2) if x is in any intersection of Basis elements there is a third basis element which contains the entire intersection. So say I have a set X and I can form a basis for a topology B for X. Does that mean that X automatically has a topology? How do these two definitions relate? If there is a topology on a set does that meant there has to be a basis? Can someone help me out here and maybe explain things a bit?