I am preparing for my first course in Topology. I read a lot of confusing articles about the correspondence between the number of topologies on a finite set X and the number of preorders on X. A preorder on a set X is a binary relation on X that is reflexive and transitive. I would like to know if the following statement is true.:

A basis for a topology on a finite set X is the collection of sets: 
{ {y|y is in every open set containing x}|x in X }. In other words, given a finite topology we can form a basis by finding the intersection of all open sets containing x for each x in X. We may need to delete some duplicate sets. Then the statement would imply that an upperbound for the minimum number of basis elements is the cardinality of X.