How many axioms there are in topology?
A "topology" for set X is a collection of subsets, U, such that:
1) X is in U
2) the empty set in in U
3) the intersection of any two sets in U is also in U
4) the union of any collection of sets in U is also in U
I suppose we could call those four requirements the "axioms" for topology, although many texts give (1) and (2) as a single statement: "The entire set, X, and the empty set are in U". Does that reduce the number of "axioms" to 3? Is there really any point in asking for the number of axioms rather than the axioms themselves?
(The sets in U are called the "open sets" in the topology. You could also define a topology for set X to be a collection of subsets, V, of X such that:
1) X is in V
2) The empty set is in V
3) The union of two sets in V is in V
4) The intersection of any collection of sets in V is in V
Using that definition, the sets in V would be the closed sets in the previous definition.)