Let X be a non empty set. A collectionof subsets of X is said to be a topology on X if the following conditions are satisfied:
(i) The union of any number of members of  belongs to.
(ii) The intersection of finite number of members of  belongs to.
(iii) The empty set and set X itself belongs to.

            In other words, a collection of subsets of a nonempty set X is said to be topology on X if it is closed under the formation of arbitrary unions and finite intersections and contains both  and X.

            The sentence “closed under the formation of arbitrary unions” means that the union of any number of members of  is in. Similarly, the sentence “closed under the formation of finite intersections” means that the intersection of any finite number of members of  is in.

            If  is a topology on X, then the pair (X,) is called a topological space. The set X is called the underlying set or the ground set and the elements of the set X are called the points of the topological space. Instead of writing (X,), we may write X for a topological space if there is no danger of confusion. In fact, Mathematicians do not bother about specifying the topology. If it is said that X is a topological space then one should himself understand that there is a topology defined on the set X.

Example:
            Let X = {a, b, c} and consider the collections
={, {a}, {c}, {ac}, X }, ={, {a}, {a,b}, {a,c}, X } are the topologies on X.