The collection of the non empty set and the set X itself is always a topology on X, and is called the indiscrete topology on X. In other words, for any non empty set X, the collection is an indiscrete topology on X, and the space is called the indiscrete topological space or simply an indiscrete space.
The power set P(X) of a non empty set X is called the discrete topology on X, and the space (X,P(X)) is called the discrete topological space or simply a discrete space.
Now we shall show that the power set of a non empty set X is a topology on X. For this, let be the power set of X, i.e. the collection of all possible subsets of X, then:
(i) The union of any number of subsets of X, being the subset of X, belongs to .
(ii) The intersection of a finite number of subsets of X, being the subset of X, belongs to .
(iii) and X, being the subsets of X, belong to .
This shows that the power set is a topology on X.