Indiscrete and Discrete Topology

Indiscrete Topology

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 \tau = \left\{ {\phi ,X} \right\} is an indiscrete topology on X, and the space \left( {X,\tau } \right) is called the indiscrete topological space or simply an indiscrete space.

Discrete Topology

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 \tau = P\left( X \right) 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 \tau .
(ii) The intersection of a finite number of subsets of X, being the subset of X, belongs to \tau .
(iii) \phi and X, being the subsets of X, belong to \tau .

This shows that the power set is a topology on X.