# 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.