# Definition of Topology

Let $X$ be a non empty set. A collection $\tau$ of 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 $\tau$ belongs to $\tau$.
(ii) The intersection of finite number of members of $\tau$ belongs to$\tau$.
(iii) The empty set $\phi$ and set $X$ itself belongs to $\tau$.

In other words, a collection$\tau$ 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 $\phi$ and $X$.

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

If $\tau$ is a topology on $X$, then the pair (X,$\tau$) 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,$\tau$), we may write $X$ for a topological space if there is no danger of confusion. In fact, mathematicians do not bother to specify 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
${\tau _1}$={$\phi$, {a}, {c}, {ac}, X }, ${\tau _1}$ = {$\phi$, {a}, {a,b}, {a,c}, X } are the topologies on $X$.