# 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 $$X$$.

Yasir Rasheed

September 21@ 3:07 pmShow that ground set which determines a unique topological space must have one element