# Definition of Topology

Let be a non empty set. A collection of subsets of is said to be a topology on if the following conditions are satisfied:

**(i)** The union of any number of members of belongs to .

**(ii)** The intersection of finite number of members of belongs to.

**(iii)** The empty set and set itself belongs to .

In other words, a collection of subsets of a nonempty set is said to be topology on if it is closed under the formation of arbitrary unions and finite intersections and contains both and .

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

If is a topology on , then the pair (X,) is called a topological space. The set is called the underlying set or the ground set and the elements of the set are called the points of the topological space. Instead of writing (X,), we may write 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 is a topological space then one should himself understand that there is a topology defined on the set .

**Example:**

Let and consider the collections

={, {a}, {c}, {ac}, X }, = {, {a}, {a,b}, {a,c}, X } are the topologies on .