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 about specifying 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.

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