Let  be a topological space, then a member of is said to be an open set in X. Thus, in a topological space, the members of  are said to be open subsets of X. Since  and full space X are always the member of, so and X are always open sets in X.

            On the other hand we can define as let  is the topological space, then the subset of X is said to be an open set of X (or in X), if .

Example: If  with topology, then and are the possible open sets of X.

            On the other hand if, then is not an open set of X. It is clear from this illustration that the open subsets of a space X depend upon the topology defined on X.

Theorems:

• Every subset of a discrete topological space is open.
• The union of any numbers of open subsets of a topological space is open.
• The intersection of any finite number of open subsets of a topological space is open.
• If Y is an open subspace of a topological space X, then each open subset of Y is also open in X.
• Every subset of a topological space is open if and only if its each singleton subset is open.