Let be a topological space, then a member of is said to be an open set in . Thus, in a topological space , the members of are said to be open subsets of . Since and full space are always members of , so and are always open sets in .
On the other hand we can define it as let be the topological space, then the subset of is said to be an open set of (or in ), if .
Example: If with topology , then and are the possible open sets of .
On the other hand if , then is not an open set of . It is clear from this illustration that the open subsets of a space depend upon the topology defined on .
• Every subset of a discrete topological space is open.
• The union of any number 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 is an open subspace of a topological space , then each open subset of is also open in .
• Every subset of a topological space is open if and only if its each singleton subset is open.